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Confidence Interval statistics problems

Confidence Interval for MEAN Normal Distribution example question

A manager at a power company monitored the employee time required to process high-efficiency lamp bulb rebates. A random sample of 40 applications gave a sample mean time of 3.8 minutes and a standard deviation of 1.2 minutes. Construct a 90% confidence interval for the mean time to process μ.

 

Solution to this Confidence Interval Statistics practice problem is given in the video below!


 


T-Distribution Confidence Interval for MEAN example

From a random sample of size 16, one has calculated the 95% confidence interval for μ (population mean) and obtained the result (39.51, 42.49). What are the (sample mean) and s (sample standard deviation) for the sample? Assume the population is normal.

 

Solution to this Confidence Interval Statistics practice problem is given in the video below!


 


Confidence Interval for PROPORTION Normal Distribution example problem

A telephone survey of 1,000 was taken after movie X came out in the national theater.

a) If 832 liked the movie, with what confidence can it be asserted that 83.2% ± 3% of the adult population liked this movie?

b) If the adult population in the nation is 150 million, what is the interval estimating the total number of people in the nation who liked the movie?

 

Solution to this Confidence Interval Statistics practice problem is given in the video below!


 

Greatest Common Factor and Least Common Multiple problems

Greatest Common Factor and Least Common Multiple example question

GCF(A, B) = 21

LCM(A, B) = 6(5)(72)

B = 3(72)

Find the value of A.

 

Solution to this Algebra GCF LCM practice problem is given in the video below!



 

Least Common Multiple example word problem

What is the least number of cards that could satisfy the following three conditions?

Condition 1: If all the cards are put in 2 equal piles, there is 1 card left over

Condition 2: If all the cards are put in 3 equal piles, there is 1 card left over

Condition 3: If all the cards are put in 5 equal piles, there is 1 card left over

 

Solution to this Algebra Least Common Multiple practice problem is given in the video below!



 

Translation Geometry problems

Translation Coordinates and Graph of Translated Image example question

Find the coordinates and the translated graph of A(-2, 3), B(4, -5), and C(0, 3) by following the rule of translation:

(x, y) → (x – 5, y + 6).

 

Solution to this Translation Geometry practice problem is given in the video below!


 


Finding Coordinate RULE of Translation given Graph of Translated Image example problem

Find the coordinate rule for the translation shown in the graph below.

 

Solution to this Translation Geometry practice problem is given in the video below!


 


Dilation Geometry problems

Dilation Formulas and Graph with Origin the Center of Dilation example question

Graph the triangle with vertices A(-2, 4), B(1, -4), C(-3, -2) and its image after dilation by scale factor k = -3. The center of dilation is the origin.

 

Solution to this Dilation Geometry practice problem is given in the video below!


 


Dilation Formulas and Graph with Center of Dilation NOT the Origin example problem

Graph the triangle with vertices A(-2, 4), B(1, -4), C(-3, -2) and its image after dilation by scale factor k = 2. The center of dilation is the point (4, 5).

 

Solution to this Dilation Geometry practice problem is given in the video below!


 


How to Find the CENTER of DILATION COORDINATES using Given Figure and its Image problem

Find the coordinates of the center of dilation if coordinates of original figure are (-4, 14), (-10, 16), (-12, 10), (-14, 12) and coordinates of its image are (2, 1), (3, 4), (-1, 2), (4, 3).

 

Solution to this Dilation Geometry practice example is given in the video below!


 


Reflection Geometry problems

Reflection Formulas example question

Draw the image of the polygon with coordinates A(-2, 3), B(0, 2), C(3, -4), D(4, 0) by reflecting it

a) through the y-axis

b) through the x-axis

c) through the origin

d) through the line y = x

 

Solution to this Reflection Geometry practice problem is given in the video below!


 


Rotation Geometry problems

Rotation Formulas example question

a) Draw the image of the triangle with coordinates A(-6, -6), B(-6, 3), C(-2, 3) by rotating it 90 degrees clockwise

b) Draw the image of the quadrilateral with coordinates A(-3, -3), B(-1, 0), C(3, 0), D(5, -3) by rotating it 90 degrees counterclockwise

c) Draw the image of the triangle with coordinates A(-3, 2), B(1, 5), C(0, 0) by rotating it 90 degrees clockwise

d) Draw the image of the triangle with coordinates A(1, -3), B(3, 3), C(6, -3) by rotating it 180 degrees clockwise or counterclockwise

e) Draw the image of the quadrilateral with coordinates A(-5, -2), B(-4, 5), C(2, 5), D(2, 0) by rotating it 90 degrees clockwise

 

Solution to this Rotation Geometry practice problem is given in the video below!


 


Projectile Motion physics problems

Projectile Motion Physics example question

A softball is thrown from the roof of the gym with a horizontal velocity of 36.6 m/s and lands 97.2 meters away. How tall is the building?

 

Solution to this Projectile Motion physics practice problem is given in the video below!


 


Solving for x in Angles and Triangles problems

Solving for X and Y in Angles example question

Find the values of x and y using the figure below. 

Use these values to find the following:

a) 

b) 

c) 

d) 

 

Solution to this value of x and y in Angles Geometry practice problem is provided in the video below!



Combined Variation and Proportion problems

Combined Variation and Proportion example question

If z varies directly as the square root of x and inversely as y, and z = 5 when x = 3 and y = 12, find z when x = 27 and y = 5.

 

Solution to this Combined Variation and Proportion practice problem is provided in the video below!



 

Joint Variation and Proportion problems

Joint Variation and Proportion example question

If z varies jointly as x and y, and z = 3 when x = 4 and y = 6, find z when x = 20 and y = 9.

 

Solution to this Joint Variation and Proportion practice problem is provided in the video below!



 

Inverse Variation and Proportion problems

Inverse Variation and Proportion example question

If y varies inversely as the square root of x, and y = 2 when x = , find y when x = 2.

 

Solution to this Inverse Variation and Proportion practice problem is provided in the video below!



 

Direct Variation and Proportion problems

Direct Variation and Proportion example question

If y varies directly as the fourth power of x, and y = 2 when x = , find y when x = 2.

 

Solution to this Direct Variation and Proportion practice problem is provided in the video below!



 

Composite Functions problems

Hard Composite Functions example question

Using the four given functions

Express the following Functions in Composite form:

 

Solution to this Composite Functions practice problem is provided in the video below!



 

Angles, Parallel Lines and Transversals problems

Angle Theorems Parallel Lines and Transversals example question

The following figure shows two parallel lines l and m and a transversal. 

Use this figure to answer the following questions:

a) 

b) 

c) 

d) 

e) 

f) 

g) 

 

Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!



Angle Theorems Parallel Lines and Transversals example problem #2

Use the following figure to find the values of x and y

 

Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!



Angle Theorems Parallel Lines and Transversals example #3

Use the figure below to solve for the values of x and y

 

Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!



Angle Theorems Parallel Lines and Transversals example question #4

Use the given figure below to find the values of x and y

 

Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!



Angle Theorems Parallel Lines and Transversals example problem #5

The figure below shows a polygon with some interior angle measures provided.

Find the values of x and y

 

Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!



Two-Way Table Data Probability problems

Two Way Table Data Conditional Probability example problem

When insurance companies establish policies for overing screening tests for diseases, one important factor is the value of the test predicting the disease. For example, for a certain type of disease, insurance companies may only cover the test costs if the test improves the prediction of having the disease by 80%. To help decide coverage policy for a new test, use the following data to help decide whether the test should be covered.

a) Find P(A) = P(Having the disease among everyone)

b) Find P(B) = P(Testing positive for everyone)

c) Find P(A and B) = P(Having the disease and testing positive)

d) Find P(A | B) = P(Having the disease given tested positive)

e) Should the test be covered? What is your conclusion? Justify your answer using the conditional probabilities above.

 

Solution to this Calculus Two Way Table Data Conditional Probability practice problem is given in the video below!



Vertex Form Quadratic Equation of a Parabola problems

Vertex Form Quadratic Equation of a Parabola example problem

Express the following Parabolic Function Equation in Vertex Form:

f(x) = 12x2 + 24x + 48

 

Solution to this Vertex Form Quadratic Function practice problem is provided in the video below!



 

Specific Heat chemistry problems

Specific Heat of Water example question

How much heat, in calories and kilocalories, does it take to raise the temperature of 814 grams of water from 18.0 degrees to 100 ºC?

 

Solution to this Specific Heat Chemistry practice problem is given in the video below!


 


Specific Heat of Iridium example problem

A 23.9-gram sample of iridium is heated to 89.7 ºC and then dropped into 20.0 grams of water in a calorimeter. The temperature of the water rises from 20.1 ºC to 22.6 ºC. Calculate the specific heat of iridium.

 

Solution to this Specific Heat Chemistry practice problem is given in the video below!


 


Particle Motion calculus problems

Average Velocity of a Particle example question

The velocity, in feet per second, of a particle moving along the x-axis is given by the function v(t) = et + tet. What is the average velocity of the particle from time t = 0 to time t = 3?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Number of Times Velocity of a Particle is Zero example problem

The position of an object attached to a spring is given by , where t is time in seconds. In the first 4 seconds, how many times is the velocity of the object equal to 0?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Times When a Particle is at Rest example

A particle moves along the x-axis so that at time t ≥ 0 its position is given by . At what time t is the particle at rest?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Height of a Particle at its Maximum Upward Velocity example question

The height h, in meters, of an object at time t is given by . What is the height of the object at the instant when it reaches its maximum upward velocity?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Acceleration of a Particle at Time t example problem

A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by v(t) = 3 + 4.1cos(0.9t). What is the acceleration of the particle at time t = 4?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


HARD Velocity Acceleration of a Particle example

At time t ≥ 0, the acceleration of a particle moving on the x-axis is a(t) = t + sin(t). At t = 0, the velocity of the particle is -2. For what value of t will the velocity of the particle be zero?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Maximum Acceleration of a Particle example question

Find the maximum acceleration attained on the interval 0 ≤ t ≤ 3 by the particle whose velocity is given by

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Position of a Particle at Time t When its Velocity is First Equal To Zero example problem

A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by . The position of the particle is 3 at time t = 0. What is the position of the particle when its velocity is first equal to 0?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Specific Velocity of a Particle using its Acceleration example

A particle moves along the x-axis so that at any time t > 0, its acceleration is given by . If the velocity of the particle is 2 at time t = 1, find the velocity of the particle at time t = 2.

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Time When the Particle is FARTHEST to the Right example question

A particle moves along the x-axis so that its acceleration at any time t is . If the initial velocity of the particle is 6, at what time t during the interval 0 ≤ t ≤ 4 is the particle farthest to the right?

 

Solution to this Particle Motion calculus practice problem is given in the video below!


 


Rational Equations problems

Solving Rational Equations with Polynomials example question

Solve the following Rational Equation provided below:

 

Solution to this Rational Equation with Polynomials practice problem is given in the video below!


Simplifying Polynomial Rational Expressions with Factoring problems

Simplifying Fractions with Polynomials example problem

Simplify the following Fraction Expression to only one term:

 

Solution to this Simplification of Fraction with Polynomials practice problem is given in the video below!


Compound Interest problems

Compound Interest example problem

Suppose you make a deposit of $8,000 in the bank account that earns you 5.5% interest. Assuming that you are not withdrawing any funds, find the amount, in dollars, that will be in your bank account 1 year from the date of your initial deposit if

a)  Interest is compounded daily

b)  Interest is compounded monthly

c)  Interest is compounded quarterly

d)  Interest is compounded continuously

 

Solution to this Compound Interest practice problem is given in the video below!


 

TRY IT YOURSELF example question

On January 1, 2021 you borrow a $15,000 loan from the bank with a 7.5% accrued interest. If you do not plan on making any payments during the loan period, find the total balance of your loan, in dollars, that you will owe to the bank on January 1, 2022 for

a)  Daily compounded interest

b)  Weekly compounded interest

c)  Monthly compounded interest

d)  Quarterly compounded interest

e)  Semi-annually compounded interest

f)  Continuously compounded interest


Daily compounded loan balance: $16,168

Weekly compounded loan balance: $16,167

Monthly compounded loan balance: $16,164

Quarterly compounded loan balance: $16,157

Semi-annually compounded loan balance: $16,146

Continuously compounded loan balance: $16,168

 
 

Function Domain and Range problems

Finding Domain of a Function example problem

Find the domain of the following functions and use number line to show finite solution:

 

 

 

 

Solution to this Function Domain practice problem is given in the video below!



Finding Range of a Function example question

Find the range of the following functions and use the xy-plane to draw the solution set:

 

 

Solution to this Function Range practice problem is given in the video below!



Find SQUARE ROOT Function Given Domain, Range, and Rate of Change example

Find the Square Root Function given the following information:

Increasing:

Range:

Rate of Change over interval is 4.

 

Solution to this Square Root Function practice problem is given in the video below!



Equation from Table of Ordered Pairs problems

Linear Quadratic Exponential Equation Comparison from Table of Ordered Pairs example question

Determine whether the given set of ordered pairs (x, y) below follows a Linear, Quadratic or Exponential Equation such as

Then, find the missing constants in that Equation.

(x, y)

(3, 81)

(4, 144)

(5, 225)

(6, 324)

(7, 441)

 

Solution to this Table of Values Linear Quadratic Exponential Equation practice problem is given in the video below!



Linear Quadratic Exponential Equation Comparison from Table of Ordered Pairs example problem

Given the set of ordered pairs (x, y) below determine whether the Graph shows a Linear, Quadratic or Exponential Equation such as 

Then, find the missing constants in that Equation.

(x, y)

(1, -3)

(2, -9)

(3, -27)

(5, -243)

 

Solution to this Table of Values Linear Quadratic Exponential Equation practice problem is given in the video below!



Tree Diagram Probability problems

Tree Diagram Spinner example problem

A fair spinner showing numbers 1 through 3 of equal area is spun twice. Find the probabilities of the following events:

a) The sum of numbers landed on is odd

b) The sum of numbers landed on is even

c) The two numbers are different

d) Given even first number, the second number is even

e) Given odd first number, the second number is odd

f) The product of the two numbers landed on is less than 6

 

Solution to this Calculus Tree Diagram Probability practice problem is given in the video below!



Tree Diagram Conditional Probability example question #2

For the given Tree Diagram, calculate the P(A | Bc).

 

Solution to this Calculus Tree Diagram Probability practice problem is given in the video below!



Tree Diagram Marbles in a Jar probability example

Suppose we have 3 jars: A, B and C. Jar A contains 2 red marbles and 1 blue marble. Jar B contains 1 red marble and 2 blue marbles. Jar C contains 2 red marbles and 2 blue marbles. We select a jar at random. The probability we select A is . The probability we select B is . The probability we select C is . From the selected jar, we choose 1 marble; all marbles in the jar have the same chance of being selected.

a) What is the probability the chosen marble is red?

b) Given that the chosen marble is red, what is the probability that the selected jar was A?

 

Solution to this Calculus Tree Diagram Probability practice problem is given in the video below!



Tree Diagram HARD Steroid Drug probability example problem

In recent years, the “Tour de France” cycling race has been plagued with accusations that cyclists have taken steroids to boost their performance. Testing for these steroids has been difficult. The level of steroids that naturally occur in the human body can vary between individuals and the levels can change throughout the day. Testing for the presence of steroids is expensive and time-consuming. Racers are randomly chosen for drug tests. Suppose that 2% of racers have taken an illegal steroid. Assume that if they took the drug, there is a 99% chance that the test will return a “positive” reading and the athlete will be disqualified.

a) Draw a Tree Diagram showing the probabilities for the sequence of events in which an athlete either takes or does not take a drug, and then is tested for the presence of the drug.

b) What is the probability that a randomly chosen cyclist will be steroid-free and will still test positive for steroids?

c) What is the probability that a randomly chosen cyclist will be disqualified due to a “positive” reading?

 

Solution to this Calculus Tree Diagram Probability practice problem is given in the video below!



Differentials problems

Total Differential example word problem

Use Differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick.

 

Solution to this Calculus Differentials practice problem is given in the video below!



Tangent Plane problems

Equation of the Plane Tangent to Surface example problem

Find an equation of the tangent plane to the given surface at the specified point:

Surface z =

Point (1, -1, 1)

 

Solution to this Calculus Tangent Plane practice problem is given in the video below!



Business Calculus Optimization problems

Demand Revenue Optimization example problem

A store has been selling 200 DVD burners a week at $350 each. A market survey indicates that for each $10 rebate offered to buyers, the number of units sold will increase by 20 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue?

 

Solution to this Business Calculus Optimization practice problem is given in the video below!



Algebra Work word problems

Printing Job word problem example

Jack can complete a printing job alone in 12 hours. With Jack’s help, Susan can complete this job in 8 hours. How long would it take her to complete this job alone?

 

Solution to this Algebra Work word practice problem is given in the video below!



 

Binomial Distribution Normal Approximation problems

Normal Approximation for Binomial Random Variable example problem

The median age of residents of the United States is 37.2 years. If a survey of 200 residents is taken, approximate the probability that at least 110 will be under 37.2 years of age.

 

Solution to this probability Binomial Distribution Normal Approximation practice problem is given in the video below!



Binomial Random Variable Normal Approximation HARD example question

The weekly amount sent by a small company for in-state travel has approximately a normal distribution with mean $1450 and standard deviation $220.

(a) What is the probability that the actual expenses will exceed $1560 in 20 or more weeks during the next year?

(b) What is the probability that the actual expenses would exceed $1500 for between 18 and 24 weeks, inclusive during the next year?

 

Solution to this probability Binomial Distribution Normal Approximation problem is given in the video below!



Double Improper Integral problems

DOUBLE Improper Integral example problem

Investigate convergence or divergence of the Double Improper Integral provided below.

 

Solution to this Calculus Double Improper Integrals practice problem is given in the video below!



Arithmetic & Geometric Sequence problems

Nth Term of a Sequence example problem

Find a formula for the Nth Term of a Sequence whose first four terms are given by

(a)  4, 7, 10, 13, …

(b)  1, -3, 5, -7, …

(c) 

 

(d)

 

Solution to these Arithmetic and Geometric Sequences practice problems is provided in the video below!



 

Sequence Terms and Summation Notation example question

Write the following Sum of the Terms of a Sequence in Summation Notation:

(a) 

(b) 

(c) 

 

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!



 

Testing For Arithmetic and Geometric Sequences example

Determine if the following Sequences are Arithmetic, geometric or Neither:

(a) 

 

(b) 

 

(c) 

 

(d) 

 

(e) 

 

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!



 

Arithmetic Sequence Common Difference Nth term example problem

For the following Arithmetic sequences, state the common difference, and write the next three terms and the Nth term:

(a) 

(b)  -8, -5, …

(c) 

 

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!



 

Geometric Sequence Common Ratio Nth term example question

For the following Geometric sequences, state the common ratio, and write the next three terms and the Nth term:

(a) 

(b)  -5, 5, -5, …

(c)  1, 1.05, …

 

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!



 

Infinite Geometric Sequence SUM example

For the following Geometric sequences, state the common ratio and find the Sum of ALL Terms (Infinite Sum), or state that the sum is undefined:

(a)  4,

 

(b) 

(c)  36, -12, 4, …

(d)  1, 0.95, …

 

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!



 

Geometric Sequence application word problem

Suppose that $0.01 were deposited into a bank account on the first day of June, $0.02 on the second day, $0.04 on the third day, and so on in a geometric sequence.

(a) How much money would be deposited at this rate on June 30th?

(b) How much money would be in the account after this last deposit?

 

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!



 

Limits problems for Precalculus

Find Limit using Table of Values problem example 

The following video provides two examples with tables containing x-values and their respective function values. Use this information to find the required limits.

 

Solution to this Precalculus Limit practice problem is provided in the video below!



 

Find Limit using Graph example question

The following video provides a complete xy-plane Graph showing the behavior of a given function. Use this information to find the limits.

 

Solution to this Precalculus Limit problem is provided in the video below!



 

Find Limit using Calculator example

Use a calculator to help you find the indicated limits for the given functions:

(a)   

 

 

 

 

(b) 

 

 

 

 

Solution to this Precalculus Limit problem is provided in the video below!



 

Find Limit using Piecewise Functions example problems

Use the following piecewise functions to find the required limits:

(a)

 

 

 

 

(b)   

 

 

 

 

Solution to these Precalculus Limit problems is provided in the video below!



 

Example problem on why some Limits may NOT exist

Investigate the following function and determine why the given Limit does not exist:

 

Solution to this Precalculus Limit problem is provided in the video below!



 

Find Limit by Rationalizing the Square Root example question

Find the indicated Limit of the given Function:

 

Solution to this Precalculus Limit problem is provided in the video below!



 

Surface Area Integration problems

Surface Area of a Solid Integration problem example

Find the Surface Area of a solid generated by revolving the curve

, from    to   , about the x-axis.

 

Solution to this calculus Surface Area of a Solid Integration practice problem is provided in the video below!



 

Arc Length of a Curve Integration problems

Arc Length of a Curve Integration problem example

Find the Arc Length of a curve     , from    to 

 

Solution to this calculus Arc Length of a Curve Integration practice problem is provided in the video below!



 

Area Between Curves Integration problems

Area Between Curves Integration problem example

Find the area bounded by two curves     and 

 

Solution to this calculus Area Between Curves Integration practice problem is provided in the video below!



 

Projectile Motion calculus problems

Projectile Motion In Two Dimensions problem example

An object is launched at angle    from the horizontal with initial speed Vo = 98 meters per second. Determine the time of flight and the horizontal range of the projectile.

 

Solution to this Calculus Projectile Motion practice problem is provided in the video below!



 

Projectile Motion “Baseball Pitcher” example question

A baseball pitcher releases the ball horizontally from a height of 6 feet with an initial velocity of 130 feet per second. Find the height of the ball when it reaches home plate 60 feet away.

 

Solution to this Calculus Projectile Motion practice problem is provided in the video below!



 

Theoretical Projectile Motion example

Show that an object dropped from a height of H feet will hit the ground at time    seconds with impact velocity    feet per second.

 

Solution to this Calculus Projectile Motion practice problem is provided in the video below!



 

Disks Washers Cylindrical Shells Volume of the Solid Integration problems

Volume of the Solid by Revolution DISK Integration Method example

Revolve the region under a curve

on the interval [0, 4] about the x-axis and find the volume of the resulting solid.

 

Solution to this Volume of the Solid by Rotation practice problem is provided in the video below!



Volume of the Solid by Rotation DISK Integration Method example problem

Find the volume of the solid resulting from revolving the region bounded by the lines

 and   

from   to 

about the y-axis.

 

Solution to this Volume of the Solid by Revolution practice problem is provided in the video below!



Volume of the Solid by Revolution WASHER Integration Method example question

Find the volume of the solid resulting from revolving the region bounded by lines

,  and  ,

about the line

 

Solution to this Volume of the Solid by Rotation practice problem is provided in the video below!



Volume of the Solid by Rotation WASHER Integration Method example #2

Find the volume of the solid resulting from revolving the region bounded by lines

,  and  ,

about the x-axis.

 

Solution to this Volume of the Solid by Revolution practice problem is provided in the video below!



Volume of the Solid by Rotation CYLINDRICAL SHELL Integration Method example problem

Revolve the region bounded by the graphs of    and    in the first quadrant about the y-axis using Cylindrical Shells and find the volume of the resulting solid.

 

Solution to this Volume of the Solid by Revolution practice problem is provided in the video below!



Volume of the Solid by Revolution WASHER & SHELL Integration Method example question

Find the volume, using Washer and Cylindrical Shell methods, of the solid formed by revolving the region bounded by curves

  and 

about the line

 

Solution to this Volume of the Solid by Rotation practice problem is provided in the video below!



The Principle of Mathematical Induction problems

The Principle of Mathematical Induction Equality problem example 

Prove the following statement by Mathematical Induction:

2 + 4 + 6 + … + 2n = n(n + 1)

 

Solution to this Mathematical Induction Proof practice problem is provided in the video below!



 

The Principle of Mathematical Induction Equality Proof example question

Prove the following sequence statement by Mathematical Induction:

3 + 7 + 11 + … + (4n – 1) = n(2n + 1)

 

Solution to this Mathematical Induction Proof practice problem is provided in the video below!



 

Exponents & Principle of Mathematical Induction Equality example

Prove the given statement by using Mathematical Induction:

1 + 3 + 32 + … + 3n – 1 =

 

Solution to this Mathematical Induction Proof practice problem is provided in the video below!



 

Exponents & Principle of Mathematical Induction Equality example problem #2

Prove the statement below by Mathematical Induction:

13 + 23 + 33 + … + n3 =

 

Solution to this Mathematical Induction Proof practice problem is provided in the video below!



 

Fractions & Principle of Mathematical Induction Equality example question

Show the given sequence statement holds true by using Mathematical Induction:

 

Solution to this Mathematical Induction Proof practice problem is provided in the video below!



 

Nth Term of Geometric Sequence Formula Proof with Principle of Mathematical Induction

Prove the following formula that finds the Nth term of a Geometric Sequence by using the Principle of

Mathematical Induction:

 

Solution to this Principle of Mathematical Induction Proof practice problem is provided in the video below!



 

EXTENDED Principle of Mathematical Induction Inequality example

Prove the following Square Root sequence inequality statement by using the Extended Principle of

Mathematical Induction:

 

Solution to this Extended Principle of Mathematical Induction Proof practice problem is provided in the video below!



 

Rotation of Axes problems

Angle of Rotation of Axes problem example 

Find an appropriate angle through which to rotate axes to eliminate the xy term in the equation

 

Solution to this Rotation of Axes practice problem is provided in the video below!



 

COORDINATE Equations of Rotation of Axes example question

Find the Transformation Equations to rotate axes through an appropriate angle to eliminate the xy term in the equation

 

Solution to this Rotation of Axes practice problem is provided in the video below!



 

New Axes of Rotation EQUATION example

Find the Equation into which

is transformed when there is rotation of the axes through an appropriate angle to eliminate the xy term in the given equation.

 

Solution to this Rotation of Axes practice problem is provided in the video below!



 

Hyperbola Equation problems

Equation of a Hyperbola problem example

Analyze and sketch Graph of the Hyperbola:

x2y2 + 6x + 34 = 0

 

Solution to this Equation of Hyperbola practice problem is provided in the video below!



 

Equation of Hyperbola ECCENTRICITY example question #2

Find the Eccentricity of the Hyperbola:

 

Solution to this Eccentricity of a Hyperbola practice problem is provided in the video below!



 

Equation of a Hyperbola example #3

Find the equation of a Hyperbola

(a) with vertices (0, ±12) and asymptotes y = ±3x

(b) with foci (3, 6) and (11, 6) and eccentricity

 

Solution to this Equation of Hyperbola practice problem is provided in the video below!



 

Ellipse Equation problems

Equation of an Ellipse problem example

Analyze and Sketch the graph of the Ellipse given by

25x2 + 16y2 + 100x – 96y = 156

 

Solution to this Equation of Ellipse practice problem is provided in the video below!



 

Eccentricity of Ellipse example question #2

Find the Eccentricity for

 

Solution to this Equation of an Ellipse practice problem is provided in the video below!



 

Equation of an Ellipse with Vertices and Eccentricity example #3

Find the equation of an Ellipse with

(a) major vertices (±4, 0) and eccentricity

(b) minor vertices (-3, 4) and (1, 4) and eccentricity

 

Solution to this Equation of Ellipse practice problem is provided in the video below!



 

Equation of Ellipse example problem #4

Use the definition of an ellipse PF1 + PF2 = 2a directly to find the equation of an ellipse with foci (0, 0) and (4, 0 ) and major axis 2a = 6.

 

Solution to this Equation of an Ellipse practice problem is provided in the video below!



 

Equation of Parabola problems

Equation of Parabola problem example

Find Equations for Parabolas in standard position

(a) with Focus at (0, 7) and Directrix the line y = -7

(b) with Focus at (, 0) and Directrix the line x =

 

Solution to this Equation of Parabola practice problem is provided in the video below!



 

Equation of Parabola example question #2

Find the Equation for Parabola in standard orientation with Focus at (-2, 3) and Directrix the line y = 1.

 

Solution to this Equation of Parabola practice problem is provided in the video below!



 

Equation of a Parabola example #3

Use the Definition of the Parabola directly to find the equation of a parabola with Focus F(2, 2) and Directrix the line x + y + 2 = 0.

 

Solution to this Equation of Parabola practice problem is provided in the video below!



 

Locus of Points problems

Loci of Points problem example

Find the locus of points P(x, y) such that the distance from P to the y-axis is 5.

 

Solution to this Locus of Points practice problem is provided in the video below!



 

Locus of Points example question #2

Find the locus of points P(x, y) such that the distance of P from P1(1, 1) is one-half the distance of P from P2(-2, -2).

 

Solution to this Loci of Points practice problem is provided in the video below!



 

Loci of Points example #3

Find the locus of points P(x, y) such that P is equidistant from both axes.

 

Solution to this Locus of Points practice problem is provided in the video below!



 

Locus of Points example problem #4

Find the locus of points P(x, y) such that P is equidistant from (5, -1) and (3, -8).

 

Solution to this Loci of Points practice problem is provided in the video below!



 

Loci of Points example question #5

Find the locus of points P(x, y) such that P is equidistant from (-5, 3) and xy + 8 = 0.

 

Solution to this hard Locus of Points practice problem is provided in the video below!



 

Hard Locus of Points example

Find the locus of points P(x, y) such that the product of their distances from (0, 4) and (0, -4) is 16.

 

Solution to this Loci of Points practice problem is provided in the video below!



 

System of NonLinear Equations problems

System of NonLinear Equations problem example

Find Real and Imaginary solutions, whichever exist, to the Systems of NonLinear Equations:

a)

 

b)   

 

Solution to these Systems of NonLinear Equations practice problems is provided in the video below!



 

HARD System of NonLinear Equations example question

Determine Real and Imaginary solutions, whichever exist, to the given System of NonLinear Equations:

 

Solution to this Hard System of NonLinear Equations practice problem is provided in the video below!



 

VERY HARD System of NonLinear Equations example

Solve the System of NonLinear Equations provided below:

 

Solution to this Very Hard System of NonLinear Equations practice problem is provided in the video below!



 

HARDEST System of NonLinear Equations example problem

Find the solutions to the System of NonLinear Equations provided below:

 

Solution to this Most Difficult System of NonLinear Equations practice problem is provided in the video below!



 

NonLinear System of Equations Word Problem

A rectangle of perimeter 100 meters is to be constructed to have area 100 square meters. What dimensions are required?

 

Solution to this NonLinear System of Equations word practice problem is provided in the video below!



 

Partial Fraction Decomposition problems

Linear Factors Partial Fraction Decomposition problem example

Find the Partial Fraction Decomposition for the following Fraction:

 

Solution to this Linear Factors Partial Fraction Decomposition practice problem is provided in the video below!



 

Linear Factors Partial Fraction Decomposition ALTERNATE Method example question

Alternate method takes into account the x-values in the equation that one can test to find the constants immediately instead of comparing coefficients.

Find the Partial Fraction Decomposition for the same example as above by using the Alternate Method:

 

Solution to this Alternate Method Linear Factors Partial Fraction Decomposition practice problem is provided in the video below!



 

QUADRATIC Factors Partial Fraction Decomposition example

Find the Partial Fraction Decomposition for the Fraction provided below:

 

Solution to this Quadratic Factors Partial Fraction Decomposition practice problem is provided in the video below!



 

HARD QUADRATIC Factors Partial Fraction Decomposition example problem

Determine the Partial Fraction Decomposition for the Fraction provided below:

 

Solution to this Difficult Quadratic Factors Partial Fraction Decomposition practice problem is provided in the video below!



 

Multiple Integral problems (complete Playlist)

The following playlist shows a variety of Calculus examples and practice problems dealing with Double Integration, Changing Order of IntegrationTriple Integration, Cylindrical Coordinates, Spherical Coordinates, and more!


Trigonometric Form of Complex Numbers problems

Complex Number Trigonometric Form Representation problem example

Express the following Complex Numbers in Trigonometric Form:

-12

-8i

2 – 2i

 

Solution to this Complex Number Trigonometric Form Representation practice problem is provided in the video below!



 

Trigonometric Complex Number Product Quotient Power Rectangular Standard Form Representation example question

Find the resulting Complex Number in standard (rectangular) form for the following:

 

 

Solution to this Trigonometric Complex Number Product Quotient Power Rectangular Standard Form Representation practice problem is provided in the video below!



 

Square Roots of a Complex Number example

Find the two Square Roots of

 

 

Solution to this Square Roots of a Complex Number practice problem is provided in the video below!



 

Complex Number Rectangular POLAR Form Conversion example problem

Express the following POLAR Complex Number in Trigonometric and Rectangular Form: 

 

Express the following Rectangular Complex Number in Trigonometric Form and POLAR Form:

5i

 

Solution to this Complex Number Polar Rectangular Form Conversion practice problem is provided in the video below!



 

Complex Number POLAR Product Quotient Power example question

Find the resulting Complex Number in POLAR form for the following:

a) 

 

b) 

 

Find the resulting Complex Number in POLAR and RECTANGULAR (standard) form:

 

Solution to this Complex Number Polar Product Quotient Power practice problem is provided in the video below!



 

POLAR Complex Number Nth Root Theorem example

Find the Four 4th (Fourth) Roots of

 

Solution to this Polar Complex Number Nth Root Theorem practice problem is provided in the video below!



 

Vector word problems

Horizontal and Vertical Components of a Force Vector problem example

A force of 46.3 pounds is applied at an angle of 34.8º to the horizontal. Resolve the force into horizontal and vertical components.

 

Solution to this Horizontal Vertical Components of a Force Vector word practice problem is provided in the video below!



 

HARD Parallel Perpendicular Weight Vector Components to Surface word problem

A weight of 75 pounds is resting on a surface inclined at an angle of 25º to the ground. Find the components of the weight parallel and perpendicular to the surface.

 

Solution to this Parallel Perpendicular Weight Vector Components word practice problem is provided in the video below!



 

VERY HARD Resultant Force Vector example question

Find the resultant of two forces, one with magnitude 155 pounds and direction N50ºW, and a second with magnitude 305 pounds and direction S55ºW.

 

Solution to this Resultant Force Vector word practice problem is provided in the video below!



 

Plane Wind vector example problem

An airplane has an airspeed of 430 miles per hour at a bearing of E45ºS (45 degrees South of East). The wind velocity is 35 miles per hour in the direction of N30ºE (30 degrees East of North). Find the ground speed and true course of the plane using vectors.

 

Solution to this Plane Wind vector word practice problem is provided in the video below!



 

Law of Cosines problems

Law of Cosines word problem example

Two sides of a parallelogram are 9 and 15 units in length. The length of the shorter diagonal of the parallelogram is 14 units. Find the length of the long diagonal.

 

Solution to this Law of Cosines word practice problem is provided in the video below!



 

HARD Law of Cosines word problem

A new car leaves an auto transport trailer for a test drive in the flat surface desert in the direction N47ºW at constant speed of 65 miles per hour. The trailer proceeds at constant rate of 50 miles per hour due East. If the car has enough fuel for exactly 3 hours of riding at constant speed, what is the maximum distance in the same direction that the car can cover in order to safely return to the trailer?

 

Solution to this Law of Cosines word practice problem is provided in the video below!



 

Law of Sines problems

Law of Sines word problem example

A radio antenna is attached to the top of the building. From a point 12.5 meters from the base of the building, on level ground, the angle of elevation of the bottom of the antenna is 47.2 degrees and the angle of elevation of the top is 51.8 degrees. Find the height of the antenna.

 

Solution to this Law of Sines word practice problem is provided in the video below!



 

Law of Sines Triangle Area PROOF problem

Show that for any triangle the area is one-half the product of any two sides and sine of the angle formed between these sides. That is, 

 

Solution to this Law of Sines Triangle Area PROOF practice problem is provided in the video below!



 




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