Deck 2: Derivatives

Question 1

Find the linearization of a suitable function, and then use it to approximate the number.
sin 0.3

Answer 1

Question 2

A point moves along the curve . When the point is at , its x-coordinate is increasing at the rate of 3 units per second. How fast is its y-coordinate changing at that instant of time?

A) units/sec
B) units/sec
C) units/sec
D) units/sec

Answer 2

units/sec

Question 3

Use the linear approximation of the function at to approximate the number .

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the differential of the function.

Question 5

Two cars start moving from the same point. One travels south at mi/h and the other travels west at mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

A)
B)
C)
D)
E)

Question 6

The circumference of a sphere was measured to be cm with a possible error of cm. Use differentials to estimate the maximum error in the calculated volume.

Question 7

Find the linearization L (x) of the function at
a.

A) x -
B) x +
C) x -
D) x +

Question 8

The top of a ladder slides down a vertical wall at a rate of m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

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

Question 9

Two sides of a triangle are m and m in length and the angle between them is increasing at a rate of rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\pi$ /3 .

A)
B)
C)
D)
E)

Question 10

Use differentials to estimate the amount of paint needed to apply a coat of paint cm thick to a hemispherical dome with diameter m.

A)
B)
C)
D)
E)

Question 11

Find the differential of the function at the indicated number.

A) dx
B) dx
C) dx
D) dx

Question 12

If two resistors with resistances and are connected in parallel, as in the figure, then the total resistance measured in ohms ( $\Omega$ ), is given by . If and are increasing at rates of and respectively, how fast is changing when and ?
Round the result to the nearest thousandth.

A)
B)
C)
D)
E)

Question 13

Compute and dy for the given values of x and .

A)
B)
C)
D)
E)

Question 14

Let y = 1/x.
a.Find $\Delta$ x and $\Delta$ y if x changes from 1 to 1.03.Round to six decimal places, if necessary.
b.Find the differential dy, and use it to approximate $\Delta$ y if x changes from 1 to 1.03.
c.Compute $\Delta$ y - dy, the error in approximating $\Delta$ y by dy.

Question 15

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0.

A)
B)
C)
D)
E)

Question 16

A plane flying horizontally at an altitude of 1 mi and a speed of mi/h 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 17

Find the differential of the function at the indicated number.

A) dx
B) dx
C) dx
D) dx

Question 18

An equation relating the variables x and y, the values of x and y, and the value of at a particular instant of time are given. Find the value of .

A)
B)
C)
D)

Question 19

Gravel is being dumped from a conveyor belt at a rate of ft /min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is ft high? Round the result to the nearest hundredth.

A)
B)
C)
D)
E)

Question 20

A car leaves an intersection traveling west. Its position 4 sec later is 26 ft from the intersection. At the same time, another car leaves the same intersection heading north so that its position 4 sec later is 26 ft from the intersection. If the speeds of the cars at that instant of time are 12 ft/sec and 10 ft/sec, respectively, find the rate at which the distance between the two cars is changing. Round to the nearest tenth if necessary.

A)15.6 ft/sec
B)3.7 ft/sec
C)3.1 ft/sec
D)36.8 ft/sec

Question 21

The quantity Q of charge in coulombs C that has passed through a point in a wire up to time t (measured in seconds) is given by . Find the current when .

A)
B)
C)
D)
E)

Question 22

A water trough is 20 m long and a cross-section has the shape of an isosceles trapezoid that is 20 cm wide at the bottom, 60 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of , how fast is the water level rising when the water is cm deep? Round the result to the nearest hundredth.

Question 23

The sides of a square baseball diamond are 90 ft long. When a player who is between the second and third base is 30 ft from second base and heading toward third base at a speed of 24 ft/sec, how fast is the distance between the player and home plate changing? Round to two decimal places.

Question 24

If a snowball melts so that its surface area decreases at a rate of , find the rate at which the diameter decreases when the diameter is cm.

Question 25

The volume of a right circular cone of radius r and height h is . Suppose that the radius and height of the cone are changing with respect to time t.
a.Find a relationship between , , and .
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 26

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 2 m/s how fast is the boat approaching the dock when it is 3 m from the dock? Round the result to the nearest hundredth if necessary.

Question 27

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is 20 ft and is increasing at the rate of ft/sec. Round to the nearest tenth if necessary.

Question 28

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure below). The point Q is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

Question 29

Find the instantaneous rate of change of the function when

A)
B)3
C)9
D)

Question 30

The top of a ladder slides down a vertical wall at a rate of 0.15 m/s . At the moment when the bottom of the ladder is 1.5 m from the wall, it slides away from the wall at a rate of 0.3 m/s. How long is the ladder?

A)3.9 m
B)3.4 m
C)4.4 m
D)2.9 m
E)2.4 m

Question 31

A water trough is 20 m long and a cross-section has the shape of an isosceles trapezoid that is 20 cm wide at the bottom, 60 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of , how fast is the water level rising when the water is cm deep? Round the result to the nearest hundredth.

Question 32

The height (in meters) of a projectile shot vertically upward from a point m above ground level with an initial velocity of 25.48 m/s is after t seconds. a) When does the projectile reach its maximum height? (b) What is the maximum height?

A)
B)
C)
D)
E)

Question 33

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of ft/s. 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 34

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s how fast is the boat approaching the dock when it is m from the dock? Round the result to the nearest hundredth if necessary.

Question 35

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is . Find the linear density when x is m.

A)
B)
C)
D)
E)

Question 36

If f is the focal length of a convex lens and an object is placed at a distance v from the lens, then its image will be at a distance u from the lens, where f, v, and u are related by the lens equation . Find the rate of change of v with respect to u.

A)
B)
C)
D)
E)

Question 37

The altitude of a triangle is increasing at a rate of while the area of the triangle is increasing at a rate of . At what rate is the base of the triangle changing when the altitude is 10 cm and the area is .

Question 38

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 39

A television camera is positioned 4,600 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 680 ft/s when it has risen 2,600 ft. If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at this moment? Round the result to the nearest thousandth.

Question 40

Let y = 1/x.
a.Find $\Delta$ x and $\Delta$ y if x changes from 1 to 1.03.Round to six decimal places, if necessary.
b.Find the differential dy, and use it to approximate $\Delta$ y if x changes from 1 to 1.03.
c.Compute $\Delta$ y - dy, the error in approximating $\Delta$ y by dy.

Question 41

A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to mm. The area is A(x). Find ( ).

Question 42

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

A)
B)
C)
D)

Question 43

Calculate .

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

Question 44

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

Question 45

Suppose the daily total cost (in dollars) of manufacturing x televisions is What is the marginal cost when x = 600? What is the actual cost incurred in manufacturing the 601st television?

A)$348.00, $348.58
B)$186.67, $186.94
C)$348.00, $348.46
D)$186.67, $186.98

Question 46

A spherical balloon is being inflated. Find the rate of increase of the surface area with respect to the radius r when r = ft.

Question 47

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference and viscosity $\eta$ =0.028.
Find the velocity of the blood at radius r =

Question 48

If

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

Question 49

Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is .
Find .

Question 50

The position function of a particle is given by When does the particle reach a velocity of 22 m/s?

Question 51

Find in terms of x and y.

A)
B)
C)
D)

Question 52

Calculate .

Question 53

Find by implicit differentiation.

Question 54

The mass of part of a wire is kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x =16m .

A)
B)
C)
D)
E)

Question 55

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

Question 56

Find the average rate of change of the area of a circle with respect to its radius r as r changes from to .

A)
B)
C)
D)
E)

Question 57

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 V of water remaining in the tank after t minutes as .
Find the rate at which water is draining from the tank after 6 minutes.

Question 58

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy⁷;

A)y = x
B)y = 1
C)y = 7x + 1
D)x =

Question 59

Find by implicit differentiation.

Question 60

Find by implicit differentiation.

A)
B)
C)
D)

Question 61

Find the second derivative of the function.

A)
B)
C)
D)

Question 62

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 63

Use the table to estimate the value of , where and . 10
10.1
10.2
10.3
10.4
10.5
10.6 4.5 5.6
4.3
2.5
9.9
7.8 6.5
5.9
4.7
4.2
5.4 6.3

Question 64

Calculate .

Question 65

Find the derivative of the function.

A)
B)
C)
D)

Question 66

If , find and .

A)
B)
C)
D)
E)

Question 67

Find the derivative of the function. g(t) = tan(cos 2t)

A)
B)
C)
D)

Question 68

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.
y - x = x = cos y

Question 69

Find the rate of change of y with respect to x at the given values of x and y. ; x = 3, y = -5

Question 70

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

Question 71

Differentiate.

Question 72

Find the derivative of the function.

Question 73

Find the derivative of the following function and calculate it for x = to the nearest tenth.

A)0.2
B)
C)1.1
D)0.3
E)0.1

Question 74

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 75

The curve with the equation is called an astroid. Find an equation of the tangent to the curve at the point ( , 1).

Question 76

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

Question 77

If u is a differentiable function of x and f (x) = |u| then . Use this to find the derivative of the following function.
F (x) = |x² - 4|

A)
B)
C)
D)

Question 78

Find

Question 79

If , find .

A)
B)
C)
D)
E)

Question 80

Find the derivative of the function.

A)
B)
C)
D)