limits-test-16-17 - web viewap calculus limits test. section i: multiple choice. time: 34....
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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)