Lawrence Woodmere Academy

AP Calculus BC Dear AP Calculus BC Student, Congratulations on passing AP Calculus AB and moving forward into AP Calculus BC. In the upcoming school year, we will be using the concepts that you previously learned in AB Calculus to expand your knowledge into the world Calculus by exploring parametric functions, polar functions, vector functions, applications of integrals, concepts of series, series of constants, and Taylor Series. To be able to move forward in BC Calculus, the basic knowledge of limits and derivatives from AB Calculus is crucial. This summer assignment is designed to allow you to continue to practice these skills and concepts throughout the time that school is not in session. The AP Calculus BC Summer Assignment packet will not require a lot of time, but it is lengthy enough that you will want to manage your time appropriately. The whole assignment should not be completed at the end of this school year, but should be worked on all summer to keep the material fresh in your mind. As AP Calculus BC students, you will need to be able to manage your time appropriately. This summer assignment is composed of two sections that review the old material from AP Calculus AB. The first part is a review of limits using both an analytical and graphical approach and the second part consists of a review of derivatives and applications of derivatives. You are expected to answer all questions on a separate sheet of paper and hand in the assignment on the first day of school. All work must be shown for each of the questions and you must provide adequate explanations for all multiple choice questions. It’s not sufficient enough to get the right answer, but you must be able to explain your answer as well. The assignment will be graded for completion and effort. You should also get the required supplies for the course which includes graph paper, notebook, pencil, and a TI-83 or TI-84. You may also want to go to a bookstore this summer and pick up an AP preparation guide for the AP exam (I recommend Baron’s).

If you have any questions, do not hesitate to e-mail me over the summer at dearley@lawrencewoodmere.org Again, welcome to AP Calculus BC!

Good luck and I look forward to seeing you in September. Sincerely, Mrs. Danielle Earley dearley@lawrencewoodmere.org

PART I - LIMITS:

1. Answer the following questions using the graph of f(x) given below:

(a) Find f(0)

(b) Find f(3)

(c) Find lim𝑥→−5

𝑓(𝑥) (d) Find lim𝑥→0+

𝑓(𝑥)

(e) Find lim

𝑥→3−𝑓(𝑥) (f) Find lim

𝑥→−3+𝑓(𝑥)

(g) List all x-values for which f(x) has a removable discontinuity. Explain what section(s) of the definition of continuity is (are) violated at these points.

(h) List all x-values for which f(x) has a nonremovable discontinuity. Explain what section(s) of the definition of continuity is (are) violated at theses points.

2. In the following problems, find the limit (if it exists) using analytic methods. (i.e. without using a

calculator).

(a) lim𝑥→−2

3𝑥2+21𝑥+30

𝑥3+8 (b) lim

𝑥→𝜋

6

1−𝑐𝑜𝑠2𝑥

4𝑥

(c) lim𝑥→4

√𝑥−3−1

𝑥−4 (d) lim

𝑥→0

[1

𝑥+1]−1

𝑥

(e) lim𝑥→0

[1

√𝑥+1]−1

𝑥 (f) lim

𝜃→0

𝑠𝑖𝑛6𝜃3

7𝜃

(g) lim𝑡→0

𝑠𝑖𝑛23𝑡2

𝑡3 (h) lim𝑥→6−

|6𝑥−36|

6−𝑥

(i) lim∆𝑥→0

sin((𝜋

6)+∆𝑥)−(

1

2)

∆𝑥 𝐻𝑖𝑛𝑡: 𝑠𝑖𝑛(𝐴 + 𝐵) = 𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐵 + 𝑐𝑜𝑠𝐴𝑠𝑖𝑛𝐵

3. Suppose 𝑓(𝑥) = {√2𝑥+1−√3

𝑥−1, 𝑥 ≥ 0

4𝑥2 + 𝑘, 𝑥 < 0

a. For what value of k will f be piecewise continuous at x = 0? Explain why this is true using one-sided limits. Hint: A function is continuous at x = c if: (i) f(c) exists (ii) lim

𝑥→𝑐𝑓(𝑥) exists and

(iii) lim𝑥→𝑐

𝑓(𝑥) = 𝑓(𝑐)

b. Using the value of k that you found in part (a), accurately graph f below. (i) Approximate the value of lim

𝑥→1𝑓(𝑥)

lim𝑥→1

𝑓(𝑥) = _________________

(c) Rationalize the numerator to find lim

𝑥→1𝑓(𝑥) analytically.

4. Analytically determine the vales of b and c such that the function of f is continuous on the entire real

number line. See the hint given in problem 3.

𝑓(𝑥) = {𝑥 + 1, 1 < 𝑥 < 3

𝑥2 + 𝑏𝑥 + 𝑐, 𝑥 < 1 𝑜𝑟 𝑥 > 3

5. Circle the correct answer and explain why the answer is the correct one. If 𝑓(𝑥) = 𝑥3 + 𝑥 − 3, and if c is the only real number such that f(c) = 0, then by the Intermediate Value Theorem, c is necessarily between (A) -2 and -1 (B) -1 and 0 (C) 0 and 1 (D) 1 and 2 (E) 2 and 3

6. Explain why each function is discontinuous and determine if the discontinuity is removable or nonremovable.

a) 2 3, 3

( )5, 3

x xg x

x x

b)

2

(3 1)( )

3 5 2

x xb x

x x

c)

2 10 25( )

5

x xh x

x

PART II - DERIVATIVES:

1. In the following problems, find the derivative of the given function by using the limit definition of the derivative. (a) 𝑓(𝑥) = 𝑥3 − 2𝑥 + 3 (b) 𝑓(𝑥) =

𝑥+1

𝑥−1

2. In the following problems, find the derivative of the given function using the power, product, quotient, and/or chain rules. (a) 𝑓(𝑥) = (3𝑥2 + 7)(𝑥2 − 2𝑥 + 3) (b) 𝑓(𝑥) = √𝑥 𝑠𝑖𝑛𝑥

(c) 𝑓(𝑥) = 𝑡3 cos(𝑡) (d) 𝑓(𝑥) =

𝑥2+𝑥−1

𝑥2−1

(e) 𝑓(𝑥) =𝑥4+𝑥

𝑡𝑎𝑛2𝑥 (f) 𝑓(𝑥) = 3𝑥2 sec3(𝑥)

(g) 𝑓(𝑥) = 3𝑥 csc(𝑥) + 𝑥 cot (𝑥)

(h) 𝑓(𝑥) = (𝑥+5

𝑥2−6𝑥)

2

(i) 𝑓(𝑥) = (𝑥3 − 2)3/2(5𝑥2 + 1)5/2 (j) 𝑓(𝑥) = 𝑥3 cot4(7𝑥)

(k) 𝑓(𝑥) = 5𝑠𝑖𝑛2(√3𝑥4 + 1)

Hint: The Intermediate Value Theorem states that if

f is a continuous function on the interval [a,b] and k

is any number between f(a) and f(b), then there

must exist at least one number c∈[a,b] such that

f(c) = k.

3. In the following problems, find an equation of the tangent line to the graph of f at the indicated point P.

(a) 𝑓(𝑥) =1+cos 𝑥

1−cos 𝑥, 𝑃 (

𝜋

2, 1) (b) 𝑓(𝑥) = (𝑥2 − 1)2/3 , 𝑃(3,4)

4. In the following problems, find the second derivative of the given function. (a) 𝑓(𝑥) = (4𝑥2 − 3𝑥)3/2 (b) ℎ(𝑥) = 𝑥3cos (𝜋𝑥)

5. Use the position function 𝑠(𝑡) = −16𝑡2 + 𝑣𝑜𝑡 + 𝑠𝑜 for free-falling objects. A ball is thrown straight down from the top of a 220-foot tall building with an initial velocity of -22 feet per second.

(a) Determine the average velocity of the ball on the interval [1,2]. (b) Determine the instantaneous velocity of the ball at t = 3. (c) Determine the time t at which the average velocity on [0,2] equals the instantaneous velocity. (d) What is the velocity of the ball when it strikes the ground?

6. If 5232 xyxy , then dx

dy

7. Circle the correct answer and explain why the answer is the correct one.

lim𝑛→0

𝑐𝑜𝑠 (𝜋6 + ℎ) − 𝑐𝑜𝑠 (

𝜋6)

ℎ=

(A) Does not exist

(B) 1

2

(C) −1

2

(D) √3

2

(E) −√3

2

8. Circle the correct answer and explain why the answer is the correct one.

Let f and g be differential functions with values for f(x), g(x), f’(x), and g’(x) shown below at x = 1 and x = 2.

x f(x) g(x) f’(x) g’(x) 1 4 -4 12 -8 2 5 1 -6 4

Find the value of the derivative of 𝑓(𝑥) ∙ 𝑔(𝑥) at x = 1.

(A) −96 (B) −80 (C) −48 (D) −32 (E) 0

9. Using the table above in question 8, Find the value of

g

f

dx

d at x = 1 is

10. Circle the correct answer and explain why the answer is the correct one.

Let 𝑓(𝑥) = {3𝑥2 + 4𝑥, 𝑥 < 1

𝑥3 + 3𝑥, 𝑥 ≥ 3 Which of the following is true?

I. f(x) is continuous at x = 1 II. f(x) is differentiable at x = 1 III. lim

𝑥→1−𝑓(𝑥) = lim

𝑥→1+𝑓(𝑥)

(A) I only (B) II only (C) III only (D) I and III only (E) II and III only

11. Circle the correct answer and explain why the answer is the correct one.

The equation of the line tangent to the curve at 𝑓(𝑥) =𝑘𝑥+8

𝑘+8 at x = -2 is 𝑦 = 𝑥 + 4. What is the value

of k? (A) - 3 (B) - 1 (C) 1 (D) 3 (E) 4

12. Circle the correct answer and explain why the answer is the correct one.

An equation of the line normal to the curve 𝑦 = √𝑥2 − 13

at the point where x = 3 is (A) 𝑦 + 12𝑥 = 36 (B) 𝑦 − 4𝑥 = 10 (C) 𝑦 + 2𝑥 = 4 (D) 𝑦 + 2𝑥 = 8 (E) 𝑦 − 2𝑥 = −4

13. Circle the correct answer and explain why the answer is the correct one.

If the nth derivative of y is denotes as 𝑦(𝑛) and 𝑦 = −𝑠𝑖𝑛𝑥, then 𝑦(14) is the same as (A) 𝑦

(B) 𝑑𝑦

𝑑𝑥

(C) 𝑑2𝑦

𝑑𝑥2

(D) 𝑑3𝑦

𝑑𝑥3

(E) 𝑁𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒

Hint: A normal line to a curve at a point is

perpendicular to the tangent line to the

curve at the same point.

14. For a, b and c, find all critical values, intervals of increasing and decreasing, any local extrema, points of inflection, and all intervals where the graph is concave up and concave down.

a) 2

445)(

32

x

xxxxf

b) 2623 23 xxxy

c) 7155)( 3 xxxf

15. The graph of the function xxxy sin25 changes concavity at x =

16. Find the equation of the line tangent to the function 74y x at 16x .

17. For what value of x is the slope of the tangent line to 7 3y x

x undefined?

PART IV: INTEGRALS – For 1 – 7, evaluate each of the following integrals:

1.

1

83

2

2dx

x

xx

2.

6/

6/

2sec

xdx

3. x

dttdx

d

1

4

4. 0

)4sin( x

t dtedx

d

5.

dxx

x

4

3

1

6. dxx

x3

2

cot

csc

7. xdxx 2sectan

8. What are all the values of k for which 02

5 k

dxx ?

9. What is the average value of 943 xxy on the interval [0, 2]?

10. If

b

a

badxxg 4)( , then

b

a

dxxg ]7)([

11. The function f is continuous on the closed interval [1, 9] and has the values given in the table. Using the subintervals [1, 3], [3, 6], and [6, 9], what is the value of the trapezoidal approximation of

9

1

)( dxxf ?

12. The table below provides data points for the continuous function )(xhy .

Use a right Riemann sum with 5 subdivisions to approximate the area under the curve of )(xhy on

the interval [0, 10].

13. A particle moves along the x-axis so that, at any time 0t , its acceleration is given by 66)( tta .

At time 0t , the velocity of the particle is -9, and its position is -27.

(a) Find )(tv , the velocity of the particle at any time 0t .

(b) For what values of 0t is the particle moving to the right? (c) Find )(tx , the position of the particle at any time 0t .

14. For a and b, find the general solution to the given differential equation.

a) 3

2

dy y

dx x

b) sindy

y xdx

15. Find the particular solution to the differential equation 2sindu

uv vdv

if (0) 1u .

16. The shaded regions, R1 and R2 shown above are enclosed by the graphs of 2)( xxf and xxg 2)( .

(a) Find the x- and y-coordinates of the three points of

intersection of the graphs of f and g. (b) Without using absolute value, set up an expression

involving one or more integrals that gives the total area enclosed by the graphs of f and g. Do not evaluate.

(c) Without using absolute value, set up an expression involving one or more integrals that gives the volume of the solid generated by revolving the region R1 about the line y = 5. Do not evaluate.

x 1 3 6 9 f(x) 15 25 40 30

x 0 2 4 6 8 10 h(x) 9 25 30 16 25 32

R1

R2

17. Let R be the region in the first quadrant under the graph of x

y1

for 94 x .

(a) Find the area of R. (b) If the line x = k divides the region R into two regions of equal area, what is the value of k? (c) Find the volume of the solid whose base is the region R and whose cross sections cut by

planes perpendicular to the x-axis are squares.

18. For each of the following, find the area described. • Be sure to sketch the picture described. Draw the functions for the given domain • Shade the region enclosed by the functions described. • Show how you calculated the total area described.

a) The area enclosed by the following lines:

• 2y x

• 3y

• 2 8y x

• The horizontal x-axis ( 0y )

b) The area enclosed by the functions

• 216y x

• The horizontal x-axis ( 0y )

c) Use 2 rectangles to estimate the area enclosed by the graphs of:

• 2xy

• 0x

• 3x

• The x-axis: 0y

Give an explanation for your estimate and tell me if the estimate is TOO HIGH or TOO LOW. (Hint: draw rectangles inside or outside of your picture and use a calculator to help you find the areas of the rectangles)

PART V : OTHER TOPICS –

1. Write the following function as the composition of TWO functions: (decompose into two functions)

Ex: 2

3( )

5h x

x x

is the composition:

( ) ( ( ))h x f g x where: 3

( )f xx

, and 2( ) 5g x x x

Let f(x) be the “big picture) and g(x) be the details within

a) 2 4( ) ( 5 6)h x x x

b) 2( ) sin( 6 1)h x x x

c) 2

6( )

(3 5)h x

x

2. Add/Subtract or Multiply/Divide the following:

a) 3 5

4 2x x

b) 2

2 5

( 3) 6 9x x x

c) 2 1

5 2

d)

3

24

93

2

x

e) 21

32x x

f) 2 3( )x x x x

g) 2xe e

h) Simplify as a single product or sum: 310k

3. Solve for y:

a) ln 2y x

b) 3 5 2 7xy y x

c) 2 23 18 2 5x y xy x

4. The balloon shown is in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder. The balloon is being inflated at the rate of 261 cubic centimeters per minute. At the instant the radius of the cylinder is 3 centimeters, the volume of the balloon is 144 cubic centimeters and the radius of the cylinder is increasing at the rate of 2 centimeters per minute. (The

volume of a cylinder with radius r and height h is 2r h , and the volume of a sphere with radius r is

3

3

4r .)

(a) At this instant, what is the height of the cylinder? (b) At this instant, how fast is the height of the cylinder increasing?

5. Solve each problem on a separate sheet of paper as if they are open ended AP problems. This means

you must include all justifications necessary as on the AP AB exam. PLEASE BE NEAT!! You have the summer to work on these and I am looking for a neat, clean, clear copy of your solution! Problems (a-c) are with calculator and (d-f) are without calculator.

a.

b.

c.

d.

e.

f.

6. Let f be the function defined by 22cos

sin)(

xfor

x

xxxf .

a. State whether f is an even function or an odd function. Justify your answer.. b. Find ).(xf

c. Write an equation for the line tangent to the graph of f at the point )).0(,0( f

7. Graphs/Properties of Functions: You may want to organize the following in a way you can save and use for a reference for the year. This section does not need to be IN THE COMPOSITION BOOK if you choose, but it can be. It must still be turned in. Fill in the following table AND GRAPH each of the following

The Function Domain Range Type of Symmetry: x-axis, y-axis, origin

Odd, Even, Neither

Intervals of Increasing/Decreasing

1. y = 1

2. y = x

3. y = x2

4. y = x3

5. y = x4

6. y = √𝑥

7. 𝑦 = √𝑥3

8. 𝑦 =

1

𝑥

9. 𝑦 =

1

𝑥2

10. y = |x|

11. 𝑦 = ⟦𝑥⟧

12. y = lnx

13. y = ex

14. y = sinx

15. y = cosx

16. y = tanx

8. For a and b, parametric equations are given. Complete the table and sketch the curve represented by

the parametric equations (label the initial and terminal points as well as indicate the direction of the

curve).

a) 4sin ,x t 2cos ,y t 0 2t

b) 2 5,x t 4 7,y t 2 3t

t 0 4

2

34

32

2

x

y

t – 2 – 1 0 1 2 3

x

y

The following Trigonometric Identities MUST be memorized

Reciprocal Identities Quotient Identities Pythagorean Identities

xx

csc

1sin

xx

sin

1csc

xx

sec

1cos

xx

cos

1sec

xx

cot

1tan

xx

tan

1cot

sintan

cos

coscot

sin

xx

x

xx

x

2 2

2 2

2 2

sin cos 1

tan 1 sec

1 cot csc

x x

x x

x x

Co-Function Identities Odd/Even Identities

sin cos2

cos sin

2

csc sec2

sec csc

2

tan cot2

cot tan

2

Odd Even

sin sin cos cos

csc csc sec sec

tan tan

cot cot

Double Angle Identities Half Angle Identities

2 2

2

2

sin 2 2sin cos

cos 2 cos sin

cos 2 2cos 1

cos 2 1 2sin

x x x

x x x

x x

x x

2

2

1 cos 2sin

2

1 cos 2cos

2

xx

xx

The Radian Measures and Coordinates MUST be memorized

Remember: coordinateyr

ysin , coordinatex

r

xcos , and

coordinatex

coordinatey

x

y

tan

ANSWERS

LIMITS:

1. (a) -1 (b) 2 (c) 0 (d) -1 (e) 1 (f) +∞ (g) x = -5,5 (h) x = -3, 0, 3

2. (a) ¾ (b) 3/8𝜋 (c) ½ (d) -1 (e) -1/2 (f) 0 (g) 0 (h) 6

(i) √3/2

3. (a) k = -1 + √3 (b) ≈ .577

(c) 1/√3 4. b = -3, c = 4 5. D

DERIVATIVES:

1. (a) 𝑓′(𝑥) = 3𝑥2 − 2

(b) 𝑓′(𝑥) =−2

(𝑥−1)2

2. (a) 𝑓′(𝑥) = 12𝑥3 − 18𝑥2 + 32𝑥 − 14

(b) 𝑓′(𝑥) = √𝑥 cos 𝑥 +sin 𝑥

2√𝑥

(c) 𝑓′(𝑡) = −𝑡3 sin 𝑡 + 3𝑡2 cos 𝑡

(d) 𝑓′(𝑥) =−𝑥2−1

(𝑥2−1)2

(e) 𝑓′(𝑥) =4𝑥3 tan 𝑥+tan 𝑥−2𝑥4𝑠𝑒𝑐2𝑥−2𝑥 𝑠𝑒𝑐2 𝑥

𝑡𝑎𝑛3𝑥

(f) 𝑓′(𝑥) = 9𝑥2𝑠𝑒𝑐3𝑥 tan 𝑥 + 6𝑥𝑠𝑒𝑐3𝑥 (g) 𝑓′(𝑥) = −3𝑥 csc 𝑥 cot 𝑥 + 3 csc 𝑥 − 𝑥 csc2 𝑥 + cot 𝑥

(h) 𝑓′(𝑥) =(2𝑥+10)(−𝑥2−10𝑥+30)

(𝑥2−6𝑥)3

(i) 𝑓′(𝑥) = 25𝑥[(𝑥3 − 2)(5𝑥2 + 1)]3/2 +9

2𝑥2(5𝑥2 + 1)5/2(𝑥3 −

2)1/2 (j) 𝑓′(𝑥) = −28𝑥3𝑐𝑜𝑡3(7𝑥) csc2(7𝑥) + 3𝑥2𝑐𝑜𝑡4(7𝑥)

(k) 𝑓′(𝑥) =60𝑥3 sin √3𝑥4+1 cos √3𝑥4+1

√3𝑥4+1

3. (a) 𝑦 − 1 = −2(𝑥 −𝜋

3)

(b) 𝑦 − 4 = 2(𝑥 − 3)

4. (a) 𝑓′(𝑥) = 12√4𝑥3 − 3𝑥 +3(8𝑥−3)2

4√4𝑥2−3𝑥

(b) ℎ′(𝑥) = −𝜋2𝑥3 cos 𝜋𝑥 − 6𝜋𝑥2 sin 𝜋𝑥 + 6𝑥 cos 𝜋𝑥 5. (a) -70 ft/s (b) -118 ft/s (c) t = 1 s (d) ≈ −120.688 ft/s 6. C 7. B 8. B 9. D 10. D 11. C