## Probability of Outcomes problems

## Coin Flip Outcome Probability example question

In an experiment, a coin is tossed until either the same side comes up twice in a row, or the coin has been tossed four times. How many outcomes are there?

a) 6

b) 8

c) 10

d) 16

e) this experiment never stops

f) none of the above

Solution to this **Outcomes Probability** practice problem is given in the video below!

## Coin Flip: Unfair Toss Probability example problem

An unfair coin with P(H) = 0.9 is tossed independently 5 times. What is the probability that the first, third, and fifth tosses are Heads, and all the others are Tails?

a) 0.0243

b) 0.00243

c) 0.243

d) 0.0729

e) 0.00729

f) none of the above

Solution to this **Outcomes Probability** practice problem is given in the video below!

## Unfair Die Toss Probability example

An unfair die is such that the outcomes 1, 2, 3, 5 are equally likely, 4 is half as likely as 2, and 6 is four times as likely as 5. What is the probability of having a 4?

a)

b)

c)

d)

e)

f) none of the above

Solution to this **Outcomes Probability** practice problem is given in the video below!

## Coin & Die Toss Outcome Probability example question

In an experiment, a coin is tossed, and if the outcome is Heads, then a die is rolled. If the outcome is Tails, the experiment stops. How many outcomes are possible in this experiment?

a) 6

b) 7

c) 12

d) 14

e) 15

f) none of the above

Solution to this **Outcomes Probability** practice problem is given in the video below!

## Choices of Pencil Colors Probability example problem

You have 4 white pencils and 6 grey pencils. You select two pencils at random. What is the probability that the two pencils are of the same color?

a)

b)

c)

d)

e)

f) none of the above

Solution to this **Outcomes Probability** practice problem is given in the video below!

## Rule of Product Probability problems

## Rule of Product Probability example question

A quiz contains one Algebra problem, one Geometry problem, and one Probability problem. There are 4 Algebra problems, 5 Geometry problems, and 2 Probability problems available to make the quiz. How many different quizzes are possible?

a) 20

b) 40

c)

d)

e) 10

f) none of the above

Solution to this **Rule of Product Probability** practice problem is given in the video below!

## Permutations & Combinations problems

## Permutation Probability example question

You are trying to form a unique string of three letters from the list **{A, B, C, D, E}** so that the first two letters are distinct. How many strings are there?

a) 10

b) 16

c) 14

d) 100

e) 125

f) none of the above

Solution to this **Permutation Probability** practice problem is given in the video below!

## Combination Probability example problem

Four tennis players decide to play one “doubles” game each day (two players per team) until all possible combinations are used. How many days will they play?

a) 1

b) 3

c) 6

d) 12

e) 24

f) none of the above

Solution to this **Combination Probability** practice problem is given in the video below!

## Combination Probability example

A Geometry class consists of 23 juniors and 44 sophomores. A committee of 5 students is chosen at random. What is the probability that no juniors are chosen?

a)

b)

c)

d)

e) 0

f) none of the above

Solution to this **Combination**** Probability** practice problem is given in the video below!

## Combination Probability example question

A jury of five must decide whether Rick will pass his Probability class. In order for Rick to pass, at least four of the jurors must be in favor. In how many ways can Rick pass his class?

a) 16

b) 6

c) 5

d) 4

e) 32

f) none of the above

Solution to this **Combination**** Probability** practice problem is given in the video below!

## Combination Probability example problem

You have 25 identical flowers in your garden, and you want to remove any 5 of them. In how many ways can you choose the flowers?

a) 25^{2}

b) 5^{25}

c)

d)

e)

f) none of the above

Solution to this **Combination**** Probability** practice problem is given in the video below!

## Permutation-Combination example

You have 35 different pancakes to choose from. You want to order a stack of three pancakes. If you care about the ordering of the pancakes you get, how many choices are possible? (For example, raspberry/blueberry/pecan is different from raspberry/pecan/blueberry).

a)

b) 75

c) 35^{3}

d) 3^{35}

e)

f) none of the above

Solution to this **Permutation-Combination**** Probability** practice problem is given in the video below!

## Permutation-Combination example problem #2

Calculate the maximum number of 3rd Great Grandparents each person has by using Factorials, Permutations and Combinations.

Solution to this **Permutation-Combination**** Probability** practice example is given in the video below!

## Percent Probability problems

## Probability with Percentages example question

A college survey shows that 10% of freshmen, 15% of sophomores, 60% of juniors, and 85% of seniors like Probability. Given that 30% of all students are freshmen, 30% are sophomores, 20% are juniors, and 20% are seniors, what percentage of all students in the college like Probability?

a) 0.635

b) 0.535

c) 0.465

d) 0.365

e) 0.235

Solution to this **Percent Probability** practice problem is given in the video below!

## Geometric Formula PROOF Integration problems

## Area of a Triangle Formula Proof example question

Use Integration techniques in order to derive the formula for the Area of a Triangle, *A*, such that

where *b* is the base and *h* is the height.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Area of a Trapezoid Formula Proof example problem

Use Integration to derive the formula for the Area of a Trapezoid, *A*, such that

where *b*_{1} and *b*_{2} are bases and *h* is the height.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Area of a Circle Formula Proof example

Use Integration to derive the formula for the Area of a Circle, *A*, such that

where *r* is the radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Area of a Circle Formula Proof example question

Use **Double** Integration to prove the formula for the Area of a Circle, *A*, such that

where *r* is the radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Sphere Formula Proof example problem

Use the **Disk** Integration method to derive the formula for the Volume of a Sphere, *V*, such that

where *r* is the radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Cone Formula Proof example

Use the **Disk** Integration method to prove the formula for the Volume of a Right Circular Cone, *V*, such that

where *r* is the radius and *h* is the height.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Torus Formula Proof example question

Use the **Washer** Integration method to derive the formula for the Volume of a Torus, *V*, such that

where *r* is the **inner** (minor) radius and *R* is the **outer** (major) radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Torus Formula Proof example problem

Use the **Shell** Integration method to prove the formula for the Volume of a Torus, *V*, such that

where *r* is the **inner** (minor) radius and *R* is the **outer** (major) radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Surface Area of a Cone Formula Proof example

Use Integration to derive the formula for the Lateral Surface Area of a Right Circular Cone, *S _{A}*, such that

where *r* is the radius and *h* is the height.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Surface Area of a Right Circular Cone Formula Proof with Double Integration example problem

Use **Double** Integrals to derive the formula for the Lateral Surface Area of a Right Circular Cone, *S _{A}*, where

where *r* is the radius and *h* is the height.

Solution to this **Calculus Geometric Formula Proof** practice problem is provided in the video below!

## Surface Area of a Sphere Formula Proof example question

Use Integration to prove the formula for the Surface Area of a Sphere, *S _{A}*, such that

where *r* is the radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Torus Formula Proof example

Use the **Cylindrical Coordinates** (Triple Integration) method to prove the formula for the Volume of a Torus, *V*, such that

where *r* is the **inner** (minor) radius and *R* is the **outer** (major) radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Sphere Formula Proof example question

Use the **Spherical Coordinates** (Triple Integration) method to derive the formula for the Volume of a Sphere, *V*, such that

where *r* is the radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Cone Formula Proof example problem

Use the **Spherical Coordinates **(Triple Integration) method to derive the formula for the Volume of a Right Circular Cone, *V*, such that

where *r* is the radius and *h* is the height.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Volume of a Torus Formula Proof example

Use the **Spherical Coordinates** (Triple Integration) method to prove the formula for the Volume of a Torus, *V*, such that

where *r* is the **inner** (minor) radius and *R* is the **outer** (major) radius.

Solution to this **Calculus Geometric Formula Proof** practice problem is given in the video below!

## Infinite Series PROOF problems

## Infinite Series Proof example question

Show that

when

Solution to this **Calculus & Precalculus Infinite Series Proof** practice problem is given in the video below!

## Fourier Infinite Series problems

## Fourier Series example question

Find the Fourier series representation of the function in the given interval.

Solution to this **Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Fourier Series example problem

Find the Fourier series representation of the given function and its interval.

Solution to this **Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Fourier Series example

Find the Fourier series of the function in the given interval.

Solution to this **Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Fourier Series problem example

Find the Fourier series representation of the function on the given interval below.

**Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Fourier Series example question #5

Find the Fourier series representation of the function with the given interval.

**Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Fourier Series example #6

Determine the Fourier series of the function in the given interval.

**Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Fourier Series example problem #7

Determine the Fourier series representation of the function in the given interval.

**Calculus & Precalculus Fourier Infinite Series** practice problem is given in the video below!

## Maclaurin & Taylor Infinite Series problems

## Maclaurin Series Interval of Convergence example question

Find the Maclaurin series (Taylor series about *c *= 0) and its interval of convergence for the function

Solution to this **Calculus & Precalculus Maclaurin Infinite Series** practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example problem

Determine the Maclaurin series (Taylor series about *c *= 0) and its interval of convergence for the function

Solution to this **Calculus & Precalculus Maclaurin Infinite Series** practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example #3

Find the Maclaurin series (Taylor series about *c *= 0) and its interval of convergence for the given function

Solution to this **Calculus & Precalculus Maclaurin Infinite Series** practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example question #4

Determine the Maclaurin series (Taylor series about *c *= 0) and its interval of convergence for the given function

**Calculus & Precalculus Maclaurin Infinite Series** practice problem is given in the video below!

## Taylor Series Interval of Convergence example problem #1

Determine the Taylor series about *c *= 1 and its interval of convergence for the given function

Solution to this **Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Interval of Convergence example #2

Determine the Taylor series about *c *= 1 and its interval of convergence for the function

Solution to this **Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Interval of Convergence example question #3

Determine the Taylor series about *c *= *e* and its interval of convergence for the following function

Solution to this **Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Radius of Convergence example problem #1

Determine the Taylor series about *c *= 0 and its radius of convergence for the given function

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Radius of Convergence example #2

Find the Taylor series about *c *= 0 and its radius of convergence for the given function

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Radius of Convergence example question #3

Determine the Taylor series about *c *= 0 and its radius of convergence for the following function

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Limit example problem #1

Use the Taylor series expansion about *c* = 0 to find the indicated Limit.

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Limit example #2

Use the Taylor series expansion about *c* = 1 to find the given Limit.

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Limit example question #3

Use the Taylor series expansion about *c* = 0 to evaluate the given Limit.

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Definite Integral example problem #1

Use the Taylor series expansion about *c* = 0 to evaluate the Definite Integral.

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Definite Integral example #2

Use the Taylor series expansion about *c* = 0 to find the Definite Integral value.

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Taylor Series Definite Integral example question #3

Use the Taylor series expansion about *c* = 1 to evaluate the following Definite Integral.

**Calculus & Precalculus Taylor Infinite Series** practice problem is given in the video below!

## Non-Power Infinite Series Interval of Convergence & Derivatives problems

## Non-Power Series Interval of Convergence & Derivatives example question

Find the interval of convergence of the non-power series

and the corresponding series of derivatives.

Solution to this **Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Non-Power Series Interval of Convergence & Derivatives example problem

Find the interval of convergence of the following non-power series

and determine the corresponding series of derivatives.

Solution to this **Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Infinite Series Functions Radius Interval of Convergence problems

## Power Series Representation of Functions example question

Find a power series representation of

about *c* = 0 and determine the radius and interval of convergence.

Solution to this **Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Representation of Functions example problem

Find a power series representation of

about *c* = 0 and determine the radius and interval of convergence.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Representation of Functions example

Find a power series representation of

about *c* = 0 and determine the radius and interval of convergence.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Representation of Functions example question #4

Find a power series representation of

about *c* = 0 and determine the radius and interval of convergence.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Representation of Functions example problem #5

Find a power series representation of

about *c* = 0 and determine the radius of convergence by integrating the derived function.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Representation of Functions example #6

Find a power series representation of

about *c* = 0 and determine the radius of convergence by taking derivative of the given function.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Representation of Functions example question #7

Find a power series representation of

about *c* = 0 and determine the radius of convergence by integrating the given function.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Function Interval of Convergence example problem

Determine the interval of convergence of the power series

and the function to which it converges.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Function Interval of Convergence example #2

Find the interval of convergence of the power series

and the function to which it converges.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Function Interval of Convergence example question #3

Determine the interval of convergence of the power series

and the function of convergence.

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example problem #1

Determine the interval and radius of convergence of the power series

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example #2

Find the interval and radius of convergence of the power series

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example problem #3

Determine the interval and radius of convergence of the given power series

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example question #4

Find the interval and radius of convergence of the given power series

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example #5

Determine the interval and radius of convergence of the power series given below

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example problem #6

Find the interval and radius of convergence of the power series given below

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example question #7

Determine the radius and interval of convergence of the power series

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Power Series Radius Interval of Convergence example #8

Find the radius and interval of convergence of the power series

**Calculus & Precalculus Power Infinite Series** practice problem is given in the video below!

## Absolute Convergence Test Infinite Series problems

## Absolute Convergence Test Series example question

Determine whether the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this **Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example problem

Determine if the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this **Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example

Determine whether the infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this **Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example question #4

Determine if the infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example problem #5

Determine whether the following infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example #6

Determine if the following infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example question #7

See if the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example problem #8

See whether the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example #9

Determine whether the infinite series given below

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example question #10

Determine whether the provided infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example problem #11

Determine whether the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example #12

Does the infinite series

converge absolutely, converge conditionally, or diverge?

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example question #13

Determine whether the infinite series given below

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example problem #14

Determine if the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Absolute Convergence Test Series example #15

Determine if the following infinite series

is absolutely convergent, conditionally convergent, or divergent.

**Calculus & Precalculus Absolute Convergence Test Series** practice problem is given in the video below!

## Alternating Infinite Series Convergence Divergence problems

## Alternating Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem

Determine if the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example

Determine whether the infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #4

Determine if the given infinite series

is convergent or divergent.

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem #5

See whether the given infinite series

converges or diverges.

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example #6

See if the given infinite series

converges or diverges.

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #7

See whether the infinite series

converges or diverges.

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem #8

Does the given infinite series

converge or diverge?

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example #9

Does the infinite series

converge or diverge?

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #10

Is the given infinite series

convergent or divergent?

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem #11

Is the infinite series

convergent or divergent?

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example #12

Determine if the infinite series

is convergent or divergent.

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #13

Determine if the following infinite series

is convergent or divergent.

**Calculus & Precalculus Alternating Series** practice problem is given in the video below!

## Limit Comparison Test Infinite Series Convergence Divergence problems

## Limit Comparison Test Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Limit Comparison Test Series** practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example problem

Determine if the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Limit Comparison Test Series** practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus Limit Comparison Test Series** practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example question #4

Determine if the infinite series

is convergent or divergent.

**Calculus & Precalculus Limit Comparison Test Series** practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example problem #5

Does the given infinite series

converge or diverge?

**Calculus & Precalculus Limit Comparison Test Series** practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example #6

Is the given infinite series

convergent or divergent?

**Calculus & Precalculus Limit Comparison Test Series** practice problem is given in the video below!

## Comparison Test Infinite Series Convergence Divergence problems

## Comparison Test Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Comparison Test Series** practice problem is given in the video below!

## Comparison Test Series Convergence Divergence example problem

Does the given infinite series

converge or diverge?

Solution to this **Calculus & Precalculus Comparison Test Series** practice problem is given in the video below!

## Comparison Test Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus Comparison Test Series** practice problem is given in the video below!

## Integral Test Infinite Series Convergence Divergence problems

## Integral Test Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Integral Test Series** practice problem is given in the video below!

## Integral Test Series Convergence Divergence example problem

Determine if the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Integral Test Series** practice problem is given in the video below!

## Integral Test Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus Integral Test Series** practice problem is given in the video below!

## Integral Test Series Convergence Divergence example question #4

Determine if the given infinite series

is convergent or divergent.

**Calculus & Precalculus Integral Test Series** practice problem is given in the video below!

## Integral Test Series Convergence Divergence example problem #5

Does the given infinite series

converge or diverge?

**Calculus & Precalculus Integral Test Series** practice problem is given in the video below!

## Harmonic Infinite Series Convergence Divergence problems

## Harmonic Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Harmonic Series** practice problem is given in the video below!

## P Series Convergence Divergence problems

## P Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus P Series** practice problem is given in the video below!

## P Series Convergence Divergence example problem

Determine if the given infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus P Series** practice problem is given in the video below!

## Telescoping Infinite Series Convergence Divergence problems

## Telescoping Infinite Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Telescoping Infinite Series** practice problem is given in the video below!

## Telescoping Infinite Series Convergence Divergence example problem

Determine whether the given infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus Telescoping Infinite Series** practice problem is given in the video below!

## Telescoping Infinite Series Convergence Divergence example

Is the infinite series

convergent or divergent?

Solution to this **Calculus & Precalculus Telescoping Infinite Series** practice problem is given in the video below!

## Multiple Terms Infinite Series Convergence Divergence problems

## Multiple terms Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Infinite Series** practice problem is given in the video below!

## Multiple terms Convergence Divergence example problem

Does the given infinite series

converge or diverge?

Solution to this **Calculus & Precalculus Infinite Series** practice problem is given in the video below!

## Nth Term Test Infinite Series Convergence Divergence problems

## Nth Term Test Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Nth Term Test Infinite Series** practice problem is given in the video below!

## Geometric Infinite Series Convergence Divergence problems

## Geometric Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Geometric Infinite Series** practice problem is given in the video below!

## Geometric Series Convergence Divergence example problem

Determine whether the infinite series

converges or diverges.

Solution to this **Calculus & Precalculus Geometric Infinite Series** practice problem is given in the video below!

## Geometric Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this **Calculus & Precalculus Geometric Infinite Series** practice problem is given in the video below!

## Geometric Series Convergence Divergence example question #4

Determine if the infinite series

is convergent or divergent.

**Calculus & Precalculus Geometric Infinite Series** practice problem is given in the video below!

## Ramanujan’s Nested Radical Theorem Proof

## Ramanujan’s Nested Radical Theorem Proof example question

Show that the given equation

holds true and follows a never-ending pattern.

Solution to this **Calculus & Precalculus Infinite Sequence** practice problem is given in the video below!

## Bounded Infinite Sequence problems

## Bounded Infinite Sequence example question

Show that the given sequence

is bounded and find its boundary value (the value that the sequence converges to).

Solution to this **Calculus & Precalculus Bounded Infinite Sequence** practice problem is given in the video below!

## Bounded Infinite Sequence example problem

Prove that the given sequence

is bounded and determine its boundary value (the value that the sequence converges to).

Solution to this **Calculus & Precalculus Bounded Infinite Sequence** practice problem is given in the video below!

## Bounded Infinite Sequence example

Show that the given sequence

is bounded and then find the boundary value that the sequence converges to.

Solution to this **Calculus & Precalculus Bounded Infinite Sequence** practice problem is given in the video below!

## Bounded Infinite Sequence example question #4

Show that the sequence

is bounded and find the boundary value the sequence converges to.

**Calculus & Precalculus Bounded Infinite Sequence** practice problem is given in the video below!

## Increasing & Decreasing Infinite Sequence problems

## Increasing & Decreasing Infinite Sequence example question

Determine whether the given sequence

increases or decreases.

Solution to this **Calculus & Precalculus Increasing & Decreasing Infinite Sequence** practice problem is given in the video below!

## Increasing & Decreasing Infinite Sequence example problem

Determine whether the sequence

increases or decreases.

Solution to this **Calculus & Precalculus Increasing & Decreasing Infinite Sequence** practice problem is given in the video below!

## Increasing & Decreasing Infinite Sequence example

Determine if the given sequence

is increasing or decreasing.

Solution to this **Calculus & Precalculus Increasing & Decreasing Infinite Sequence** practice problem is given in the video below!

## Infinite Sequences Convergence Divergence problems

## Sequence Limit Definition Convergence Proof example question

Show that the sequence

converges to 1 by Sequence Limit Definition.

Solution to this **Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence with Squeeze Theorem example problem

Show that the sequence

converges by using the **Squeeze Theorem**.

Solution to this **Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example

Determine whether the sequence

converges or diverges.

Solution to this **Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example question #2

Determine whether the given sequence

converges or diverges.

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example problem #3

Determine whether the following sequence

converges or diverges.

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example #4

Determine whether the sequence

is convergent or divergent.

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example question #5

Determine whether the given sequence

is convergent or divergent.

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example problem #6

Determine whether the following sequence

is convergent or divergent.

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example #7

Does the sequence

converge or diverge?

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Sequence Convergence Divergence example question #8

Does the given sequence

converge or diverge?

**Calculus & Precalculus Infinite Sequence Convergence** practice problem is given in the video below!

## Conic Section Proof problems

## Parabola Directrix Conic Section Proof example question

Show that in polar coordinates the equation of a conic section with (one) focus at the pole and directrix,

the line

can be written as

.

Solution to this **Conic Section Proof** practice problem is given in the video below!

## Trigonometric Identity Proof problems

## Trigonometric Identity Proofs example problem #1

Show that

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proof example question #2

Show that the following equation is valid.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proofs example #3

Prove the validity of the following equation.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proof example problem #4

Prove that

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proofs example question #5

Prove the following identity.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proof example #6

Show that the following identity holds.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proofs example problem #7

Prove the following identity.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proof example question #8

Verify the given identity.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proofs example #9

Show that the identity given below is valid.

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Trigonometric Identity Proof example problem #10

Verify the given three Trigonometric Identities:

(a)

(b)

(c)

Solution to this **Trigonometric Identity Proof** practice problem is given in the video below!

## Power Rule of Derivative PROOF & Binomial Theorem

## Derivative Power Rule PROOF example question

Show that

using **Limits** and **Binomial Theorem**.

Solution to this Calculus **Differentiation Power Rule** practice problem is given in the video below!

## Hyperbolic & Inverse Hyperbolic Functions Derivative problems

**Derivative of Hyperbolic Functions example question**

Find the derivative of the following Hyperbolic functions.

Solution to these Calculus **Derivative of Hyperbolic Functions** practice problems is given in the video below!

**Derivative of Hyperbolic & Inverse Hyperbolic Functions example problem**

Find the derivative of the following Hyperbolic & Inverse Hyperbolic functions.

Solution to these Calculus **Derivative of Hyperbolic & Inverse Hyperbolic Functions** practice problems is given in the video below!

## Derivative of Inverse Trigonometric Functions problems

**Derivative of Inverse Trigonometric Functions example question**

Find the derivative of the following Inverse Trig functions.

Solution to these Calculus **Derivative of Inverse Trigonometric Functions** practice problems is given in the video below!

**Derivative of Inverse Trigonometric Functions example problem**

Find the derivative of the given Inverse Trig functions.

Solution to these Calculus **Derivative of Inverse Trigonometric Functions** practice problems is given in the video below!

## Derivative of Inverse Functions problems

**Derivative of Inverse Functions example question**

Find the derivative of the **Inverse** of the following function.

Solution to this Calculus **Derivative of Inverse Functions** practice problem is given in the video below!

**Derivative of Inverse Functions example problem**

Find the derivative of the **Inverse** of the given function.

Solution to this Calculus **Derivative of Inverse Functions** practice problem is given in the video below!

## Logarithmic Differentiation problems

**Logarithmic Differentiation example question**

Find the derivative of the following functions.

Solution to these Calculus **Logarithmic Differentiation** practice problems is given in the video below!

## Derivative of Logarithmic Functions problems

**Derivative of Natural Logarithmic Functions example question**

Find the derivative of the following functions.

Solution to these Calculus **Derivative of Logarithmic Functions** practice problems is given in the video below!

**Derivative of Common Logarithmic Functions example problem**

Find the derivative of the functions given below.

Solution to these Calculus **Derivative of Logarithmic Functions** practice problems is given in the video below!

## Derivative of Exponential Functions problems

**Derivative of Exponential Functions example question**

Find the derivative of the following functions.

Solution to these Calculus **Derivative of Exponential Functions** practice problems is given in the video below!

**Derivative of Exponential Functions example problem**

Find the derivative of the functions provided below.

Solution to these Calculus **Derivative of Exponential Functions** practice problems is given in the video below!

**Derivative of Complex Exponential Functions example**

Find the derivative of the given functions.

Solution to these Calculus **Derivative of Exponential Functions** practice problems is given in the video below!

## Power Rule, Product Rule, Quotient Rule, Chain Rule Derivative problems

**Power & Product Rule derivative example question**

Find the derivative of the function

by using Power Rule **or** Product Rule.

Solution to this Calculus **Power Product Quotient Chain Rule Derivative** practice problem is given in the video below!

**Quotient Rule derivative example problem**

Find the derivative of the function

by using the Quotient Rule.

Solution to this Calculus **Power Product Quotient Chain Rule Derivative** practice problem is given in the video below!

**Chain Rule derivative example**

Find the derivative of the function

by using the Chain Rule.

Solution to this Calculus **Power Product Quotient Chain Rule Derivative** practice problem is given in the video below!

## Limit Definition of a Derivative problems

**Limit Definition of a Derivative example problem**

Find the derivative of the function

by using **Formal Limit Definition of a Derivative**.

Solution to this Calculus **Limit Definition of Derivative Differentiation** practice problem is given in the video below!

**Equation of the Tangent Line using Limit Definition of a Derivative example question**

Find the Equation of the Tangent Line to the graph of

at the point **(5, 1) **by using the Limit Definition of a Derivative.

Solution to this Calculus **Limit Definition of Derivative Differentiation** practice problem is given in the video below!

## Linear Approximation Differentiation problems

**Linear Approximation example question**

Find the given values of the function

by using Linear Approximation.

a) sin(0.1)

b) sin(1.0)

c)

Solution to this Calculus **Linear Approximation Differentiation** practice problem is given in the video below!

## Curve Sketching Differentiation problems

**Polynomial Function Curve Sketching example question**

Sketch the curve represented by the given function by finding critical values (local minimum and local maximum points) and inflection points by finding first and second Derivatives.

Solution to this Calculus **Curve Sketching Differentiation** practice problem is given in the video below!

**Rational Function Curve Sketching example problem**

Sketch the curve represented by the function below by finding critical values (local minimum and local maximum points) and inflection points by determining the first and second Derivative functions.

Solution to this Calculus **Curve Sketching Differentiation** practice problem is given in the video below!

**Exponential Function Curve Sketching example**

Sketch the curve represented by the given function by finding critical values (local minimum, local maximum points and general extrema) and inflection points by analyzing first and second Derivative functions.

Solution to this Calculus **Curve Sketching Differentiation** practice problem is given in the video below!

## Functions with Roots & Fractions Integration by Substitution problems

**Antiderivative example question**

Find the following Antiderivatives by using **U Substitution**.

Solution to these Calculus **Integration of Functions with Roots & Fractions** practice problems is given in the video below!

**Antiderivative example problem**

Find the following Antiderivative by using **U Substitution**.

Solution to this Calculus **Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example**

Find the following Antiderivative using **U Substitution**.

Solution to this Calculus **Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #4**

Find the Antiderivative by using **U Substitution**.

Solution to this Calculus **Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #5**

Find the given Antiderivative by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #6**

Solve the given Integrals by using **U Substitution**.

Solution to these Calculus **Integration of Functions with Roots & Fractions** practice problems is given in the video below!

**Antiderivative example question #7**

Solve the Integrals by using **U Substitution**.

Solution to these Calculus **Integration of Functions with Roots & Fractions** practice problems is given in the video below!

**Antiderivative example problem #8**

Solve the Integral using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #9**

Find the given Antiderivative using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #10**

Find the following Antiderivative by **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #11**

Find the Antiderivative by **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #12**

Solve for the given Antiderivative by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #13**

Find the Antiderivative below by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #14**

Find the Antiderivative below using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #15**

Find the given Antiderivative by **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #16**

Find the following Antiderivative given below with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #17**

Find the Antiderivative given below with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #18**

Solve the following Integral using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #19**

Find the given Antiderivative with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #20**

Find the Antiderivative by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #21**

Find the given Antiderivative by **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #22**

Find the Antiderivative using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #23**

Solve the given Integral by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #24**

Find the Antiderivative function by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #25**

Solve for the Antiderivative by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #26**

Solve for the Antiderivative with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example #27**

Derive the Antiderivative function by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #28**

Derive the Antiderivative function using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example problem #29**

Derive the Antiderivative function with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Hard Antiderivative example #30**

Solve for the Antiderivative function by **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Antiderivative example question #31**

Solve for the Antiderivative function with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Definite Integral example problem #1**

Find the Definite Integral value by using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Definite Integral ****example #2**

Find the Definite Integral value using **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Definite Integral ****example question #3**

Solve the Definite Integrals with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

**Definite Integral ****example problem #4**

Solve the Definite Integrals below with **U Substitution**.

**Integration of Functions with Roots & Fractions** practice problem is given in the video below!

## Hyperbolic & Inverse Hyperbolic Functions Integration by Substitution problems

**Hyperbolic Antiderivative example question**

Find the following Antiderivatives by using **U Substitution**.

Solution to these Calculus **Integration of Hyperbolic Functions** practice problems is given in the video below!

**Inverse Hyperbolic Antiderivative example problem**

Find the given Antiderivative with **U Substitution**.

Solution to this Calculus **Integration of Inverse Hyperbolic Functions** practice problem is given in the video below!

**Hyperbolic Definite Integral example**

Find the following Definite Integral value by using **U Substitution**.

Solution to this Calculus **Integration of Hyperbolic Functions** practice problem is given in the video below!

**Hyperbolic Definite Integral example question #2**

Evaluate the Definite Integrals below by using **U Substitution**.

Solution to these Calculus **Integration of Hyperbolic Functions** practice problems is given in the video below!

## Inverse Trigonometric Functions Integration by Substitution problems

**Antiderivative example question**

Find the following Antiderivatives by using **U Substitution**.

Solution to these Calculus **Integration of Inverse Trigonometric Functions** practice problems is given in the video below!

**Antiderivative example problem**

Find the following Antiderivatives by using **U Substitution**.

Solution to these Calculus **Integration of Inverse Trigonometric Functions** practice problems is given in the video below!

**Definite Integral example **

Find the given Definite Integral values using **U Substitution**.

Solution to these Calculus **Integration of Inverse Trigonometric Functions** practice problems is given in the video below!

## Trigonometric Functions Integration by Substitution problems

**Antiderivative example question**

Find the following Antiderivatives by using **U Substitution.**

Solution to these Calculus **Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

**Antiderivative example problem**

Solve the following Antiderivatives by using **U Substitution.**

Solution to these Calculus **Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

**Antiderivative example **

Find the given Antiderivatives by using **U Substitution.**

Solution to these Calculus **Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

**Antiderivative example question #4**

Find the given Antiderivative by using **U Substitution.**

Solution to this Calculus **Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #5**

Find the following Antiderivative by using **U Substitution.**

Solution to this Calculus **Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #6**

Solve the following Integral by using **U Substitution.**

Solution to this Calculus **Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #7**

Solve the given Integral by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #8**

Solve the Integral given below by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #9**

Find the following Antiderivative by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #10**

Find the given Antiderivative by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #11**

Solve the Integral below by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #12**

Find the following Integral function by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #13**

Find the Antiderivative by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #14**

Find the Antiderivative below by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #15**

Find the given Antiderivative below by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #16**

Find the given Antiderivative by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #17**

Find the following Antiderivatives by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

**Antiderivative example #18**

Find the given Antiderivatives by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

**Antiderivative example question #19**

Solve the following Integral by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #20**

Solve the Integral by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #21**

Solve the given Antiderivative by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #22**

Solve for the given Antiderivative by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #23**

Find the following Antiderivative using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #24**

Solve the following Integral by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #25**

Find the given Antiderivative by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #26**

Solve the Integral below by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #27**

Find the following Antiderivative with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #28**

Solve the following Integral with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #29**

Find the following Antiderivative via **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #30**

Solve the given Integral below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #31**

Solve the Integral with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #32**

Solve the Integral below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #33**

Solve the following Integral with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #34**

Solve the given Integral by using **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #35**

Solve the Integral with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example #36**

Solve the given Integral below by **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #37**

Find the Antiderivative below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Antiderivative example problem #38**

Solve the given Integral below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Hard Antiderivative example #39**

Solve the given Integral below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Hard Antiderivative example question #40**

Solve the Integral with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problem is given in the video below!

**Definite Integral example problem**

Solve the given Definite Integrals below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

**Definite Integral example question**

Find the given Definite Integral values below with **U Substitution.**

**Integration of Trigonometric Functions by Substitution** practice problems is given in the video below!

## Logarithmic Functions Integration by Substitution problems

**Antiderivative example question**

Find the following Antiderivatives by using **U Substitution.**

Solution to these Calculus **Integration of Logarithmic Functions by Substitution** practice problems is given in the video below!

**Antiderivative example problem**

Find the given Antiderivatives by using **U Substitution.**

Solution to these Calculus **Integration of Logarithmic Functions by Substitution** practice problems is given in the video below!

**Definite Integral example**

Solve the given Definite Integrals by using **U Substitution.**

Solution to these Calculus **Integration of Logarithmic Functions by Substitution** practice problems is given in the video below!

## Exponential Functions Integration by Substitution problems

**Antiderivative example question**

Find the following Antiderivatives by using **U Substitution.**

Solution to these Calculus **Integration of Exponential Functions by Substitution** practice problems is given in the video below!

**Antiderivative example problem**

Solve the following Integrals by using **U Substitution.**

Solution to these Calculus **Integration of Exponential Functions by Substitution** practice problems is given in the video below!

**Antiderivative example**

Solve for the following Antiderivative by using **U Substitution.**

Solution to this Calculus **Integration of Exponential Functions by Substitution** practice problem is given in the video below!

**Antiderivative example question #4**

Find the given Antiderivatives below by using **U Substitution.**

Solution to these Calculus **Integration of Exponential Functions by Substitution** practice problems is given in the video below!

**Definite Integral example problem**

Find the following Definite Integral value by using **U Substitution.**

Solution to this Calculus **Integration of Exponential Functions by Substitution** practice problem is given in the video below!

**Definite Integral example**

Find the following Definite Integral values by using **U Substitution.**

**Integration of Exponential Functions by Substitution** practice problems is given in the video below!

**Definite Integral example question**

Solve the given Definite Integral by using **U Substitution.**

Solution to this Calculus **Integration of Exponential Functions by Substitution** practice problem is given in the video below!

## Integration by Trigonometric Substitution problems

**Antiderivative example question**

Find the following Antiderivative by using **Trigonometric Substitution.**

Solution to this Calculus **Integration by Trigonometric Substitution** practice problem is given in the video below!

**Antiderivative example problem**

Solve for the following Antiderivative by using **Trigonometric Substitution.**

Solution to this Calculus **Integration by Trigonometric Substitution** practice problem is given in the video below!

**Antiderivative example**

Solve the following Integral by using **Trigonometric Substitution.**

Solution to this Calculus **Integration by Trigonometric Substitution** practice problem is given in the video below!

**Antiderivative example question #4**

Solve the given Integral by using **Trigonometric Substitution.**

**Integration by Trigonometric Substitution** practice problem is given in the video below!

**Definite Integral example problem**

Find the given Definite Integral value by using **Trigonometric Substitution.**

**Integration by Trigonometric Substitution** practice problem is given in the video below!

## Eigenvalues & Eigenvectors problems in Linear Algebra

**Eigenvalue of Matrix example question**

Find the corresponding Eigenvalue of the following matrix *A* given the Eigenvector **x**.

*A* =

**x **=

Solution to this **Linear Algebra** practice problem is given in the video below!

**Basis for Eigenspace of Matrix example problem**

Find the Characteristic Equation and Bases for the Eigenspaces of the given matrix *A*.

*A* =

Solution to this **Linear Algebra** practice problem is given in the video below!

## General Vector Spaces problems in Linear Algebra

**Real Vector Space example question**

Determine whether the given set equipped with the given operations is a vector space.

The set of all triplets of real numbers with the standard vector addition but with scalar multiplication defined by

*k*(*x*, *y*, *z*) = (*k*^{2}*x*, *k*^{2}*y*, *k*^{2}*z*)

Solution to this **Linear Algebra** practice problem is given in the video below!

**Subspace example problem**

Determine whether the following are subspaces of *R*^{3}:

(a) All vectors of the form (*a*, 0, 0)

(b) All vectors of the form (*a*, 1, 0)

(c) All vectors of the form (*a*, *b*, *c*) where *b* = *a* + *c *

(d) All vectors of the form (*a*, *b*, *c*) where *c* = *a* *b *

(e) All vectors of the form (*a*, *a*, 0)

Solution to these **Linear Algebra** practice problems is given in the video below!

**Linear Combination example**

Which of the following are linear combinations of **u** = (1, 3, 2) and **v** = (1, 0, 4)?

(a) (0, 3, 6)

(b) (3, 9, 2)

(c) (0, 0, 0)

(d) (1, 6, 16)

Solution to these **Linear Algebra** practice problems is given in the video below!

**Vector Span example question**

Determine whether the given vector spans *R*^{3}:

**v**_{1} = (1, 2, 3)

**v**_{2} = (2, 0, 0)

**v**_{3} = (2, 1, 0)

Solution to this **Linear Algebra** practice problem is given in the video below!

**Linear Independence example problem**

Determine whether the following vectors in *R*^{4} are linearly dependent:

(1, 2, 2, 1)

(3, 6, 6, 3)

(4, 2, 4, 1)

Solution to this **Linear Algebra** practice problem is given in the video below!

**Basis of Vector Set example**

Determine whether the following set of vectors form a basis in *R*^{3}:

(1, 0, 0)

(2, 2, 0)

(3, 3, 3)

Solution to this **Linear Algebra** practice problem is given in the video below!

**Basis & Dimension of Solution Space example question**

Find a basis for the solution space of the homogeneous linear system, and find the dimension of that space.

2*x*_{1} + *x*_{2} + 3*x*_{3} = 0

*x*_{1} + 5*x*_{3} = 0

*x*_{2} + *x*_{3} = 0

Solution to this **Linear Algebra** practice problem is given in the video below!

**Change of Basis example problem**

Consider the bases

*B *= {**u**_{1}, **u**_{2}} and *B’* = {**u**_{1}‘, **u**_{2}‘} for *R*^{2},

where

**u**_{1 }=

**u**_{2 }=

**u**_{1}‘ =

**u**_{2}‘ =

Find the transition matrix from* B* to *B’*.

Solution to this **Linear Algebra** practice problem is given in the video below!

**Subspace Basis example**

Find a basis for the subspace of *R*^{4} spanned by the given vectors

(2, 4, 2, 3)

(2, 2, 2, 4)

(1, 3, 1, 1)

Solution to this **Linear Algebra** practice problem is given in the video below!

**Rank of Matrix example question**

Are there values of *r* and *s* for which the given matrix has rank 1 or rank 2?

Solution to this **Linear Algebra** practice problem is given in the video below!

## Matrix Inverse problems in Linear Algebra

**2×2 Matrix Inverse using Determinant example**

A =

D =

B =

Find Inverses of all these 3 matrices, A^{-1}, D^{-1}, B^{-1}, by using their respective determinants.

Solution to this **Linear Algebra** practice problem is given in the video below!

**Matrix Inverse using Identity Matrix example problem**

Find the Inverse, if it exists, of the following square matrices by using their Identity Matrix.

Solution to these **Linear Algebra Identity Matrix Inverse** practice problems is given in the video below!

**System Solution using Matrix Inverse example question**

Find the solution to the following system of linear equations by using the product of the Inverse of a coefficient matrix and augmented matrix column.

3*x*_{1} + 5*x*_{2} = 2

*x*_{1} + 2*x*_{2} = 3

Solution to this **Linear Algebra Matrix Inverse** practice problem is given in the video below!

**Find Linear Equation using Matrix Inverse example**

Use Matrix Inverse to find an Equation of a Line ** y** =

**+**

*mx*

*b*when

Solution to this **Matrix Inverse Linear Equation** practice problem is given in the video below!

**Find Quadratic Equation using Matrix Inverse example question**

Use Matrix Inverse to find a Quadratic Equation of a Curve ** y** =

**+**

*ax*^{2}**+**

*bx*

*c*when

Solution to this **Matrix Inverse Quadratic Equation** practice problem is given in the video below!

**Minors and Cofactors example problem**

Find the minors and cofactors of the following matrix.

Solution to this **Linear Algebra** practice problem is given in the video below!

**Matrix Determinant with Arrow Technique example**

Find the determinant of the following 3 x 3 matrix by using the Arrow Technique.

Solution to this **Linear Algebra** practice problem is given in the video below!

**System Solution using Cramer’s Rule example question**

Find the solution to the following system of linear equations by using Cramer’s Rule.

7*x*_{1} 2*x*_{2} = 3

3*x*_{1} + *x*_{2} = 5

Solution to this **Linear Algebra** practice problem is given in the video below!

**Matrix Inverse using Adjoint Method example problem**

Find the Inverse of the following Matrix by using the Adjoint Method.

Solution to this **Linear Algebra** practice problem is given in the video below!

## Matrix Operations problems in Linear Algebra

**Defined & Undefined Matrix Operations example question**

**A** is a 4 x 5 matrix

**B** is 4 x 5 matrix

**C** is a 5 x 2 matrix

**D** is a 4 x 2 matrix

**E** is a 5 x 4 matrix

Compute the following matrix expressions (whenever possible):

BA =

AC + D =

AE + B =

AB + B =

E(A + B) =

E(AC) =

E^{T}(A) =

(A^{T} + E)D =

Solution to these **Linear Algebra** practice problems is given in the video below!

**Matrix Operations example problem**

**A** =

**B** =

**D** =

**E** =

Compute the following matrix expressions (whenever possible):

AB =

BA =

**tr**(4E^{T} D) =

Solution to these **Linear Algebra** practice problems is given in the video below!

## Gaussian & Gauss-Jordan Elimination problems in Linear Algebra

**Gaussian Elimination example question**

Solve the system of linear equations by using **Gaussian Elimination**.

*x*_{1} + 2*x*_{2} 3*x*_{3} = 6

2*x*_{1} *x*_{2} + 4*x*_{3} = 1

*x*_{1} *x*_{2} + *x*_{3} = 3

Solution to this **Linear Algebra** practice problem is given in the video below!

**Gauss-Jordan Elimination example problem**

Solve the following system of linear equations by using **Gauss-Jordan Elimination**.

*x*_{1} + 2*x*_{2} 3*x*_{3} = 6

2*x*_{1} *x*_{2} + 4*x*_{3} = 1

*x*_{1} *x*_{2} + *x*_{3} = 3

Solution to this **Linear Algebra** practice problem is given in the video below!

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