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# Category Archives: Application of Differentiation

## 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*) =

*e*+

^{t}*te*. What is the average velocity of the particle from time

^{t}*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(0.9

*t*). 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!

## 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!

## Optimization calculus problems

**Inscribed Rectangle Optimization example question**

Find the area of the largest rectangle that can be inscribed in a circle of radius 4.

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

**Rain Gutter ****Optimization ****example problem**

A rain gutter is to be constructed from a metal sheet of width 30 centimeters by bending up one-third of the sheet on each side through an angle *θ*. How should angle *θ* be chosen so that the gutter will carry the maximum amount of water?

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

**Distance ****Optimization ****example**

Find the point on the curve *y* = cos(*x*) closest to the point (0,0)

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

**Running Track ****Optimization ****example question**

A running track is to be built around a rectangular field, with two straightaways and two semicircular curves at the ends, as indicated in the figure. The length of the track is to be 400 meters. Find the dimensions that will maximize the area of the enclosed rectangle.

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

**Ticket Cost ****Optimization ****example problem**

Suppose that group tickets to a concert are priced at $40 per ticket if 20 tickets are ordered, up to a maximum of 50 tickets. (For example, if 22 tickets are ordered, the price is $38 per ticket). Find the number of tickets that maximizes the total cost of the tickets.

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

**Rectangular Ad ****Optimization ****example**

An advertisement consists of a rectangular printed region plus 1-inch margins on the sides and 2-inch margins at the top and bottom. If the area of the printed region is to be 92 square inches, find the dimensions of the printed region and overall advertisement that minimize the total area.

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

**Soda Can ****Optimization ****example question**

A soda can is to hold 12 fluid ounces. Find the dimensions that will minimize the amount of material used in its construction, assuming that the thickness of the material is uniform.

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

**Graph Optimization example problem**

Suppose that

For each *x* in the interval (0,1), consider the rectangle formed in the following manner:

a) Its right side is the line segment connecting the points **(***x*, *f*(*x*)**)** and **(***x*, *g*(*x*)**) **

b) Its left side lies along the *y*-axis

Which value of *x* in the interval (0,1) results in the rectangle of largest area?

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

**Optimization ****Proof example**

Show that the Rectangle of Maximum Area for a given Perimeter is always a Square.

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

**Optimization ****Proof example question**

Show that the Rectangle of Minimum Perimeter for a given Area is always a Square.

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

## Equations of Lines Tangent to Curve & Passing Point Implicit Differentiation problems

**Implicit Differentiation HARD example question**

Find **Equations of Lines** passing through the point (12, 3) and** Tangent** to the Curve

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

## Equation of the Tangent Line Implicit Differentiation problems

**Implicit Differentiation Equation of the Tangent Line example question**

Find the **Equation of the Tangent Line** to the Curve

at the point (-1, -2)

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

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