Teaneck Public Schools
OFFICE OF CURRICULUM AND INSTRUCTION 100 Elizabeth Avenue Teaneck, New Jersey 07601
Phone: (201) 833-5093 Fax: (201) 833-5495

AMS 12/AP Calculus BC Summer Assignment

Google Classroom Class Code

j6bf7bd

Please complete and return the first day of class either as a hard copy or electronically via Google Classroom. Please note the following: For the multiple choice questions, please show all work that lead to your selected answer on a separate sheet. You will receive credit for all work correctly shown. Join Google Classroom to receive regular updates over the summer.

Here is a link to the AP Notes from the first three chapters of the Calculus text book:(https:// drive.google.com/drive/folders/1Dr9wuEdB65RqmEF6ks29GgDuB9t0HkUR?
usp=sharing). Please review and answer the questions at the end of each section. Be prepared to discuss these questions in class during the first week of school.

If you have any questions please email me at kchung@teaneckschools.org. Additional references: Previous work from your AMS 10/11 classes

Chapter P: Preparation for Calculus

Name: __________________________	Date: _____________

1. Sketch the graph of the equation: x = y3 – 4y A)

B)

C)

D)

E) none of the others

2. Find the points of intersection of the graphs of the equations:

x = y2 – 27

y = x –3 A) (9,3),(–2,–8) B) (9,6),(–2,–5)

C) (9,12),(–2,–5)

D) (–9,6),(2,–5) E) (9,–6),(2,–5)

3. Find the sales necessary to break even (R =C) if the cost C of producing x units is

C = 5.5 x +50,000 and the revenue R for selling x units is R =3.25x. Round your answer to

the nearest integer.
A) x ≈15,602 units                                              D) x ≈ 7904 units B) x ≈15,638 units                                              E) x ≈ 7910 units

C) x ≈15,596 units

1 2	3 1
⎛	⎞	⎛	⎞
⎜	⎟	⎜	⎟
4. Find the slope of the line passing through the points ⎝− 2, 3 ⎠ and ⎝− 4,6 ⎠. A) 1	B) 2	C) 3	D) –2	E) –1

5. A moving conveyor is built to rise 1 meter for every 7 meters of horizontal change. Suppose the conveyor runs between two floors in a factory. Find the length of the conveyor if the vertical distance between floors is 10 meters. Round your answer to the nearest meter.
A) 70 meters	B) 81 meters	C) 61 meters	D) 59 meters	E) 71 meters

6. Find the y-intercept of the line x+4y = 24.

A) (0,4) B) (0,6) C) (6,0) D) (4,0) E) (0,24)

5 16	2	13
⎛	⎞	⎛	⎞
⎜	⎟	⎜	⎟
3
3
3
3
3
7. Find an equation of the line that passes through the points ⎝ 6, 9 ⎠ and ⎝ 11 ,– 33⎠. A) y =10 x	B) y =10 x –1	C) y =10 x – 2	D) y =10 x + 1	E) y =10 x + 2

8. Write an equation of the line that passes through the point (–7,2) and is perpendicular to the line x+ y = 8.

A) x− y +9 = 0 B) x− y +5 = 0
C) x+ y +9 = 0

D) x+ y −5 = 0

E) x+ y −8 = 0

1
1
9. Find an equation of the line through the points of intersection of y = x2 and y =14x− x2 . A) y =14x	B) y = 7x	C) y = x −14	D) y = −14x	E) y = x+7

1
1
10. A real estate office handles an apartment complex with 50 units. When the rent is $790 per month, all 50 units are occupied. However, when the rent is $835, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent p and the demand x is linear. Write a linear equation giving the demand x in terms of the rent p.

A) x = 15(1585− p) B) x = 45(1540+ p)
C) x = 45(1585− p)

D) x = 15(1495+ p)

1
E) x = 15(1540− p)

1
– 10
x
( )
,
g x
=
11. Let g(x)=	x – 6 . Evaluate the expression g(x)− g(10) and then simplify the result. 1         g(x)− g(10)

x – 6	x – 10

A)

B)

C)

2 x – 6 − x + 6 2(x +10)(x – 6)

2 x – 6 + x +6 2(x – 10)(x – 6)

2 x – 6 − x + 6 2(x – 10)(x – 6)

D)

E)

2 x – 6 + x +6 2(x +10)(x – 6)

undefined

12. Use the graph of y = f (x) given below to find the graph of the function y = f (x)– 5.

A)

B)

C)

D)

E)

π
⎛	⎞
⎜	⎟
13. Specify a sequence of transformations for the function h(x)= sin⎝x+ 8 ⎠+2 that will yield the graph of h from the graph of the function f (x)=sin x.

8
⎛	⎞
⎜	⎟
8
A) The function h(x)= sin⎝x+ π ⎠+2 is a vertical shift π unit downwards, followed by a horizontal shift 2 units to the right.

8
⎛	⎞
⎜	⎟
8
B) The function h(x)= sin⎝x+ π ⎠+2 is a horizontal shift π unit to the left, followed by a vertical shift 2 units upwards.

8
⎛	⎞
⎜	⎟
8
C) The function h(x)= sin⎝x+ π ⎠+2 is a horizontal shift π unit to the right, followed by a vertical shift 2 units downwards.

8
⎛	⎞
⎜	⎟
8
D) The function h(x)= sin⎝x+ π ⎠+2 is a vertical shift π unit upwards, followed by a horizontal shift 2 units to the left.

8
⎛	⎞
⎜	⎟
8
E) The function h(x)= sin⎝x+ π ⎠+2 is a horizontal shift π unit to the left, followed by a horizontal shift 2 units to the right.

14. An open box of maximum volume is to be made from a square piece of material 26 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). Write the volume V as a function of x, the length of the corner squares.

A) V = x2 +(26−2x)	D) V = x 26−2x) B) V = x(26−2x)2	E) V = x2     26−2x) C) V = x+(26− x)2

15. A V8 car engine is coupled to a dynamometer and the horsepower y is measured at different engine speeds x (in thousands of revolutions per minute). The results are shown in the table below. Use the model y = −1.806x3 +14.58x2 +16.4x+30 to approximate the horsepower

when the engine is running at 3500 revolutions per minute. Round your answer to two decimal places.

x	1	2	3	4	5	6 y            60            105           160           220           245          265
A) 193.74 hp	B) 198.80 hp	C) 196.28 hp	D) 188.57 hp	E) 191.17 hp

Chapter 1: Limits and Their Properties

Name: __________________________	Date: _____________

1. Consider the function f (x)= x and the point P(4,2) on the graph of f. Estimate the slope m of the tangent line of f at P(4,2). Round your answer to four decimal places.
A) m = 0.1790	B) m = 0.2500	C) m = 0.1667	D) m = 0.1250	E) m = 0.4633

2
x –
x
x→
2. Complete the table and use the result to estimate the limit. lim x2 – 6 2+8

x	1.9	1.99	1.999	2.001	2.01	2.1 f(x)
A) –0.500000	B) 0.000000	C) –0.125000	D) 0.125000	E) –0.875000

3. Determine the following limit. (Hint: Use the graph of the function.)

x→1
lim(6− x)

A) 7	B) 6	C) 1	D) 5	E) Does not exist

4. Let f (x) = 4 + 4x2 and g(x) = x + 6 . Find the limits:

(a) lim f (x) x→5

(b) lim g(x) x→5

(c) lim g( f (x)) x→2

A) 104, 11, 26	B) 5, 5, 2	C) 8,5, 20	D) 100, 6, 10	E) 8, 5, 26

5. Find the limit:

π
1
1
limsin x x→6

A) − 2	B)

1

2

C)

1

4

D) Does not exist	E) − 4

6. Suppose that lim f (x) = 3 and lim g(x) = 7. Find the following limit: x→c	x→c

→
⎣	⎦
lim⎡ f (x)g(x) ⎤ x     c

A) –4	B) 10	C) 0	D) 21	E) 2187

7. Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point.

lim
x3 +1000 x→–10	x + 10

A) 300 , x2 – 10x + 100 B) 100 , x2 +10x +100
C) 100 , x2 – 10x – 100

D) Limit does not exist.

E) –300 , x2 – 10x + 100

8. Find the limit (if it exists):

2
49
x
−
1
lim – x – 7 x→–7

A) 28	B) –14	C)

1

14

D) –7	E)

1

28

+
−
9. Use the graph as shown to determine the following limits, and discuss the continuity of the function at x = 4.

(i) lim f (x) x→4

(ii) lim f (x) x→4

(iii) lim f (x) x→4

A) 4 , 4, 4, Not continuous B) 2 , 2 , 2 , Not continuous C) 1 , 1 , 1 , Continuous

D) 2 , 2 , 2 , Continuous
E) 1 , 1 , 1 , Not continuous

⎢ ⎥
+
⎣ ⎦
10. Find the limit (if it exists). Note that f (x) = ⎣x⎦ represents the greatest integer function. lim (–3⎢x⎥ –9)
x→–5

A) –6	B) 9	C) Does not exist	D) 6	E) –9

1
11. Find the x-values (if any) at which the function f (x) = 3x2 + x –13 is not continuous. Which of the discontinuitites are removable?

A) Continuous everywhere

B) x = –13. Removable

1
C) x = – 6 . Removable.

D) x = – 6 . Not removable. E) both B and C

−
+
12. Find constants a and b such that the function

⎪
⎧10,	x ≤ –9

⎪
f (x) = ⎨ax+b, –9< x <1 ⎩–10,	x ≥1

is continuous on the entire real line. A) a = 2 , b = 0
B) a = 2 , b = –8 C) a = 2 , b = 8

D) a = –2 , b = 8 E) a = –2 , b = –8

x
2
13. Determine whether f (x) = x2 −100 approaches ∞ or −∞as x approaches –10 from the left and from the right by completing the tables below.

x	–10.5	–10.1 f(x)

x	–9.999	–9.99 f(x)

–10.01	–10.001

–9.9	–9.5

A)	 lim f (x) = ∞ x→–10

−
B)	 lim f (x) = –∞ x→–10

+
C)	 lim f (x) = –∞ x→–10

D)	 lim f (x) = ∞ x→–10

E) Both A and C

F) Both B and D

16
x
−
2
14. Find the vertical asymptotes (if any) of the function f (x) = x2 + 12x + 32 . A) x = 8	B) x = –8	C) x = –4	D) x = 4	E) x = –32

15. Find the limit:

0
1
−
x→
⎛	⎞
⎜	⎟
lim ⎝x2 + x ⎠
A) 0	B) ∞	C) 1	D) –1	E) –∞

2
3	1
Chapter 2: Differentiation

Name: __________________________	Date: _____________

7
1. Find the derivative of the following function using the limiting process.

f (x) = x+8

A)	f ′(x) =	 7 x+8

2
D)	f ′(x) = −	 7 x+8

(	)
(	)
(	)
2
B)	f ′(x) = −	 7	E) None of the above x –8

(	)
2
C)	f ′(x) =	 7 x –8

2. Find the derivative of the function.

f (x) = 6x2 – 4cos(x)

A)	f ′(x) = 6x+4sin(x) B)	f ′(x) =12x – 4sin(x) C)	f ′(x) =12x+4sin(x)

D)	f ′(x) =12x – 4cos(x)

E) None of the above

3. Find the slope of the graph of the function at the given value.

4	7
h(v)= v7 −v8 when v = 2

3	1
A) h'(2) =	 4	+	7 7(2)7         8(2)8

3	1
B) h'(2) =		4	−	7 7(2)−7        8(2)−8

3	1
C) h'(2) =		4	+	7 7(2)−7        8(2)−8

D) h'(2) = −4 +	7 7(2)7         8(2)8

3	1
E) h'(2) =	 4	−	7 7(2)7        8(2)8

4. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent.

2
4
y(x) = (x−x )2

A) −4
B) 4 and −2 C) −4 and 2 D) 2
E) There are no points at which the graph has a horizontal tangent.

5. A projectile is shot upwards from the surface of the earth with an initial velocity of 118 meters per second.

What is its velocity after 4 seconds?

What is its velocity after 10 seconds?

The position function is s(t) = −4.9t2 +v0t + s0 .

A) Its velocity after 4 seconds is –246 meters per second. After 10 seconds its velocity is 20 meters per second.
B) Its velocity after 4 seconds is –246 meters per second. After 10 seconds its velocity is –438 meters per second.
C) Its velocity after 4 seconds is 78.8 meters per second. After 10 seconds its velocity is –438 meters per second.
D) Its velocity after 4 seconds is 78.8 meters per second. After 10 seconds its velocity is 20 meters per second.
E) None of the above.

6. Use the product rule to differentiate.

Q(x) = x4 cos x

A) Q'(x) = −x4 cos x+4x3 sin x B) Q'(x) = −x4 sin x−4x3 cosx
C) Q'(x) = x4 sin x+4x3 cosx

D) Q'(x) = −x4 sin x+4x3 cosx

E) Both A and B

4
+
49
7
7
2401
147
2401
7. Use the quotient rule to differentiate the following function and evaluate R'(1).

R(x) = x4x6 A) R'(1) = –12

B) R'(1) = 12

C) R'(1) = –12

12
D) R'(1) = 49

12
E) R'(1) = 343

7
8. Find the derivative of the trigonometric function.

g(x) = x6 tan x

A) g'(x) = x5 sec2 x−6x5 tan x B) g'(x) = x6 sec2 x+6x5 tan x
C) g'(x) = 6x5 tan x− x5 sec2 x

D) g'(x) = x6 sec2 x+5x5 tan x

E) None of the above

9. Find an equation of the tangent line to the graph of f at the given point.

f (t) = (t −5)(t2 −6), at (2,6)

A) y =14t +34 B) y = 22t – 14
C) y = –14t −22

D) y = –14t +34

E) y = –14 +34t

10. Given the derivative below find the requested higher-order derivative.

4
f ′′(x) =3x7 , f (iv) (x)

A)	f (iv) (x) = –6120 x–25

4
7
–25 B)	f (iv) (x) = 2401x 7 C)	f (iv) (x) = –6120 x–24

–24 D)	f (iv) (x) = 2401x 7

E) None of the above

7
5
9
9
=
= −
=
=
=
11. Find the derivative of the function.

3 f (t) = (7+5t)13

–10 A)	f ′(t) = 13(7+5t) 13

–10 B)	f ′(t) =5(7+5t) 13

15
–10 C)	f ′(t) = 13(7+5t) 3

–10 D)	f ′(t) = 13(7+5t) 13

15
–10 E)	f ′(t) = 13(7+5t) 13

12. Evaluate the derivative of the function at the given point.

2t +5
5
⎛	⎞
⎜	⎟
3
3
f (t) = 5t −1 ,	⎝5,8⎠

A)     f '(5) = 64 B)     f '(5) = – 8
C)	f '(5) = 8

D)	f '(5) = – 64

27
E)	f '(5) = – 676

13. Find the second derivative of the function.

f (x) = (5x4 +4)7
A)	f ′′(x) =140x2 (4+5x4 )5 (12+140x4 ) B)	f ′′(x) =140x2 (4+5x4 )5 (12+135x4 ) C)	f ′′(x) =140x2 (4+5x)5 (12+135x4 )

D)	f ′′(x) =140x2

E)	f ′′(x) =140x2

4+5x3 )5 (12+135x4 ) 4+5x4 )5 (12−135x4 )

14. Find dy/dx by implicit differentiation.

x8 +2x+6xy− y5 =9

A)

B)

C)

dy	8x7 +2−6y dx        5y4 −6x dy     7x7 +2+6y dx        5y4 −6x dy	8x7 +2+6y
dx	4y4 −6x

D)

E)

dy	8x7 +2+6y dx           5y4 −6x dy     8x7 +2+6y
dx	5y4 −6x

–
=
–
=
=
=
–
=
15. Find an equation of the tangent line to the graph of the function given below at the given point.

(y −6)2 = 5(x−3),	(12.80,–1.00).

(The coefficients below are given to two decimal places.)
A) y = 0.4x+3.57	D) y = –0.36x+3.57 B) y = –0.36x+34.83	E) y = 2.64x+34.83 C) y = 2.64x−34.83

16. A point is moving along the graph of the function

1
y = 5x2 +6

such that dx/dt = 2 centimeters per second.

Find dy/dt whenx =3.

A)

B)

C)

D)

E)

dy	10

dt	867 dy         20

dt	867 dy      20

dt	867 dy      10

dt	867 dy	20

dt	17

3
3
3
17. Shadow Length A man 6 feet tall walks at a rate of 5 ft per second away from a light that is 15 ft above the ground (see figure). When he is 11 ft from the base of the light find the following.

(a) The rate the tip of the shadow is moving.
(b) The rate the length of his shadow is changing.

(a) A)
25

6 B)
25

6 C)
25

3

ft per minute

ft per minute

ft per minute

(b)

40 ft per minute

10 ft per minute

40 ft per minute

D)

25

3

ft per minute

10

3

ft per minute

E) None of the above

18. Find the slope of the tangent line to the graph of the function at the given point.

f (x) = –2x2 +3, (–3,–15)

A) 12	B) –2	C) –3	D) –18	E) None of the above

Chapter 2: Differentiation - Free Response Questions

Name: __________________________	Date: _____________

1. The position of a weight attached to a spring is described by s(t) = e−2t sin3t.

Determine and graph the velocity function for positive values of t and find the approximate first time when the velocity is zero. Find the approximate position of the weight the first time the velocity is zero. Round answers to tenths.

2. Prove that 5 x3 + 4x – 2 = 0 has exactly one solution.

3. Find the error in the following incorrect calculation.
2
+ 7	2 +7	2	2
x	x	x
lim1−cosx = lim sin x = lim cosx = 1 x→0	x→0	x→0

( )	)
4. Determine which function "dominates", where we say that the function f(x) dominates

the function g(x) as x →∞ if lim f x = lim g x = ∞ and either x→∞	x→∞
( )	( )
g x	f x
lim f (x = ∞ or lim g(x = 0.Show all work.
x→0	x→0

xp or px

5. Find the x-coordinates of all extrema and sketch the graph of f (x) = x3 +7x2 −49x−30 showing global and local behavior.

6. Sketch a graph of a function with the following properties:

f '(−5) = 0; f '(0) = 0; f '(5) = 0; f '(x) > 0 if x >5; f '(x) < 0 if 0< x <5.

7. Using the critical numbers of f (x) =8 + 2x2 – x3 , use the Second Derivative Test to determine all local extrema.

2
3
8. Determine the following significant features by hand and sketch the graph of f (x) = xx−1.
a). intercepts b). asymptotes c). extrema
d). inflection points

2
x
9. Sketch the graph of f (x) = x4+2 while answering the following questions. a.	What is the domain and range of f (x)?

b.	What are the intercepts of f (x)?

c.	What, if any, are the equation(s) of vertical asymototes of f (x)? d.	What, if any, are the local min(s) and local max(s) of f (x)?
e.	What, if any, are the inflection point(s) of f (x)?

f.	What, if it exists, is the equation of the horizontal asymotote of f (x)?

10. Below is a graph of f ′(x). Sketch a graph of a plausible f (x).