Name:____________________________________________________________________________________

AP Calculus ABLimits Test

Multiple Choice Score:

_______/17

Free Response Score:_______/18

Overall Grade:

A

What not to do:

AP Calculus Limits TestSection I: Multiple Choice

Time: 34 MinutesNumber of problems: 17

No Calculator

1. If f ( x )={ cos ( x ) for 0<x≤ 2x2 cos ( x ) for 2<x≤ 4 , then

limx→2

f ( x ) is

(A) cos(2) (B) cos(8) (C)

cos(16) (D) 4 (E) nonexistent

2. The graph of the function f is shown in the figure above. Which of the following statements about f is true?

(A) limx→b

f ( x )=1

(B) limx→a

f ( x )=2

(C) limx→b

f ( x )=2

(D) limx→a

f ( x )= limx→b

f ( x )

(E) limx→a

f ( x ) does not exist

3. Let f(x) and g(x) be continuous functions such that limx→ c

f ( x )=−3and

limx→ c

g( x )=4. Determine

limx→ c

−2 ( f ( x )+g( x ))

(A) -2(B) 2(C) -1(D) -24(E) Impossible to determine

f ( x )=¿ {x+2 if x<3¿ {−3 if x=3¿ ¿¿¿4. Let f be the function given above. Which of the following statements are true about f?

I. limx→3

f ( x ) exists

II. f (3) exists

III. f is continuous at x = 3

(A) None(B) I only(C) II only(D) I and II only(E) I, II and III

5. Let f be a function that is continuous on the closed interval [2 , 4] with f (2 )=10 and f ( 4 )=30 . Which of the following is guaranteed by the Intermediate Value Theorem?

(A) f (3 )=20

(B) f ( x )=23 has at least one solution in the open interval (2 , 4).

(C) f ' ( x )=10 has at least one solution in the open interval (2 , 4).

(D) f ' ( x )>0 for all x in the open interval (2 , 4).

(E) f attains a maximum on the open interval (2 , 4).

Use the following table to answer questions 6 – 7:

x f ( x ) f ' ( x ) g ( x ) g ' ( x )-1

-2-2 1 4

0 3 -1 -2 1/31 -4 0 0 02 5 4 -1 -3

6. Which of the following represents an equation for the line normal to g ( x ) at x = 0?

(A) y=−1

3x−2

(B) y=1

3x+2

(C) y=−3 x+2

(D) y=−3x−2

(E) y=3 x+2

7. If the value of f (1 .9 ) is approximated using the line tangent to the graph of f ( x ) at x=2 , then f (1 .9 )≈¿ ¿

(A) -1.3(B) 0.7(C) 4.5(D) 4.6

(E) 5.4

8. Let j be a continuous function such that j (1) = 2 and j (3) = 7. Which of the following must be true for the function j on the interval 1 ≤ x ≤ 3?

I. j (2 )=4 . 5

II. The average rate of change of j on the interval [1 , 3] is

52 .

III. j (c )=2 π for some c in the interval [1 , 3].

(A) I only(B) II only

(C) III only(D) II and III only(E) I, II and III

f ( x )=¿{ (2 x−1 ) ( x−2 )x−2

for x≠2 ¿ ¿¿¿

9. Let f be the function defined above. For what value of k is f continuous at x = 2?

(A) 0 (B) 1 (C)

2 (D) 3 (E) 5

10. The graph of the function f is shown above. Which of the following statements is false?

(A) limx→2

f ( x ) exists.

(B) limx→3

f ( x ) exists.

(C) limx→4

f ( x ) exists.

(D) limx→5

f ( x ) exists.

(E) The function f is continuous at x = 3.

11. Let f be the function defined by f ( x )=2 x2−3 x . Which of the following is an equation of the line tangent to the graph of f at the point where x = -1?

(A) y=5(B) y=−x+4(C) y=−x+6(D) y=−7x−2(E) y=−7x+12

12. The line y = 5 is a horizontal asymptote to the graph of which of the following functions?

(A)y=cos (5x )

x(B)

y= 1x+5

(C) y=15 x−4

1+3x(D)

y= 5 x2

1−x2

(E) y=5 x

13. If f ( x )= x2−5x+6

x2−4 , which of the following statements is a true statement about f (x)?

I. f (x) has a removable discontinuity at x = 2.

II. f (x) has an infinite discontinuity at x = -2.

III. f (x) has a horizontal asymptote at y = 1.

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

14.. If f ( x )=√x+3 , then f ' ( x )=?

(A) f ' ( x )= 1

2√ x+3

(B) f ' ( x )= 1

√x+3

(C) f ' ( x )=√x+3

(D) f ' ( x )=2√x+3(E) f ' ( x ) is undefined

x 0 1 2f(x) 1 k 2

15.. The function f is continuous on the closed interval [0,2] and has values that are given in the table above. The

equation f ( x )=52 must have at least two solutions in the interval [0,2] if k =?

(A) 0

(B) 12

(C) 1(D) 2(E) 3

16. The graph of the function f is shown above. Which of the following statements must be false?

(A) f(0) exists(B) f(x) is continuous for 0 < x < 4(C) f is not continuous at x = 0

(D) lim

x→−0+f (x )

exists

(E) lim

x→−2+f ( x )

exists

17. If a≠ 0

, then limx→a

x2−a2

x 4−a4 is

(A) y= 1a2

(B) 12 a2

(C)

16 a2

(D) 0 (E) nonexistent

DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO

AP Calculus Limits TestSection III: Free Response

Time: 15 MinutesNumber of problems: 1

Calculator Allowed

Question One

t (hours) 0 1 3 4 7 8 9L(t) (people) 132 164 182 172 100 57 0

Concert tickets for the big NewTones show went on sale at noon (t = 0) yesterday and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a continuous function L(t) for 0 ≤ t ≤ 9. Values of L(t) at various times t are shown in the table above.

(a) Find the average rate of change for each of the following time intervals: [0 , 1] , [1 , 3] , [3 , 4]. Which interval had the greatest average rate of change? Show the work that leads to your answer.

(b) Write an equation for a secant line that you could use to approximate the number of people waiting in line between 4 pm (t = 4) and 7 pm (t = 7). Use your equation to estimate the number of people waiting in line at 5:30 pm (t = 5.5). Show the computations that lead to your answer. Indicate units of measure.

(c) For 0 ≤ t ≤ 9, what is the fewest number of times at which L(t) must equal 135? Justify your answer.

(d) The quadratic function P (t )=−5 t2+30 t+135 is used to model the number of people waiting in line at time

t. Write an equation for P ' ( t ) and show the work that leads to your answer.

Question One(a)

(b)

(c)

(d)

AP Calculus Limits TestSection IV: Free Response

Time: 15 MinutesNumber of problems: 1

No Calculator

Question Two

g( x )=¿ {5 x4−4 x2+37 x2+6

if x<0¿ {−12

if x=0 ¿ ¿¿¿

(a) Find limx→0

g( x ). Show the computations that lead to your answer.

(b) Is g(x) continuous at x = 0? Justify your answer.

(c) Evaluate limx→∞

g ( x ) and

limx→−∞

g (x )

(d) Let m(x) = 7x2 +6. Write the equation of the line normal to m(x) at x = -1. Show the work that leads to your answer.

Question Two(a)

(b)

(c)

(d)

Question Three

Graph of f

Let f(x) be given by the graph shown above.

(a) Find limx→2−

f (x ),

limx→2+

f (x ) and

limx→2

f ( x ).

(b) Find limx→4−

f ( x ),

limx→4+

f ( x ) and

limx→4

f ( x ).

(c) Write all intervals for which f(x) continuous?

(d) Write a piecewise defined function for f(x):

Question Three(a)

(b)

(c)

(d)