Deck 3: Differentiation Rules

Question 1

Find all points at which the tangent line is horizontal on the graph of the function.

Answer 1

Question 2

Calculate

Answer 2

Question 3

Differentiate the function.

Answer 3

Question 4

Find the derivative function.

Question 5

Find

Question 6

The position function of a particle is given by When does the particle reach a velocity of

Question 7

by implicit differentiation.

Question 8

A 18-ft ladder leaning against a wall begins to sl ide.How fast is the angle between the ladder and the wall changing at the instant of time when the bottom of the ladder is 9 ft from the wall and sliding away from the wall at the rate of 4 ft/sec? Round theAnswer to the nearest hundredth.

Question 9

Find the first and the second derivatives of the function.

Question 10

Find an equation of the tangent line to the curve.

Question 11

Differentiate the function.

Question 12

Two chemicals react to form another chemical.Suppose that the amount of chemical formed in time (in hours) is given by where is measured in pounds.
a.Find the rate at which the chemical is formed when Round to two decimal places.
b.How many pounds of the chemical are formed eventually?

Question 13

Evaluate

Question 14

A circle's radius is increasing.Find the rate of change of the area of the circle with respect to the radius when

Question 15

Let a.Find the derivative of b.Find the point on the graph of where the tangent line to the curve is horizontal.
c.Sketch the graph of nd the tangent line to the curve at the point found in part (b).
d.What is the rate of change of at this point?

Question 16

Find the rate of change of with respect to x at the given values of and

Question 17

A baseball diamond is a square with side 90 ft.A ba tter hits the ball and runs toward first base with a speed of At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth.

Question 18

If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then Torricelli's Law gives the volume of water remaining in the tank after minutes as Find the rate at which water is draining from the tank after minutes.

Question 19

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves.Show that the curves of the given equations are orthogonal.

Question 20

is the position of a body moving along a coordinate line; is measured in feet and in seconds, where Find the position, velocity, and speed of the body at the indicated time. t = 1

Question 21

Differentiate the function.

Question 22

Refer to the law of laminar flow.Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference and viscosity Find the velocity of the blood at radius

Question 23

Find the derivative of the function.

Question 24

Find, correct to three decimal places the area of the region above the hyperbola below the axis, and between the lines

Question 25

Differentiate.

Question 26

Calculate

Question 27

Evaluate the integral.

Question 28

Two chemicals react to form another chemical.Suppose that the amount of chemical formed in time (in hours) is given by where is measured in pounds.
a.Find the rate at which the chemical is formed when Round to two decimal places.
b.How many pounds of the chemical are formed eventually?

Question 29

Calculate

Question 30

The volume of a right circular cone of radius and height is Suppose that the radius and height of the cone are changing with respect to time a.Find a relationship between b.At a certain instant of time, the radius and height of the cone are 12 in.and 13 in.and are increasing at the rate of 0.2 in./sec and 0.5 in./sec, respectively.How fast is the volume of the cone increasing?

Question 31

Find

Question 32

Use logarithmic differentiation to find the derivative of the function.

Question 33

The volume of a cube is increasing at a rate of How fast is the surface area increasing when the length of an edge is

Question 34

Differentiate.

Question 35

in terms of

Question 36

Differentiate.

Question 37

Differentiate.

Question 38

If and find

Question 39

Find the equation of the tangent line to the given curve at the specified point.

Question 40

Find the derivative of the function.

Question 41

Select the correct Answer: for each question.
Find the derivative of the function.

A)
B)
C)
D)

Question 42

Select the correct Answer: for each question.
Find the derivative of the function.

A)
B)
C)
D)
E)

Question 43

Select the correct Answer: for each question.
Find by implicit differentiation.

A)
B)
C)
D)

Question 44

Select the correct Answer: for each question.
Water flows from a tank of constant cross-sectional area 50 through an orifice of constant cross-sectional area located at the bottom of the tank.Initially, the height of the water in the tank was 20 ft, and t sec later it was given by the equation How fast was the height of the water decreasing when its height was 2 ft?

A)
B)
C)
D)

Question 45

Select the correct Answer: for each question.
Find an equation of the tangent line to the given curve at the indicated point.

A)
B)
C)
D)

Question 46

Select the correct Answer: for each question.
Differentiate the function.

A)
B)
C)
D)

Question 47

Select the correct Answer: for each question.
Find the second derivative of the function.

A)
B)
C)
D)

Question 48

Select the correct Answer: for each question.
Use the definitions of the hyperbolic functions to find the following limit.

A)
B)
C)
D)
E)None of these

Question 49

Select the correct Answer: for each question.
Find the tangent line to the ellipse at the point

A)
B)
C)
D)
E) None of these

Question 50

Select the correct Answer: for each question.
Compute and dy for the given values of and

A)
B)
C)
D)
E)

Question 51

Select the correct Answer: for each question.
Find in terms of

A)
B)
C)
D)
E)

Question 52

Select the correct Answer: for each question.
Find the derivative of the function.

A)
B)
C)
D)

Question 53

Select the correct Answer: for each question.
Find an equation of the line tangent to the graph of at the point where

A)
B)
C)
D)

Question 54

Select the correct Answer: for each question.
Find an equation of the tangent line to the curve at the point (4,1).

A)
B)
C)
D)
E)

Question 55

Select the correct Answer: for each question.
Find the derivative of the function.

A)
B)
C)
D)

Question 56

Select the correct Answer: for each question.
Find the given derivative by finding the first few derivatives and observing the pattern that occurs.

A)
B)
C)
D)
E)None of these

Question 57

Select the correct Answer: for each question.
Use logarithmic differentiation to find the derivative of the function.

A)
B)
C)
D)
E)

Question 58

Select the correct Answer: for each question.
Find the instantaneous rate of change of the function when

A)
B)3
C)9
D)

Question 59

Select the correct Answer: for each question.
Suppose the daily total cost (in dollars) of manufacturing televisions is What is the marginal cost when What is the actual cost incurred in manufacturing the 201st television?

A)
B)
C)
D)

Question 60

Select the correct Answer: for each question.
Use logarithmic differentiation to find the derivative of the function.

A)
B)
C)
D)
E)

Question 61

Select the correct Answer for each question.
Find the derivative of the function.

A)
B)
C)
D)

Question 62

Select the correct Answer for each question.
Find in terms of

A)
B)
C)
D)
E)

Question 63

Select the correct Answer for each question.
Find

A)
B)
C)
D)
E)

Question 64

Select the correct Answer for each question.
A plane flying horizontally at an altitude of 1 mi and a speed of passes directly over a radar station.Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

A)
B)
C)
D)
E)

Question 65

Select the correct Answer for each question.
If

A)
B)
C)
D)
E)

Question 66

Select the correct Answer for each question.
Use the definition of the derivative to find the derivative of the function.

A)
B)
C)
D)

Question 67

Select the correct Answer for each question.
An equation relating the variables and the values of x and y, and the value of at a particular instant of time are given.Find the value of

A)2
B)-2
C)4
D)-4

Question 68

Select the correct Answer for each question.
If

A)
B)
C)
D)
E)
F)-12

Question 69

Select the correct Answer for each question.
If

A)
B)
C)
D)
E)

Question 70

Select the correct Answer for each question.
Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point.

A)y = 1
B)y = x
C)
D)y = 2x + 1

Question 71

Select the correct Answer for each question.
Differentiate:

A)
B)
C)
D)
E)

Question 72

Select the correct Answer for each question.
The equation of motion is given for a particle, where is in meters and is in seconds.Find the acceleration after seconds.

A)
B)
C)
D)
E)

Question 73

Select the correct Answer for each question.
Determine the values of for which the given linear approximation is accurate to within 0.07 at

A)
B)
C)
D)
E)

Question 74

Select the correct Answer for each question.
Find equations of the tangent lines to the curve that are parallel to the line

A)
B)
C)
D)
E)

Question 75

Select the correct Answer for each question.
Find the derivative of the function.

A)
B)
C)
D)

Question 76

Select the correct Answer for each question.
Suppose that Find

A)24
B)17
C)
D)140
E)20

Question 77

Select the correct Answer for each question.
Find the given integral.

A)
B)
C)
D)

Question 78

Select the correct Answer for each question.
Differentiate:

A)
B)
C)
D)
E)

Question 79

Select the correct Answer for each question.
A curve passes through the point and has the property that the slope of the curve at every point is twice the of What is the equation of the curve?

A)
B)
C)
D)
E)

Question 80

Select the correct Answer for each question.
Calculate

A)
B)
C)
D)
E)