Deck 3: Differentiation

Question 1

An isosceles triangle is inscribed in a semicircle,as shown in the diagram,and it continues to be inscribed as the semicircle changes size.The area of the semicircle is increasing at the rate of 1 cm²/sec when the radius of the semicircle is 6 cm.
​
​How fast is the radius of the semicircle increasing when the radius is 6 cm? Include units in your answer.​ Round your answer to 3 decimal places.

Answer 1

0.053 cm/sec

Question 2

​What does the limit statement represent? ​

A)​0
B)​
C)​ ,if

D)​6

Answer 2

​ ,if

Question 3

​If ,find in terms of and . ​

A)​
B)​
C)​
D)​

Answer 3

​

Question 4

​Find the second derivative of if . ​

A)​
B)​
C)​
D)​

Question 5

​For what values of and is the piecewise function differentiable? ​

A)​ and

B)​ and

C)​ and​

D)​ and

Question 6

​Find for . ​

A)​
B)​
C)​
D)​

Question 7

​How fast is the perimeter of the semicircle increasing when the radius is 6 cm? Include units in your answer.Round your answer to 3 decimal places.

Question 8

Complete two iterations of Newton's Method for the function using the initial guess .Round all numerical values in your answer to four decimal places. ​

A)​

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1





2





​
​
​
​
B)​






1





2





​
​
C)​






1





2





​
​
D)​






1





2





​
​
E)​






1





2





​

Question 9

​How fast is the area of the isosceles triangle increasing when the radius is 6 cm? Include units in your answer.Round your answer to 3 decimal places.

Question 10

​In right triangle ,point is moving along a leg of the right triangle toward point at a rate of cm/sec and point is moving toward point at a rate of cm/sec along a line containing the other leg of the right triangle,as illustrated in the triangle shown below.What is the rate of change in the area of ,with respect to time,at the instant when cm and cm? ​ ​

A)-4.48 cm²/sec
B)-3.36 cm²/sec
C)-3.2 cm²/sec
D)​​-4 cm²/sec

Question 11

​How fast is the shaded region increasing when the radius is 6 cm? Include units in your answer.Round your answer to 3 decimal places.

Question 12

​If the nth derivative of is denoted as and ,then is the same as ​

A)​
B)​
C)​
D)​

Question 13

​Find for . ​

A)​
B)​
C)​
D)​

Question 14

​Find the derivative of the function . ​

A)​
B)​
C)​
D)​

Question 15

​When the height of a cylinder is 18 cm and the radius is 6 cm,the circumference of the cylinder is increasing at a rate of cm/min,and the height of the cylinder is increasing four times faster than the radius.How fast is the volume of the cylinder changing? ​

A)​ cm³/min
B)​ cm³/min
C)​ cm³/min
D)​ cm³/min

Question 16

Complete two iterations of Newton's Method for the function using the initial guess 0.8.Round your answers to four decimal places. ​

A)1.0251,1.0001
B)1.0251,1.0005
C)1.0251,1.0004
D)1.0248,1.0002
E)1.0250,1.0003

Question 17

​If ,then ​

A)​
B)​
C)​
D)​

Question 18

​If , and ,what is the equation of the tangent line to at ? ​

A)​
B)​
C)​
D)​

Question 19

​Find if . ​

A)​
B)​
C)​
D)​

Question 20

​For what positive value of does the tangent line to the curve intersect the -axis at the point ? ​

A)​4.076
B)3.375
C)​3.638
D)4.383

Question 21

Use Newton's Method to approximate the zero(s)of the function accurate to three decimal places. ​

A)0.486
B)-0.486
C)-0.528
D)0.517
E)0.433

Question 22

Find the rate of change of the distance D between the origin and a moving point on the graph of if centimeters per second. ​

A)
B)
C)
D)
E)

Question 23

Assume that x and y are both differentiable functions of t.Find when and for the equation . ​

A)
B)
C)
D)
E)

Question 24

A spherical balloon is inflated with gas at the rate of 900 cubic centimeters per minute.How fast is the radius of the balloon increasing at the instant the radius is 70 centimeters? ​

A) cm/min
B) cm/min
C) cm/min
D) cm/min
E) cm/min

Question 25

Apply Newton's Method to approximate the x-value of the indicated point of intersection of and .Continue the process until two successive approximations differ by less than 0.001.[Hint: Let .] Round your answer to three decimal places. ​
​ ​

A)5.494
B)8.725
C)6.494
D)7.725
E)4.641

Question 26

The radius r of a sphere is increasing at a rate of 9 inches per minute.Find the rate of change of the volume when r = 11 inches. ​

A) in³/min
B) in³/min
C) in³/min
D) in³/min
E) in³/min

Question 27

The radius,r,of a circle is increasing at a rate of 3 centimeters per minute.​Find the rate of change of area,A,when the radius is 6. ​

A)
B)
C)
D)
E)

Question 28

Approximate the fixed point of the function between and to two decimal places.[A fixed point of a function f is a value of x such that .] ​

A)0.65
B)0
C)0.74
D)1.00
E)0.91

Question 29

Use Newton's Method to approximate the x-value of the indicated point of intersection of the two graphs accurate to three decimal places.Continue the process until two successive approximations differ by less than 0.001.[Hint: Let .]

A)2.651
B)2.703
C)2.660
D)2.764
E)2.600

Question 30

All edges of a cube are expanding at a rate of 9 centimeters per second.How fast is the volume changing when each edge is 2 centimeters? ​

A)486
B)72
C)36
D)108
E)162

Question 31

A point is moving along the graph of the function such that centimeters per second.Find when x = 1. ​

A)
B)
C)
D)
E)

Question 32

Approximate the positive zero(s)of the function to three decimal places.Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. ​

A)-1.304
B)1.202
C)1.179
D)-1.202
E)1.253

Question 33

Suppose that the total number of arrests T (in thousands)for all males ages 14 to 27 in 2006 is approximated by the model where x is the age in years (see figure).Approximate the two ages to one decimal place that had total arrests of 300 thousand. ​ ​

A)16.1;22.9
B)17.4;21.3
C)15.1;25.9
D)16.1;25.9
E)15.1;21.3

Question 34

Use Newton's Method to approximate the zero(s)of the function accurate to three decimal places. ​

A)19.314
B)-19.346
C)19.251
D)-19.314
E)19.365

Question 35

A point is moving along the graph of the function such that centimeters per second.Find when . ​

A)
B)
C)
D)
E)

Question 36

A manufacturer of digital audio players estimates that the profit for selling a particular model is where P is the profit in dollars and x is the advertising expense in 10,000's of dollars (see figure).Find the smaller of two advertising amounts that yield a profit P of $1,900,000.Round your answer to the nearest dollar. ​ ​

A)$532,519
B)$291,308
C)$271,308
D)$460,892
E)$315,369

Question 37

Use Newton's Method to approximate the x-value of the indicated point of intersection of the two graphs accurate to three decimal places.Continue the process until two successive approximations differ by less than 0.001.[Hint: Let .] ​ ​ ​
​ ​

A)0.444
B)0.371
C)0.432
D)0.393
E)0.476

Question 38

Assume that x and y are both differentiable functions of t .Find when and for the equation . ​

A)
B)
C)
D)
E)

Question 39

Use Newton's Method to approximate the zero(s)of the function accurate to three decimal places. ​

A)-0.486
B)0.444
C)-0.539
D)0.486
E)-0.437

Question 40

A conical tank (with vertex down)is 20 feet across the top and 18 feet deep.If water is flowing into the tank at a rate of 20 cubic feet per minute,find the rate of change of the depth of the water when the water is 12 feet deep. ​

A)
B)
C)
D)
E)

Question 41

Find an equation of the tangent line to the graph of at the point . ​

A)
B)
C)
D)
E)

Question 42

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 43

A ladder a = 20 feet long is leaning against the wall of a house (see figure).The base of the ladder is pulled away from the wall at a rate of b = 4 feet per second.Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 19 feet from the wall.Round your answer to three decimal places. ​ ​​

A)0.528
B)0.641
C)4.058
D)7.604
E)0.339

Question 44

An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find when miles.Round your answer to three decimal places. ​

A)-17.647
B)2.941
C)17.647
D)0.044
E)-8.824

Question 45

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 46

An airplane is flying in still air with an airspeed of 266 miles per hour.If it is climbing at an angle of 22°,find the rate at which it is gaining altitude.Round your answer to four decimal places. ​

A)112.4165
B)86.6011
C)120.7615
D)100.7692
E)99.6454

Question 47

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 48

In a free-fall experiment,an object is dropped from a height of a = 256 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.At what time will the object reach the ground level? ​ ​

A)3 seconds
B)6 seconds
C)7 seconds
D)5 seconds
E)4 seconds

Question 49

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure).When he is 5 feet from the base of the light,at what rate is the length of his shadow changing? ​ ​

A)
B)
C)
D)
E)

Question 50

A ladder a = 20 feet long is leaning against the wall of a house (see figure).The base of the ladder is pulled away from the wall at a rate of b = 4 feet per second.Consider the triangle formed by the side of the house,the ladder,and the ground.Find the rate at which the area of the triangle is changed when the base of the ladder is 11 feet from the wall.Round your answer to two decimal places. ​ ​

A)1.48
B)33.41
C)153.81
D)18.92
E)53.68

Question 51

Find the slope-intercept form of the equation of the line tangent to the graph of when . ​

A)
B)
C)
D)
E)

Question 52

A ladder a = 20 feet long is leaning against the wall of a house (see figure).The base of the ladder is pulled away from the wall at a rate of b = 4 feet per second.How fast is the top of the ladder moving down the wall when its base is 13 feet from the wall? Round your answer to two decimal places. ​ ​

A)5.00
B)-6.86
C)-3.42
D)6.86
E)-6.00

Question 53

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 54

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure).When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving? ​ ​​

A)
B)50
C)
D)
E)

Question 55

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 56

An airplane flies at an altitude of ​h​ = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x. ​ ​

A)
B)
C)
D)
E)

Question 57

Find at the point for the equation . ​

A)
B)
C)
D)
E)

Question 58

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 59

Find at the point for the equation . ​

A)
B)
C)
D)
E)

Question 60

In a free-fall experiment,an object is dropped from a height of ​a = 144 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.Find the rate of change of the angle of elevation of the camera when t = 1.Round your answer to four decimal places. ​​ ​​

A)-0.6006
B)-0.0581
C)0.0601
D)0.0581
E)-0.0601

Question 61

Find by implicit differentiation. ​ ​

A)
B)
C)
D)
E)

Question 62

Evaluate for the equation at the given point .Round your answer to two decimal places if necessary. ​

A)
B)
C)
D)
E)

Question 63

Find by implicit differentiation given that . ​​

A)
B)
C)
D)
E)

Question 64

Find by implicit differentiation. ​ ​

A)
B)
C)
D)
E)

Question 65

Evaluate for the equation at the given point .Round your answer to two decimal places if necessary. ​

A)
B)
C)
D)
E)

Question 66

Find by implicit differentiation. ​ ​​

A)
B)
C)
D)
E)

Question 67

Find by implicit differentiation. ​ ​

A)
B)
C)
D)
E)

Question 68

Use implicit differentiation to find . ​ ​

A)
B)
C)
D)
E)

Question 69

Use implicit differentiation to find . ​ ​

A)
B)
C)
D)
E)

Question 70

Use implicit differentiation to find at the point . ​ ​

A)
B)
C)
D)
E)

Question 71

Find by implicit differentiation. ​ ​

A)
B)
C)
D)
E)

Question 72

Find the slope of the tangent line at the given point .Round your answer to two decimal places if necessary. ​

A)0.67
B)2
C)1
D)1.67
E)3

Question 73

Find an equation of the tangent line to the graph of the function given below at the given point. , (The coefficients below are given to two decimal places. )

A)
B)
C)
D)
E)

Question 74

Use implicit differentiation to find . ​ ​

A)
B)
C)
D)
E)

Question 75

A petrol car is parked 50 feet from a long warehouse (see figure).The revolving light on top of the car turns at a rate of 30 revolutions per minute.How fast is the light beam moving along the wall when the beam makes an angle of with the perpendicular from the light to the wall.

A)
B)
C)
D)
E)

Question 76

Find by implicit differentiation. ​ ​

A)
B)

Question 77

Evaluate for the equation at the given point .Round your answer to two decimal places if necessary. ​

A)
B)
C)
D)
E)

Question 78

A petrol car is parked ​a​ = 40 feet from a long warehouse (see figure).The revolving light on top of the car turns at a rate of 30 revolutions per minute.Write θ as a function of x. ​ ​

A)
B)
C)
D)
E)

Question 79

Find by implicit differentiation given that .Use the original equation to simplify your answer. ​

A)
B)
C)
D)
E)

Question 80

Find by implicit differentiation given that . ​

A)
B)
C)
D)
E)