Deck 3: Topics in Deifferentiation

Question 1

Find .

Answer 1

Question 2

Evaluate. .

Answer 2

-oo

Question 3

Answer true or false.

Answer 3

True

Question 4

Evaluate. .

Question 5

Answer true or false.

Question 6

Evaluate.

Question 7

Evaluate.

Question 8

Answer true or false. .

Question 9

Evaluate.

Question 10

Evaluate. .

Question 11

Evaluate. .

Question 12

Evaluate.

Question 13

Evaluate. .

Question 14

Answer true or false. .

Question 15

Answer true or false. .

Question 16

Evaluate. .

Question 17

Evaluate. .

Question 18

Evaluate.

Question 19

Evaluate. .

Question 20

Evaluate. .

Question 21

Evaluate.

Question 22

Let y = x³ - 6. Find $\Delta$ y if $\Delta$ x = 3 and the initial value of x is x = 2.

Question 23

Answer true or false. A circular spill is spreading so that when its radius r is 1 m, dr = 0.005 m. The corresponding change in the area covered by the spill, A, is, to the nearest hundredth of a square meter, 0.008 m².

Question 24

Answer true or false. A cube is expanding as temperature increases. If the length of the cube is changing at a rate of dx = 0.6 mm when x is 3.5 m, the volume is experiencing a corresponding change of 22,050,000 mm³.

Question 25

A small suspended droplet of radius 10 microns is growing. If dr = 0.004 micron find the change in the volume, dV, to the nearest thousandth of a cubic micron.

A) 1.257 cubic microns
B) 1.676 cubic microns
C) 1.600 cubic microns
D) 5.027 cubic microns
E) 1256.637 cubic microns

Question 26

Evaluate.

Question 27

y = x⁸. Find the formula for dy.

A) dy = 8x⁷
B) dy = 8(x + dx)⁷
C) dy = 8x⁷ - 8(dx)⁷
D) dy = (x + dx)⁸ - x⁸
E) dy = 8x⁷dx

Question 28

If y = x⁴, find the formula for $\Delta$ y.

A)
B)
C)
D)
E)

Question 29

y = cos x. Find the formula for dy.

A) dy = -sin (x + dx)
B) dy = -sin (x + dx) - sin x dx
C) dy = -sin x dx
D) dy = -sin (x + dx) + sin x dx
E) dy = -sin (x + dx) - sin dx

Question 30

Let . Find $\Delta$ y at x = 4 if $\Delta$ x = 1.1.

A) 1.100
B) 0.275
C) 0.250
D) 0.550
E) 0

Question 31

Let . Find dy at x = 3 if dx = 0.032.

A) -0.0118
B) 0.1066
C) 0.0118
D) 0.3703
E) -0.0177

Question 32

If , find the formula for $\Delta$ y.

A)
B)
C)
D)
E)

Question 33

Use dy to approximate starting at x = 9.

A) 3.003
B) 9.003
C) 2.997
D) 3.001
E) 3.167

Question 34

y = sin x cos x. Find the formula for dy.

A) dy = -sin x cos x dx
B) dy = (-sin ²x + cos ²x) dx
C) dy = -sin ²(x + dx) + cos ²(x + dx)
D) dy = (-sin ²(x + dx) + cos ²(x + dx)) - (-sin ² x + cos ²x)
E) dy = sin (x +dx) cos (x +dx) - sin x cos x

Question 35

If y = tan x, find the formula for $\Delta$ y.

A)
B)
C)
D)
E)

Question 36

Let y = 4x⁴. Find dy at x = 4 if dx = -0.01.

A) 1024
B) -40.96
C) -163.84
D) -102,400
E) -10.24

Question 37

A particle moves according to s = t³. Find ds if t = 5 and dt = 4.

A) 75
B) 100
C) 300
D) 500
E) 60

Question 38

Let y = x³ - 3. Find dy if dx = 1 and the initial value of x is x = 3.

Question 39

Approximate sin 30.075°.

A) 0.4993
B) 0.5003
C) 0.5013
D) 0.5023
E) 0.5033

Question 40

Answer true or false. The formula for dy is D y = f(x)dx.

Question 41

Use a differential to approximate .

Question 42

A circular hole 6 inches in diameter and 12 feet deep is to be drilled out of a glacier. The diameter of the hole is exact but the depth of the hole is measured with an error of ±1%. Estimate the percentage error in the volume of ice removed. ( is the volume of a cylinder of diameter d and height h.)

Question 43

Use a differential to approximate cos 65°.

Question 44

Use a differential to approximate sin 36°.

Question 45

Let y = x⁴ and dx = 0.05 at x = 4. Find dy.

Question 46

When a cubical block of metal is heated, each edge increases by 0.1% per degree increase in temperature. Use differentials to estimate the percentage increase in the surface area and volume of the block per degree increase in temperature.

Question 47

Use a differential to approximate tan 41°.

Question 48

Let . Find $\Delta$ y if $\Delta$ x = 1 and the initial value of x is x = 1.

Question 49

The surface area of a sphere is given by S = 4 $\pi$ r² where r is the radius of the sphere. The radius is measured to be 6cm with an error of ±0.07cm. Use differentials to estimate the error in the calculated surface area.

Question 50

When a spherical ball of metal is heated, the radius of the sphere increases by 0.3% per degree increase in temperature. Use differentials to estimate the percentage increase in the surface area and volume of the ball per degree increase in temperature.

Question 51

The area of a circle is to be computed from a measured value of its diameter. Estimate the maximum permissible percentage error in the measurement if the percentage error in the area must be kept within 0.5%.

Question 52

Use a differential to approximate .

Question 53

Let . Find dy if dx = 2 and the initial value of x is x = 2.

Question 54

The surface area S of a cube is to be computed from a measured value of its side x. Estimate the maximum permissible percentage error in the side measurement if the percentage error in the surface area must be kept to within ±1%.

Question 55

Let y = 2x² and dx = 0.04 at x = 4. Find dy.

Question 56

The magnetic force F acting on a particle is given by , where r is the distance from the magnetic source and k is a constant. r is measured to be 4cm with a possible error of ±3%. Use differentials to estimate the error in the calculated value of F.

Question 57

The magnetic force F acting on a particle is given by , where r is the distance from the magnetic source and k is a constant. r is measured to be 2cm with a possible error of ±8%. Estimate the percentage error in F and r.

Question 58

Find the formula for dy if y = x⁶.

Question 59

The pressure P, the volume V, and the temperature T of an enclosed gas are related by the Ideal Gas Law, PV = kT where k is a constant. With the temperature held constant, the volume of the gas is calculated from a measured value of its pressure. Estimate the maximum permissible error in the pressure measurement if the percentage error in the volume must be kept to within ±3%.

Question 60

Use a differential to approximate (2.97)⁵.

Question 61

Answer true or false. Suppose z = 3yx. Then dz/dt = 3x(dy/dt)+3y(dx/dt).

Question 62

The power in watts for a circuit is given by P = I ²R. How fast is the power changing if the resistance, R, of the circuit is 1,400 $\Omega$ , the current, I, is 1.5A, and the current is decreasing with respect to time at a rate of 0.025 A/s. (Assume R is a constant.)

A) 105 w/s
B) 4,200 w/s
C) 26.25 w/s
D) 157.5 w/s
E) -105 w/s

Question 63

Answer true or false. If z = 2x³y², then .

Question 64

The number of board feet of lumber in a log that is 10 feet long is given by the formula , where D is the diameter of the tree in inches. In a tree that will produce a 10 foot log and has a current diameter of 22 inches, the diameter is changing at a rate of 3.45 inches per year. How fast is the volume increasing?

A) 22.5 ft³/year
B) 38.8125 ft³/year
C) 77.625 ft³/year
D) 11.3 ft³/year
E) 202.5 ft³/year

Question 65

Answer true or false. If A = 3 $\pi$ r³, then .

Question 66

Answer true or false. Suppose z = 4yx. Then dz/dt = 4(dy/dt)(dx/dt).

Question 67

The volume of a cylinder is given by V = $\pi$ r²h. Find in terms of . (Assume that r is a constant.)

A)
B) 1
C)
D)
E)

Question 68

Find the formula for dy if .

Question 69

Answer true or false. If sin $\theta$ = 8xy, then .

Question 70

Answer true or false. Water is running out of an inverted conical tank so the height is changing at a rate of 4 ft/s. The height of the water in the tank is changing at 4 ft/s if the height is currently 10 ft and the radius is 8 ft.

Question 71

Gravitational force is inversely proportional to the distance between two objects squared. If F = at a distance d = 2m, how fast is the force diminishing if the objects are moving away from each other at 0.1 m/s?

A) 1 N/s
B) -1 N/s
C) -10 N/s
D) 10 N/s
E) 0 N/s

Question 72

Find the formula for $\Delta$ y if y = csc x.

Question 73

Answer true or false. If A = 8xy, then

Question 74

A point P is moving along a curve whose equation is . When P = (9, 41) , y is increasing at a rate of 2 units/s. How fast is x changing?

A) 0.439 units/s
B) 738 units/s
C) 8.111 units/s
D) 9.111 units/s
E) 10.111 units/s

Question 75

Find the formula for $\Delta$ y if y = 3x⁴ - 9.

Question 76

Answer true or false. A cube is expanding, so .

Question 77

Find the formula for dy if y = cot x.

Question 78

Suppose z = x² + y⁵. Then dz/dt =

A)
B)
C)
D)
E)

Question 79

Answer true or false. A plane is approaching an observer with a horizontal speed of 140ft/s and is currently 7,000ft from being directly overhead at an altitude of 12,000ft. The rate at which the angle of elevation, $\theta$ , is changing with respect to time, d $\theta$ /dt = (1/x)dy/dt. Where x is the horizontal distance and y is the altitude.

Question 80

A 10-ft ladder rests against a wall at $\pi$ /4 radians. If it were to begin to slip, when the bottom of the ladder is moving at 0.08 ft/s, how fast would the top of the ladder be moving down the wall? (How fast would the height of the upper end of the ladder on the side of the wall be changing?)

A) 0.074 ft/s
B) 0.071 ft/s
C) 0.069 ft/s
D) 0.080 ft/s
E) 0.088 ft/s