Deck 5: More Applications of Differentiation

Question 1

A balloon is 100 metres off the ground and rising vertically at the constant rate of 3 metres per second just as an automobile passes beneath it travelling along a straight, level road at the constant rate of 72 kilometres per hour. How fast is the distance between them changing one second later? (rounded to the nearest hundredth of a metre)

A) 6.76 metres per second
B) 43.71 metres per second
C) 20.22 metres per second
D) 10.00 metres per second
E) 1.28 metres per second

Answer 1

6.76 metres per second

Question 2

An aircraft is climbing at a 30-degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is 750 kilometres per hour?

A) 400 kilometres per hour
B) 350 kilometres per hour
C) 375 kilometres per hour
D) 250 kilometres per hour
E) 425 kilometres per hour

Answer 2

375 kilometres per hour

Question 3

A lamp is shining on top of a vertical light post that is 540 centimetres tall. A girl is running away from the base of the light post at a constant rate so that her shadow is increasing at the rate of 15 cm/s and the tip of her shadow is moving away from the base at the rate of 90 cm/s. How tall is she?

Answer 3

Let x be length of shadow, y be the distance from the tip of the shadow to the base of the light post and L be the girl's height (as shown in the figure below).
From similarity, = ; hence 540 x = L y; L is a constant!
Differentiating both sides with respect to time we obtain 540 = L
Substituting = rate of change of shadow = +15 cm / s and
= rate of change of distance between the tip of the shadow and the base of the light post = +
90 cm /s, we obtain 540(15) = L (90). Therefore
L = = 90 cm.
The girl is 90 cm tall (about 3 feet ).

Question 4

A plane flying horizontally at an altitude of 1 kilometre and a speed of 500 kilometres per hour passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing at the instant when the plane is 2 kilometres away from the station.

A) 350 kilometres per hour
B) 462 kilometres per hour
C) 402 kilometres per hour
D) 433 kilometres per hour
E) 382 kilometres per hour

Question 5

A water trough is 10 m long and has vertical cross-sections perpendicular to its long axis with the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height of 50 cm. If the trough is being filled with water at the rate of 0.2 cubic metres per minute, how fast is the water level rising when the water is 30 cm deep?

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

Question 6

Water is leaking out of an inverted conical tank at a rate of 0.1 cubic metres per minute at the same time the water is being pumped into the tank at a constant rate. The tank has height 6 metres and the diameter at the top is 4 metres. If the water level is rising at a rate of 20 centimetres per minute, find the rate at which water is being pumped into the tank at the instant when the water is 2 metres deep.

A) 0.179 /min
B) 0.279 /min
C) 0.379 /min
D) 0.479 /min
E) 0.365 /min

Question 7

A boat is pulled toward a dock by a rope with one end attached to the bow of the boat and the other end passing through a ring attached to the dock 2 metres higher than the bow of the boat. If the rope is pulled in at the rate of 0.5 metres per second, how fast is the boat approaching the dock when 3 metres of rope are out? (round to the nearest hundredth metre)

A) 1.15 metres per second
B) 0.67 metres per second
C) 2.38 metres per second
D) 0.97 metres per second
E) 1.68 metres per second

Question 8

In a classical equation from chemistry, the pressure of a gas is proportional to the product of the volume of the gas and the temperature of the gas. In symbols, P = kVT for some constant k. How fast is the pressure changing at an instant when the volume is 20 mL and increasing at 3 mL/min and the temperature is at 21 degrees C and is decreasing at a rate of 2 degrees per minute? Assume that k = 0.8 kilograms per mL-degree.

A) increasing at 18.4 kilograms/min
B) increasing at 14.4 kilograms/min
C) decreasing at 23.5 kilograms/min
D) decreasing at 12.8 kilograms/min
E) increasing at 12.7 kilograms/min

Question 9

At a certain instant the length of one leg of a right triangle is 6 m and is increasing at 2 m/min while the length of the other leg is 8 m and is decreasing at 3 m/min. How fast is the area of the triangle changing at that instant?

A) decreasing at 1 /min
B) decreasing at 2 /min
C) unchanging, holding steady
D) increasing at 2 /min
E) increasing at 1 /min

Question 10

The power P (watts) in an electric circuit is related to the circuit's resistance R (ohms) and the current I (amperes) by the equation P = R I². If the power of a circuit is a constant 40 watts and the resistance R is decreasing at the constant rate of 4 ohms /s, at what rate is the current I changing at the instant the current is 2 amperes?

A) The current is decreasing at the rate of amperes / s.
B) The current is increasing at the rate of amperes / s.
C) The current is decreasing at the rate of amperes / s.
D) The current is increasing at the rate of amperes / s.
E) The current is decreasing at the rate of amperes / s.

Question 11

A lamp is 3 metres high on a post located 5 metres from a vertical wall. A 2 metre tall man walks toward the wall from the lamppost on a path perpendicular to the wall. He is walking at a rate of 1 metre per second. When he is 1 metre from the wall, how fast is the shadow of his head moving up the wall?

A) 0.31 metres per second
B) 0.52 metres per second
C) 1.29 metres per second
D) 0.96 metres per second
E) 0.36 metres per second

Question 12

The height of a right circular cylinder is increasing at the rate of 4 cm/s and its radius is decreasing at the rate of 2 cm/s. At what rate is the lateral surface area of the cylinder changing when the height is 3 centimetres and the radius is 1 centimetre?
The lateral surface area of a cylinder of base radius r and height h is given by S = 2 $\pi$ r h

A) decreasing at 16 $\pi$ /s
B) decreasing at 4 $\pi$ /s
C) increasing at 4 $\pi$ /s
D) increasing at 20 $\pi$ /s
E) increasing at 16 $\pi$ /s

Question 13

A plane is flying horizontally at an altitude of seven kilometres and a speed of 800 kilometres per hour. At time t = 0 the plane passes over a tracking station on the ground. How fast is the angle of elevation of the plane as measured at the tracking station changing 18 seconds later?

A) increasing at 86 rad/h
B) increasing at 66 rad/h
C) decreasing at 86 rad/h
D) decreasing at 66 rad/h
E) decreasing at 62 rad/h

Question 14

Water is leaking out of an inverted conical cup at a rate of 2 cubic centimetres per second. The radius of the cone is six centimetres and the height is 10 centimetres. Find the rate at which the water depth is changing at time t when the depth of the water is 3 centimetres.

A) decreasing at cm/sec
B) decreasing at cm/sec
C) decreasing at cm/sec
D) decreasing at cm/sec
E) decreasing at cm/sec

Question 15

Find correct to four decimal places using Newton's Method.

A) 2.6672
B) 2.6627
C) 2.6698
D) 2.6684
E) 2.6667

Question 16

Find the roots of the equation x³ - 5x - 3 = 0 correct to three decimal places using Newton's Method.

A) -1.834, -0.657, 2.491
B) -1.834, 0.657, 2.491
C) -1.824, -0.667, -2.501
D) -1.824, -0.667, 2.589
E) -1.834, -0.667, 2.491

Question 17

Find the solution of the equation + 5x = 0 to four decimal places using Newton's Method.

A) 0.1486
B) -0.2486
C) -0.1486
D) -0.1506
E) -0.1473

Question 18

Find the roots of the equation ln x - = 0 to four decimal places using Newton's Method.

A) 1.2445
B) 1.9445
C) 1.3445
D) 1.0445
E) 1.0543

Question 19

Find the solutions of the equation cos x - x⁴ = 0 to 4 decimal places using Newton's Method.

A) 0.8241, -0.6421
B) 0.8906, -0.8906
C) 0.8241, -0.8241
D) 0.6421, -0.6421
E) 0.8954, -0.8954

Question 20

Newton's Method with initial approximation x₀ is used to approximate a real root of the equation x⁴ - 2 = 0. The value of Newton's Method iteration x₁ is equal to

A)
B)
C)
D)
E) ±

Question 21

Use Newton's Method to find the x coordinate of a point on the curve y = x³ + 1 such that the tangent line to the curve at that point passes through the point (2, 0). Give your answer accurate to three decimal places.

A) 3.061
B) 3.050
C) 3.054
D) 3.058
E) 3.049

Question 22

Use Newton's Method to find the solution of the equation x + tan x = 0 on the interval [2, 3] accurate to four decimal places.

A) 2.0288
B) 2.0293
C) 2.0278
D) 2.0272
E) 2.0284

Question 23

Newton's Method with the initial approximation x₁ = 1 is used to approximate the real root of the equation x³ + 3x - 1 = 0. Determine the value of x₃, the third iteration of Newton's Method.

A)
B)
C)
D)
E)

Question 24

The function f(x) = ( - 5) has one critical point in the interval x > 0. Find this critical point with error within 0.005 by applying Newton's Method to an appropriate function.

A) 3.4023
B) 3.2240
C) 3.1278
D) 3.4259
E) none of the above

Question 25

Find all the roots of the equation sin x = x².

A) ± 0.876726
B) 0.876726
C) 0.876833
D) 0, 0.876726
E) 0, 0.876833

Question 26

Give the iteration formula for finding the roots of the equation sin x - x² = 0 using Newton's Method.

A) = -
B) = -
C) = +
D) = -
E) = +

Question 27

Use Newton's Method to find the roots of the equation cos x = x.

A) 0.739335
B) 0.739161
C) 0.739085
D) 0.739204
E) 0.739185

Question 28

Suppose Newton's Method applied to f(x) = is used to "find"the root of the equation = 0 with initial guess x₀ = r $\neq$ 0. What result does the first iteration of the method yield? The second iteration? The nth iteration? Why do these not converge to the obvious root x = 0 no matter how close the initial guess r was to that root?

Question 29

By considering the locations of its critical points, determine how many real zeros the function must have. Find them, correct to six decimal places.

A) one zero, -1.624717
B) one zero, -1.424717
C) one zero, -1.324717
D) three zeros, -1.424717, -0.436712, and 0.688212
E) two zeros, -1.424717, and -0.436712

Question 30

Evaluate .

Question 31

Evaluate .

A)
B)
C)
D)
E)

Question 32

Use the table of values below to evaluate

A) -
B) - 2
C)
D) -
E)

Question 33

Evaluate the limit .

A) 3
B) 0
C) -3
D) 1
E) $\infty$

Question 34

Evaluate .

A) -1
B) $\infty$
C) 3
D) 1
E) - $\infty$

Question 35

Evaluate the limit .

A) 1
B)
C) 0
D) $\infty$
E)

Question 36

Evaluate the limit x ln x.

A) - $\infty$
B) 1
C) 0
D) $\infty$
E) -1

Question 37

Evaluate .

A) 1
B) e
C)
D) $\infty$
E) -6

Question 38

Evaluate the limit .

A) -2
B) 2
C) 0
D) 1
E) -1

Question 39

Evaluate the limit .

A)
B)
C) 1
D)
E)

Question 40

Evaluate the limit .

A) -2
B) -1
C) 0
D) $\infty$
E) 1

Question 41

Evaluate the limit .

A) - $\infty$
B) $\infty$
C) 1
D) 0
E) -1

Question 42

Evaluate the limit . -

A)
B)
C)
D) 1
E) -

Question 43

Evaluate x ( $\pi$ - 2 (7x)).

A)
B) 0
C)
D) -
E) $\infty$

Question 44

Evaluate the limit .

A) -
B) -3
C)
D) -ln 3
E)

Question 45

Evaluate the limit .

A)
B) 1
C)
D) -
E) -

Question 46

Find all local extreme values of the function f(x) = 2 + 3 - 12x + 13 and their locations.

A) local maximum 33 at x = -2, local minimum 26 at x = 1
B) local maximum 26 at x = -2, local minimum 33 at x = 1
C) local maximum 26 at x = -1, local minimum 17 at x = 2
D) local maximum 17 at x = -1, local minimum 26 at x = 2
E) no local extrema

Question 47

Find all local extreme values of the function f(x) = - 6 + 12x - 5 and their locations.

A) local maximum 3 at x = 2, local minimum -61 at x = -2
B) local maximum -61 at x = -2, local minimum 3 at x = 2
C) local maximum 3 at x = -2, local minimum -61 at x = 2
D) local maximum -61 at x = 2, local minimum 3 at x = -2
E) no local extrema

Question 48

Find any extreme values of the function f(x) = - 7 and their locations.

A) local (and absolute) minimum -7 at x = 5/2, no local maxima
B) local (and absolute) maximum -7 at x = 5/2, no local minima
C) local (and absolute) minimum -7 at x = 5/2, local maximum -2 at x = 0
D) local (and absolute) maximum -7 at x = 5/2, local minimum -2 at x = 0
E) no absolute or local extrema

Question 49

At what values of t does the function g(t) = have extreme values?

A) absolute maximum at t = 1, absolute minimum at t = -1, no other local extrema
B) absolute maximum at t = 1, absolute minimum at t = 0, no other local extrema
C) absolute maximum at t = 2, absolute minimum at t = 0, no other local extrema
D) absolute maximum at t = 2, absolute minimum at t = -2, no other local extrema
E) no absolute or local extrema

Question 50

Find any extreme values of the function f(x) = x .

A) absolute minimum 0 at x = 1, no maxima
B) absolute minimum 0 at x = 0, local maximum 2 at x = 2
C) absolute minimum 0 at x = 0, no local maxima
D) absolute minimum 0 at x = 1, local maximum 2 at x = 2
E) no absolute or local extrema

Question 51

Find the absolute maximum and absolute minimum values of the function f(x) = 2 - 3 , -2 $\le$ x $\le$ 2.

A) maximum 4, minimum -28
B) maximum 2, minimum -26
C) maximum 3, minimum -29
D) maximum 4, minimum -12
E) no absolute extrema

Question 52

Find the absolute maximum and absolute minimum values of the function f(x) = - , -4 $\le$ x $\le$ -1.

A) maximum -20, minimum -67
B) maximum -20, minimum -49
C) maximum -49, minimum -67
D) maximum -4, minimum -67
E) no absolute extrema

Question 53

Find the absolute maximum and absolute minimum values of the function f(x) = - 1, 0 $\le$ x $\le$ 2.

A) maximum 2, minimum -1
B) maximum 2, minimum 0
C) maximum 2, minimum -2
D) maximum 3, minimum -2
E) no absolute extrema

Question 54

Find the minimum values of the function f(x) = - .

A) -
B) 0
C) -
D) -
E) no local or absolute minima

Question 55

For what value of k will f(x) = 2x - 3k , have a local minimum at x = 1?

A) k =
B) k = -
C) k = 3
D) k = 0
E) k = -3

Question 56

Find the extreme values of f(x) = x - 2 sin x on [0, $\pi$ ].

A) maximum $\pi$ , minimum -
B) maximum , minimum -
C) maximum , minimum - 2
D) maximum + , minimum -
E) no extreme values

Question 57

Find the extreme values of f(x) = x + cos x on [0, π].

Question 58

Find the extreme values of f(x) = x + cos x on [0, $\pi$ ].

A) maximum + 1, minimum - 1
B) maximum , minimum $\pi$ -
C) maximum $\pi$ - , minimum
D) maximum + , minimum -
E) no extreme values

Question 59

The function f(x) = 4 + k - has a maximum value at x = 2. Find k.

A) k = 8
B) k = -8
C) k = 4
D) k = -4
E) k = 0

Question 60

Find the absolute maximum and minimum values (if any) of f(x) = .

A) minimum -1, maximum 1
B) minimum -1, no maximum
C) no minimum, maximum 1
D) minimum -1, maximum 0
E) no absolute maximum or minimum

Question 61

Find the maximum value of the function f(x) = .

A)
B)
C) 1
D)
E) no maximum value

Question 62

Let g(x) = 3(x - 1)^2/3 - x. Which of the following statements is true?

A) g has a local maximum at (1, -1) and a local minimum at (9, 3).
B) g is increasing on the interval (- $\infty$ , 9).
C) g has a local maximum at (9, 3) and a local minimum at (1, -1).
D) g is decreasing on the interval (1, 9).
E) g is decreasing on the interval (- $\infty$ , 9).

Question 63

Find and classify all the local extrema of the function f(x) = x - sin2x.

A) local minima at x = k $\pi$ + and local maxima at x = k $\pi$ - , where k is an integer
B) local maxima at x = k $\pi$ + and local minima at x = k $\pi$ - , where k is an integer
C) local maxima at x = k $\pi$ + and local minima at x = k $\pi$ - , where k is an integer
D) local minima at x = k $\pi$ + and local maxima at x = k $\pi$ - , where k is an integer
E) local minima at x = 2k $\pi$ + and local maxima at x = 2k $\pi$ - , where k is an integer

Question 64

Determine the concavity of f(x) = - 24 + 6x + 18 and identify any points of inflection.

A) concave downwards on (- $\infty$ , 8), upwards on (8, $\infty$ ); inflection at x = 8
B) concave downwards on (- $\infty$ , -8), upwards on (-8, $\infty$ ); inflection at x = -8
C) concave upwards on (- $\infty$ , 8), downwards on (8, $\infty$ ); inflection at x = 8
D) concave upwards on (- $\infty$ , -8), downwards on (-8, $\infty$ ); inflection at x = -8
E) concave upwards on (- $\infty$ , $\infty$ ); no inflection points

Question 65

Determine the concavity of f(x) = cos x + sin x on [0, 2 $\pi$ ] and identify any points of inflection.

A) concave down on [0, 3 $\pi$ /4) (7 $\pi$ /4, 2 $\pi$ ], concave on (3 $\pi$ /4, 7 $\pi$ /4); inflection points at x = 3 $\pi$ /4 and
B) concave up on [0, 3 $\pi$ /4) 11ee7b09_453f_4329_ae82_2b7669cf7fd7_TB9661_11 (7 $\pi$ /4, 2 $\pi$ ], concave down on (3 $\pi$ /4, 7 $\pi$ /4); inflection points at x = 3 $\pi$ /4 and
C) concave down on [0, $\pi$ /4) 11ee7b09_453f_4329_ae82_2b7669cf7fd7_TB9661_11 (5 $\pi$ /4, 2 $\pi$ ], concave up on ( $\pi$ /4, 5 $\pi$ /4); inflection points at x = $\pi$ /4 and
D) concave up on [0, $\pi$ /4) 11ee7b09_453f_4329_ae82_2b7669cf7fd7_TB9661_11 (5 $\pi$ /4, 2 $\pi$ ], concave down on ( $\pi$ /4, 5 $\pi$ /4); inflection points at x = $\pi$ /4 and
E) concave down on (0, 3 $\pi$ /4) 11ee7b09_453f_4329_ae82_2b7669cf7fd7_TB9661_11 (7 $\pi$ /4, 2 $\pi$ ), concave up on (3 $\pi$ /4, 7 $\pi$ /4); inflection points at x = 3 $\pi$ /4 and

Question 66

Find all inflection points of the graph of f(x) = 3 - 5 + 13x.

A) (0, 0) and (1, 11)
B) (0, 0) only
C) (1, 11) only
D) (-1, -21) and (1, 11)
E) (-1, -21), (0, 0), and (1, 11)

Question 67

Let f(x) = 18 + 9 . The first and the second order derivatives of f are given by (x) = and (x) = , respectively. Determine:(a) intervals of concavity.(b) the x and y coordinates of the inflection points.

Question 68

Find any extreme values and points of inflection of the function f(x) = x⁴ - 4 + 10.

A) absolute minimum -17 at x = 3; local maximum 10 at x = 0; inflection at x = 2
B) absolute minimum -17 at x = 3; inflections at x = 0 and x = 2
C) absolute maximum 10 at x = 0; inflections at x = 2 and x = 3
D) absolute maximum -17 at x = 3; inflections at x = 0 and x = 2
E) absolute maximum -17 at x = 3; local minimum 10 at x = 0; inflection at x = 2

Question 69

Using the second derivative test, classify the critical points of the function f(t) = t³ - t² - t + 2 and locate any points of inflection.

A) local max at t = -1/3, local min at t = 1, inflection at t = 1/3
B) local min at t = -1/3, local max at t = 1, inflection at t = 1/3
C) local max at t = 1/3, local min at t = -1, inflection at t = -1/3
D) local min at t = 1/3, local max at t = -1, inflection at t = -1/3
E) none of the above

Question 70

Let g be a polynomial function such that (x) = (x + 2)( - 10x -24). Find the x-coordinate of all inflection points of the graph of g.

A) only -2
B) -2 and 12
C) - 2 , 4, and 6
D) only 12
E) -2, - 4, and -6

Question 71

Find the inflection points of the graph of f(x) = where c is a nonzero constant.

A)
B)
C)
D)
E) There are no inflection points.

Question 72

The function f(x) = 3x⁵ + Ax⁴ + Bx³ has inflection points at x = 0, x = -1, and x = 1. Find the values of the constants A and B.

A) A = 0, B = -10
B) A = 0, B = 10
C) A has any value, B = -10
D) A has any value, B = 10
E) A = 10, B = 0

Question 73

Determine the concavity of f(x) = cos x + sin x on [0, 2π] and identify any points of inflection.

Question 74

At what value(s) of x does the graph of f(x) = x have inflections?

A) ±
B) 0 and ±
C) ±
D) 0 and ±
E) 0

Question 75

Which of the following statements best describes the graph of the function f(x) = ?

A) The graph is concave up on (0, $\infty$ ).
B) The graph is concave down on (0, $\infty$ ).
C) The graph is concave up on (0,e) and concave down on (e, $\infty$ ).
D) The graph is a straight line.
E) The graph is concave down on (0,e) and concave up on (e, $\infty$ ).

Question 76

Determine the concavity and inflections of f(x) = .

A)
B)
C)
D)
E) concave up on (0, $\infty$ )

Question 77

Let P(x) be a polynomial in x, let k be a positive integer, and let be a number such that is a factor of (x) but is not a factor of (x). For what values of k is it possible that P(x) has a local maximum or minimum value at x = 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11?

A) even values of k
B) k = 1 only
C) k = 2 only
D) odd values of k
E) k = 0 only

Question 78

What are the asymptotes of the graph of y = ?

A) horizontal asymptote at y = 2, vertical asymptotes at x = -1 and x = 2
B) horizontal asymptote at y = 2, vertical asymptotes at x = 1 and x = -2
C) horizontal asymptote at y = , vertical asymptotes at x = 1 and x = 2
D) oblique asymptote at y = -x - 2, vertical asymptotes at x = -1 and x = 2
E) oblique asymptote at y = x - 2, vertical asymptotes at x = -1 and x = 2

Question 79

Find the equations of all horizontal asymptotes of f(x) =

A) y = -1 and y = 5
B) y = -1 and y = 1
C) y = 3
D) y = -1 and y = 3
E) y = 1

Question 80

Find the equations of the vertical asymptotes of f(x) =

A) x = 0, x = 12, and x = -2
B) x = 0, x = 6, and x = 4
C) x = 0 and x = 12
D) x = 12 and x = -2
E) x = 0, x = -8, and x = -3