Assignments Differentiation



AP Calculus Assignments: Derivative Concepts

|Day |Topic |Assignment |

|1 |Average Rate of Change |HW Derivative Concepts - 1 |

|2 |Slope of a Curve |HW Derivative Concepts - 2 |

|3 |The derivative at a point; nDeriv |HW Derivative Concepts - 3 |

|4 |The derivative function |HW Derivative Concepts - 4 |

|5 |Differentiability and continuity |HW Derivative Concepts - 5 |

|6 |Increasing/decreasing functions, stationary points |HW Derivative Concepts - 6 |

|7 |Concavity, inflection points |HW Derivative Concepts - 7 |

|8 |Practice with graphs **QUIZ** |HW Derivative Concepts - 8 |

|9 |Estimating derivatives from tables and graphs |HW Derivative Concepts - 9 |

|10 |Derivative interpretation |HW Derivative Concepts - 10 |

|11 |Review and practice **QUIZ** |HW Derivative Concepts – 11 |

|12 |Review |HW Derivative Concepts - Review |

|13 |**TEST** | |

AP Calculus HW: Derivative Concepts - 1

1. a. Write the formula for average rate of change of a function f over the interval [a, b].

b. Give the geometric (graphical) interpretation of rate of change.

c. Memorize both of the above.

2. Find the average rate of change of [pic] on the interval [1, 16].

3. The average rate of change of a function ƒ on the interval [–2, 6] is –1.75. If ƒ(6) = –3, find ƒ(–2).

4. The average rate of change of a function ƒ on the interval [a, b] is k. Find an expression for f(b) in terms of a, b, k and f(a).

5. The average rate of change of a function f on the interval [0, 6] is 3 and the average rate of change on the interval [6, 10] is 2. What is the average rate of change on the interval [0, 10]?

6. If the average rate of change of a function f on the interval [2, 5] is 3 and f(2) = 1, which of the following must be true?

a. f(5) > f(2) b. f is increasing on [2, 5] c. f does not have a root in [2, 5]

d. f(3) = 4 e. f(5) = 10

f. The equation of the secant line to f on the interval [2, 5] is y = 3x – 5.

7. What must be true if the average rate of f on [a, b] is 0?

8. Approximately a billion years ago, an alien spacecraft crash-landed on Mars. The surviving aliens were never able to leave but managed to salvage enough materials from their ship to form a small colony. The population of the colony over time is given in the table below.

|Years since crash | 50 |100 |150 |200 |250 |300 |350 |400 |450 |

|Population |137 |162 |187 |212 |236 |259 |272 |254 |150 |

a. Find the average rate of change of the colony’s population between the years 50 and 300.

b. Write a linear equation to describe the alien population for the years 50 to 300.

c. Using your equation from part (b), estimate the number of aliens that survived the crash of the ship.

9. The graph of Isabugga’s walk from today’s notes is shown at right.

a. What do we mean by Izzy’s instantaneous velocity?

b. At time t = 10, was Izzy’s velocity greater than or less than 25 cm/min? How do you know?

c. At time t = 21.333, was Izzy’s velocity greater than or less than 25 cm/min?

d. At time t = 29, was Izzy’s velocity greater than or less than 25 cm/min?

e. Was Izzy traveling faster at time t = 12.5 or t = 17.5?

(This assignment continued on the next page.)

10. When a model rocket is launched, it burns propellant for a brief time, accelerating it upward. Several seconds after burnout, a parachute opens to slow the rocket’s fall. The graph of the rocket’s velocity as a function of time is shown at right.

a. How long did the rocket engine burn?

b. What was the rocket’s average acceleration (rate of change of velocity) during the burn stage?

c. Describe what happened between burnout and the opening of the

parachute.

d. What was the rocket’s average acceleration during the time between burnout and the opening of the parachute? Is this what you would expect? Why?

e. When did the rocket reach its maximum height above the ground?

11. On a quiz, Kenny needed to solve the equation x3 + 4x = 7. He tried to do it algebraically. What happened?

AP Calculus HW: Derivative Concepts - 2

1. Use your calculator to approximate the slope of [pic] at the point x = 2 and write an equation of the tangent line to the graph of f at that point.

2. A certain function f has f(4) = 3 and the slope of the tangent line to f at x = 4 is –1.25.

a. Write the equation of the tangent line to the graph of f at x = 4.

b. Use the tangent line equation to estimate the value of f(3.7).

3. The graph at right shows the position of an object moving along the x-axis as a function of time. Use the graph to answer the following.

a. Write the points A – F in order from least to greatest velocity.

b. Write the points A – F in order from least to greatest speed.

c Was the velocity increasing or decreasing at A? At C? At F?

d. At what point does the velocity appear to change from increasing to

decreasing? From decreasing to increasing?

4. The graph at right shows the vertical position of an elevator as a function of time.

a. Describe what happens.

b. What was the elevator's instantaneous velocity

at t = 10 seconds?

c. Sketch a velocity versus time graph for the

motion of the elevator for 0 ( t ( 20.

5. The graph at right shows the position versus time curve for a particle moving along a straight line. Use the graph to estimate the following.

a. What is the average velocity of the particle

1) on the interval 0 ( t ( 3?

2) on the interval 2 ( t ( 4?

2) on the interval 4 ( t ( 8?

b. At what times does the particle have instantaneous velocity of zero?

c. When is the instantaneous velocity of the particle

1) a maximum? 2) a minimum?

6. For each of the following graphs of, sketch a rough graph of what the slope of the function should look like.

a. b. c.

7. Kenny was given the graph from problem number 5 above and asked whether the particle’s velocity was greater at time t = 3 or at time t = 5.

a. Kenny first said at t = 3 because the y-value is higher there. What happened?

b. Kenny then said at t = 3 because the graph is steeper there. What happened?

AP Calculus HW: Derivative Concepts - 3

1. Find each derivative without using your calculator; then use nDeriv to check your answer.

a. Find f '(2) for ƒ(x) = x3. b. Find f '(3) for ƒ(x) = [pic]. c. Find f '(4) for ƒ(x) = [pic].

2. Each limit represents the derivative of some function f at some number a. Find the function f and the number a for each of the following:

a. [pic] b. [pic] c. [pic] d. [pic]

3. A bungee jumper jumps off a 110 foot tower and bounces at the end of her tether. Her height is approximated by h = 50 + 60 e–0.2tcos (1.257 t) where h is in feet and t is time in seconds since she jumped.

a. Find the jumper's velocity one second after she jumps. (Do it the easy way!)

b. Find her velocity three seconds after she jumps.

c. What do the signs in parts a and b indicate?

4. a. For the function ƒ(x) = x2 – 4x, find the derivative at an arbitrary point x = a; in other words, find [pic].

b. Replace a by x in your answer from part a above and graph both ƒ(x) and [pic] on the same set of axes.

c. Is [pic] a function of x? How do you know?

d. If [pic] represents the slope of ƒ, does the graph of [pic] "agree" with the graph of ƒ?

5. Kenny couldn’t believe his teacher was serious about memorizing the formulas for the derivative so he didn’t bother. What happened?

AP Calculus HW: Derivative Concepts - 4

1. Match the graph of each function (a) – (d) with the graph of its derivative (1) – (4) below.

(a) (b) (c) (d)

(1) (2) (3) (4)

2. For each of the functions shown, sketch a graph of the derivative.

a. b. c. d.

3. Find the derivative of the given function using the definition of the derivative.

a. [pic] b. [pic]

4. Recall the three general ways a function can be discontinuous. At a point of discontinuity, can a function have a derivative? Explain or illustrate.

5. Graph the function ƒ(x) = |x – 2| + 1.

a. Is the function continuous at x = 2?

b. Does the function have a derivative at x = 2? Explain.

6. Graph the function ƒ(x) = . Does it have a derivative at x = 2? Explain. Hint: it may help to zoom in several times on the point (2, 0).

7. Kenny was asked about the derivative of the function shown at right. Kenny said the derivative was negative and increasing. What happened?

AP Calculus HW: Derivative Concepts - 5

1. Briefly (no more than two words each) identify two ways a continuous function could fail to be differentiable at a point. Illustrate each with a diagram.

2. A function f is graphed at right. Note that f has a horizontal

tangent at x = 7 and a vertical tangent at x = 9. Name the

x-values where

a. f is discontinuous.

b. f is not differentiable.

3. Consider the function [pic].

Is f continuous at x = 0? Justify your answer.

4. Consider the function [pic].

Is f continuous at x = 0? Justify your answer.

5. Read Increasing and Decreasing Functions and Derivatives which is at the end of the HW packet. (If you chose not to read this, try not to ask questions next class that make it appear that you are lazy and/or illiterate.) Then, as a preview of coming attractions, answer the following.

The function [pic] is continuous for all x. f '(x) < 0 on (((, (2) and (2, () and f '(x) > 0 on

((2, 2).

a. On what interval(s) is f increasing? Justify your answer.

b. On what interval(s) is f decreasing? Justify your answer.

6. Kenny was given a function g that was continuous at the point x = 7. Kenny concluded that g'(7) must exist. What happened?

AP Calculus HW: Derivative Concepts – 6

1. The derivative f ' of a function f , with domain [0, 10] is graphed at right. Use the graph to answer the following:

a. On what intervals is f increasing? decreasing?

b. Where is f increasing the fastest? decreasing the fastest?

c. At what points is f stationary?

d. Classify the stationary points as relative maxima, relative

minima, or neither.

(This assignment continued on the next page.)

2. The derivative [pic] of a function g with domain [0, 8] is graphed at right. From the graph, determine

a. On what interval(s) is g increasing? Decreasing?

b. Which is greater g(0) or g(1)? Explain

c. Which is greater, g(3) or g(6)? Explain.

d. Where does g have a relative minimum? Explain.

e. Where does g have a relative maximum? Explain.

f. Find an upper and lower bound for the difference g(5) – g(4).

3. The derivative of a continuous function f is given by [pic].

a. Find the critical points of f.

b. On what interval(s) is f increasing? decreasing? Justify your answers.

4. The derivative of a continuous function f is given by f '(x) = x3 – x2 – 6x.

a. Find the critical points of f.

b. On what interval(s) is f increasing? Decreasing? Justify your answers.

5. For each derivative graphed below draw a possible graph of the original function ƒ.

a. b. c.

6. Sketch a graph of the derivative of each of the following functions.

a. b. c.

7. Kenny kept mixing up the words increasing and positive. Afterward, he’d tell the teacher “You knew what I meant.” What happened?

AP Calculus HW: Derivative Concepts – 7

1. For each function graphed below, give the signs of both the first and second derivatives.

a. b. c.

d. e. f.

2. The graph of the derivative [pic] of a function ƒ is given at right. From the graph, determine on what interval(s) the graph of ƒ is concave up and on what interval(s) the graph is concave down.

3. The graph at left represents the

second derivative of a function g.

From the graph, determine

a. On what interval(s) is the graph of

g concave up? concave down?

b. Where does the graph of g have points

of inflection?

4. In the diagrams a – d below represent the graphs of four different functions while (1) – (4) represent the second derivatives of those functions. Match each graph with its correct second derivative.

a. b. c. d.

(1) (2) (3) (4)

(This assignment continued on the next page.)

5. A continuous function f has first derivative [pic] and second derivative [pic].

a. Determine the intervals on which f is increasing and decreasing and find and classify the relative extrema of f.

b. Determine the intervals on which f is concave up and concave down and find the inflection points of f.

6. a. Sketch a graph of a function that is concave up and draw the tangent line to the graph at an arbitrary point.

b. If the tangent line is used to estimate function values near the point of tangency, will the estimates be over or under estimates?

c. Does the answer to part (b) above depend on whether the function is increasing or decreasing?

d. If the graph of a function is concave down and the tangent line is used to estimate function values near the point of tangency, will the estimates be over or under estimates?

7. You may already know this from Physics. If you do, great. If you don’t, learn it.

Velocity is a vector; it has a size and a direction. For motion along a line, this means that velocity is a signed quantity. v = 2 m/s means an object is moving 2 meters per second to the right (or up); v = –2 m/s means the object is moving 2 meters per second to the left (or down). Velocity is the rate of change (derivative) of position.

Speed is a scalar; it has (non-negative) size but no direction. A speed of 2 m/s means an object is moving 2 meters per second but does not tell which way. Speed = |v|.

For each of the position – time graphs shown below, tell

1) whether the velocity is increasing or decreasing at the point indicated on the graph and

2) whether the speed is increasing or decreasing at the same point.

a. b. c. d.

8. Kenny was asked what he could say about the slope and the derivative of the function shown. He said “The slope is increasing because the derivative is positive.”

What happened?

AP Calculus HW: Derivative Concepts - 8

1. Match each of the 12 graphs in a - l with the correct graph A - L of its derivative.

a. b. c. d.

e. f. g. h.

i. j. k. l.

A. B. C. D.

E. F. G. H.

I. J. K. L.

(This assignment continued on the next page.)

AP Calculus HW: Derivative Concepts - 8 (continued)

2. The graph of a function ƒ is shown. Sketch the graphs of both ƒ' and ƒ".

a. b.

3. The graphs below show the position, velocity, and acceleration of a car as a function of time in minutes. Identify which is which.

Note: velocity is the rate of change of position and acceleration is the rate of change of velocity.

4. The graph at right shows the derivative of a function f. Based on the graph,

a. On what intervals is ƒ increasing?

b. On what intervals is ƒ concave up?

c. Where does ƒ have a relative maximum? relative minimum?

d. Which is greater, ƒ(2) or ƒ(3)? Explain.

e. If f(2) = 8, between what two integers is the value of f(3)?

Explain.

f. If ƒ(–3) = 5, between what two integers is the value of ƒ(–2)? Explain.

5. Kenny was given the graph of the derivative of a function, f '. Unfortunately, he didn’t pay attention and thought it was a graph of the function f. What happened?

AP Calculus HW: Derivative Concepts - 9

1. A high performance sports car accelerates from 0 to 60 mph in 5 seconds. Its velocity (in ft/s) is given at one-second intervals in the table below.

| |Time (s) |0 |1 |2 |3 |4 |5 |

| |Velocity (ft/s) |0 |30 |52 |68 |80 |88 |

Estimate the acceleration of the car at a. t = 2, b. t = 0, and c. t = 5 seconds.

2. The table below gives the life expectancy E of men born in the US during certain years t.

| |Year (t) |1900 |1910 |1920 |1930 |1940 |

|f(x) |0.171 |0.288 |0.357 |0.384 |0.375 |0.366 |

The best approximation of [pic] according to this table is

(A) 0.12 (B) 1.08 (C) 1.17 (D) 1.77 (E) 2.88

Use the graph of y = f(x) shown at right to answer questions 15 and 16.

10. [pic] is most closely approximated by

(A) 0.3

(B) 0.8

(C) 1.5

(D) 1.8

(E) 2

11. The rate of change of f is least at x (

(A) –3.0

(B) –1.3

(C) 0.5

(D) 2.7

(E) 4.0

(This assignment continued on the next page.)

12. A function f has the derivative shown at right. Which of the following statements must be false?

(A) f is continuous at x = a

(B) f(a) = 0

(C) f has a vertical asymptote at x = a

(D) f has a jump discontinuity at x = a

(E) f has a removable discontinuity at x = a

13. The function f whose graph is shown at right has

f ' = 0 at x =

(A) 2 only

(B) 2 and 5

(C) 4 and 7

(D) 2, 4 and 7

(E) 2, 4, 5, and 7

14. A differentiable function f has the values shown. Which of the following is the best estimate of f '(1.5)?

|x |1.0 |1.2 |1.4 |1.6 |

|f(x) |8 |10 |14 |22 |

(A) 8 (B) 12 (C) 18 (D) 40 (E) 80

15. Kenny believed that because multiple choice problems did not require work to be shown, it meant that no work had to be done (all problems could be done mentally). What happened?

(A) Kenny died horribly.

(B) Kenny died horribly.

(C) Kenny died horribly.

(D) Kenny died horribly.

(E) All of the above.

AP Calculus Review: Derivative Concepts

1. a. Write both forms of the definition of a derivative at a point a.

b. Give two interpretations of the derivative of a function at a point a.

c. Give two reasons why a continuous function might fail to have a derivative at a point a.

2. Each of the limits below represents ƒ'(a) for some function f and some number a. Find ƒ(x) and a for each.

a. [pic] b. [pic] c. [pic] d. [pic]

3. A function f is graphed at right. F has a vertical tangent at x = 2 and horizontal tangents at x = –3, x = 1 and x = 3.

a. Where does f have discontinuities?

b. At what points does f fail to be differentiable?

c. Make a sketch of the derivative of f.

4. Sketch the graph of differentiable functions ƒ, g and h that satisfy the given conditions:

a. ƒ(0) = 1, [pic], [pic] for x < 0,

and [pic] for x > 0.

b. g(x) > 0 for all x and [pic] for all x. c. h(x) > 0, [pic], and [pic] for all x > 0.

5. $1000 is compounded continuously at a rate of r%. After 10 years the balance is B where B = ƒ(r).

a. What does the statement ƒ(5) = 1649 mean in the context of the problem?

b. What does the statement [pic] mean in the context of the problem?

c. What are the units of [pic]?

6. a. How is the derivative of f related to intervals where f is increasing? Is this a biconditional?

b. How can we find a relative maximum using the derivative?

c. What does concave up mean? How are derivatives related to intervals where f is concave up?

d. How can we find inflection points of f?

7. The graph of the derivative of a continuous function f is shown at right.

a. Find and classify the relative extrema of f.

b. Where does f have inflection points?

c. On what interval(s) is f increasing?

d. On what interval(s) is f concave up?

e. What happens to f at x = –1?

f. Which is greater: f(2) or f(3)? Why?

g. Which is greater: f(–5) or f(–4)? Why?

h. If f(4) = 6, estimate f(5) and f(3).

i. If f(1) = –1, estimate f(3).

(This assignment continued on the next page.)

8. A rocket powered car is being tested in the California desert. Starting at rest, the car accelerates to the velocities shown in the table where t is in seconds since the car started and v is in meters per second.

|t (seconds) |1 |2 |3 |5 |10 |20 |30 |40 |

|v (meters per second) |21 |68 |135 |186 |214 |226 |228 |230 |

a. Estimate the car’s acceleration at time t = 2 seconds. Give units.

b. Estimate the car’s acceleration at time t = 40 seconds.

c. Write a linear equation to estimate the car’s velocity for times t > 40 seconds if it continues to accelerate at the same rate.

d. Suppose the car fails catastrophically (it basically flies apart) at velocities greater than 250 m/s. What is the last time when the driver can safely eject?

e. Estimate how many meters the car travels in the interval 30 ( t ( 40 seconds.

9. When a skydiver jumps out of a plane, two main forces are affecting her speed. Gravity tries to speed her up, while air resistance tries to slow her down. The effect of gravity is (roughly) constant while the effect of air resistance increases at greater speeds. Suppose v(t) represents the skydivers speed in feet per second (measured downward so v > 0) as a function of time t in seconds since she left the plane.

a. What is the sign of v' ? Justify your answer.

b. If v(5) = 150 and |v'(5)| = 12, estimate v(7).

b. What is the sign of v" ? Justify your answer.

c. Is your estimate of v(7) from part (b) an overestimate or an underestimate? Justify your answer.

10. A driver is accelerating on the highway. His speed in feet per second at three different times (in seconds) is given by v(4) = 82.400, v(4.01) = 82.423 and v(7) = 88.250.

a. What was his average acceleration on the interval 4 ( t ( 7 seconds? Include units.

b. Estimate his instantaneous acceleration at t = 4.

c. If v" does not change sign during the interval 4 ( t ( 7, what is its sign? Justify your answer.

11. Kenny didn’t really understand a lot of the stuff in this unit but figured he could afford one poor test grade and the course would move on to other topics. What happened?

Answers to selected problems

HW - 1

1a. [pic] b. The slope of the secant line from [pic] to [pic] 2. –2/3

3. 11 4. [pic] 5. 2.6 6. a, e and f are true 7. f(a) = f(b)

8a. 0.488 aliens/yr b. P = 0.488t + 112.6 c. 112 or 113

9a. Izzy’s instantaneous velocity is her velocity at a particular instant in time. Izzy’s instantaneous velocity (usually just called velocity for brevity) is the slope of her position-time graph at a particular time. We’ll talk more next class about just we mean by the slope of a curve. b. Greater. At t = 10, Izzy’s position graph is increasing faster than Buford’s; it has a greater slope. c. Less. d. Greater. e. 12.5

10a. 2 sec b. 95 ft/s2 c. See solutions d. –31.8 ft/s2 e. 8 sec

HW – 2

1. m = 0.5; y ( 1 = 0.5(x ( 2) 2a. y – 3 = –1.25(x – 4) b. 3.375

3a. E, D, F, A, C, B b. F, A, C, D, B, E (possibly open to debate)

c. Increasing, decreasing, increasing d. B and E, respectively

4a. Elevator starts at rest for about 2.5 seconds, then rises at a constant speed for

15 seconds, then returns to rest. b. 4 m/s c. See graph at right.

5a1. 0 cm/s a2. -5 cm/s a3. 1.25 cm/s b. At t = 0, 2, 4, and 8 s.

c1. t ( 1 s c2. t ( 3 s

6a. b. c.

.

7. Kenny died horribly. Twice in one day.

HW – 3

1a. 12 b. 1/4 c. –1/2

2a. [pic], a = 1 b. [pic], a = 2 c. [pic], a = 1 d. [pic], a = 0

3a. –61.766 ft/s b. 29.689 ft/s c. Going down at t = 1, up at t = 3

4a. [pic]

HW – 4

1a. (2) b. (4) c. (1) d. (3)

2 a. b. c. d.

3a. [pic] b. [pic]

4. No. 5a. Yes b. No, there is not a unique tangent line at x = 2

6. No, the tangent line is vertical so its slope is undefined

HW – 5

1. A vertical tangent or a “corner.”

2a. x = 3, x = 12, x = 14 b. x = 1, x = 3, x = 5, x = 9, x = 12, x = 14

3. No because [pic] does not exist.

4. Yes. [pic].

5a. “f is increasing on (((, (2) and (2, () because f ' > 0 on those intervals” is acceptable for the AP test. “f is increasing on (((, (2] and [2, () because f ' > 0 on (((, (2) and (2, ()” is a better answer (if a little longer).

b. “f is decreasing on ((2, 2) because f ' < 0 on that interval” is acceptable. “f is decreasing on [(2, 2] because f ' < 0 on ((2, 2)” is better.

HW – 6

1a. Increasing on [0, 5.5] and [9, 10]; decreasing on [5.5, 9]

b. Increasing fastest at x = 4; decreasing fastest at x = 7 c. x = 1, 5.5, and 9

d. x = 1: neither; x = 5.5: relative max; x = 9: relative min

2a. Increasing on [3, 6], decreasing on [0, 3] and [6, 8] b. g(0) c. g(6) d. x = 3

e. x = 6 f. [pic]

3a. x = 0, x = 5 b. f increasing on [0, 5] b/c f ' > 0 on (0, 5); f decreasing on (–(, 0] and [5, () b/c f ' < 0

on (–(, 0) and (5, ()

4a. x = –2, x = 0 and x = 3 b. f decreasing on (–(, –2] and [0, 3] b/c f ' < 0 on (–(, –2) and (3, (); f increasing on [–2, 0] and [3, () b/c f ' > 0on (–2, 0) and (3, ()

5a. b. c.

6. a. b. c.

7. For once, Kenny didn’t die. But he kept losing lots of points on his AP Calc quizzes and tests.

HW – 7

1a. +, + b. 0, 0 c. –, 0 d. –, + e. +, – f. –, –

2. Concave up on [–2, 2] and down on [–5, –2] and [2, 5]

3a. Concave up on [3, 6] and down on [0, 3] and [6, 8] b. IPs at x = 3 and x = 6

4a. (4) b. (3) c. (1) d. (2)

5a. f is decreasing on (–∞, 6] because f ' < 0 on (–∞, 6).

f is increasing on [6, ∞) because f ' > 0 on (6, 8) and (8, ∞).

f has a relative minimum at x = 6 b/c f ' changes from negative to positive there.

b. f is concave up on (–∞, 8) and (12, ∞) because f " > 0 on those intervals.

f is concave down on (8, 12) because f " < 0 on that interval.

f has inflection points at x = 8 and x = 12 because f " changes sign at those points.

6a. See graph at right b. Under. c. No d. Over

7a. Both decreasing b. both increasing c. v incr, speed decr d. v decr, speed incr

HW – 8

1a. E b. K c. D d. G e. F f. J g. C h. L i. A j. I k. B l. H

2a. b

.

3. C is position, A is velocity and B is acceleration

4a. [–1, 6] b. [–3, 2], [4, 5] c. Relative max at x = 6; relative min at x = –1

d. f(3) since f ' > 0 on (2, 3) e. 10 < f(3) ................
................

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