AP CALCULUS BC - CHAPTER 3 TEST - OPEN ENDED SECTION
PERIOD 3
Name: ________________________________
1.
Let f(x) be a differentiable function such that f(2)=4 and f'(2)=-1/2. What is the approximation for f(2.1) found
by using the linearization at x=2?
A.
2.95
B.
3.95
C.
4.05
D.
4.1
2.
4
3
Let g(x) be the function defined by g ( x)  x  4 x . How many relative extrema does g have?
A.
3.
0
B.
1
C.
2
Let f be a function with first derivative defined by f '( x ) 
D.
3
3x 2  6
for x>0. It is known that f(1)=9 and f(3)=11.
x2
What value of x in the open interval (1, 3) satisfies the MVT for f on the closed interval [1, 3]?
A.
4.
6
B.
3
C.
2
D.
1
The derivative of the function f(x)=ln(x) is f'(x)=1/x. Consider a triangle in the xy-plane. Two vertices of the
triangle are on the x-axis at (1, 0) and (5, 0), and a third vertex is on the graph of y=ln(2x)-0.5x+5 for x in the
interval [1/2, 8]. What is the maximum area of such a triangle?
A.
19/2
B.
2ln(2)+9
C.
2ln(4)+8
D.
2ln(16)+2
5.
The graph of f', the derivative of f, is shown below. Which of the following could be the graph of f?
6.
Let f be a function with a second derivative given by f "( x)  x 2  x  4  x  7  . What are the x-coordinates of
the points of inflection of the graph of f?
A.
0 only B.
4 only C.
0 and 7 only
D.
7.
A.
B.
C.
D.
4 and 7 only
E.
0, 4, and 7
8.
9.
10.
11.
12.
13.
14.
15.
AP CALCULUS BC - CHAPTER 3 TEST - OPEN ENDED SECTION
PERIOD 5/6
Name: ________________________________
1.
2.
3.
4.
Given f(x) = x3 find a tangent line approximation, T(x), at the point (3,27). Using T(x), approximate
f(x) = x3 at 3.1.
A.
E.
29.08
B.
None of these
29.7
C.
29.8
D.
29.75
5.
6.
The product of two positive numbers is 363. Minimize the sum of the first and three times the second.
A.
B.
33 and 11
11 3 and 11 3
C.
22 and
D.
E.
30.67 and 12
None of these
Given f(x) =
A.
B.
C.
D.
E.
7.
x2
, which of the following statements are true?
x 1
f(x) has a vertical asymptote at x = -1 and a horizontal asymptote
f(x) has a slant asymptote at y = x – 1 and a vertical asymptote at x=-1
f(x) has a slant asymptote at y = x and a vertical asymptote at x=-1
f(x) has a horizontal asymptote at y = 0
f(x) has a horizontal asymptote at y = 1
Determine the function whose graph has vertical asymptotes at x = +2 and no horizontal asymptote.
A.
B.
C.
D.
E.
8.
33
2
x
(x - 2)2
x
f(x) = 2
x 4
x3
f(x) = 2
x 4
3x
f(x) = 2
x 4
f(x) =
none of these
Find the values of x that give relative extrema for the function f(x) = 3x 5 – 5x3.
A.
relative maximum: x = 0; relative minimum x = 5 3
B.
C.
D.
E.
relative maximum: x = +1; relative minimum x = 0
relative maximum: x = 0; relative minimum x = +1
relative maximum: x = -1; relative minimum x = 1
none of these
9.
Find the differential of y = sec3x
A.
sec3xdx
B.
3sec3xtan3xdx
C.
3sec23xdx
D.
3tan23xdx
E.
none of these
10.
Determine whether the Mean Value Theorem applies to f(x) = 3x – x2 on the interval [2, 3]. If the
“MVT” does apply, find all values of c in [2, 3] such that f’(c) =
A.
B.
C.
D.
E.
11.
MVT applies; c = 2 3
MVT applies; c = 5 2
f(b)  f(a)
ba
. If it does not apply, state why.
MVT does not apply; f(2) = f(3)
MVT does not apply; f(x) is not differentiable on [2, 3]
none of these
Let f be a function with first derivative defined by f '( x ) 
3x 2  6
for x>0. It is known that f(1)=9 and f(3)=11.
x2
What value of x in the open interval (1, 3) satisfies the MVT for f on the closed interval [1, 3]?
A.
12.
13.
6
B.
3
C.
2
D.
1
The derivative of the function f(x)=ln(x) is f'(x)=1/x. Consider a triangle in the xy-plane. Two vertices of the
triangle are on the x-axis at (1, 0) and (5, 0), and a third vertex is on the graph of y=ln(2x)-0.5x+5 for x in the
interval [1/2, 8]. What is the maximum area of such a triangle?
A.
19/2
B.
2ln(2)+9
C.
2ln(4)+8
D.
2ln(16)+2
14.
Let f be a function with a second derivative given by f "( x)  x 2  x  4  x  7  . What are the x-coordinates of
the points of inflection of the graph of f?
A.
0 only B.
4 only C.
0 and 7 only
D.
15.
A.
B.
C.
D.
4 and 7 only
E.
0, 4, and 7
AP CALCULUS BC - CHAPTER 3 TEST - OPEN ENDED SECTION
PERIOD 10/11
Name: ________________________________
1.
The graph of f', the derivative of f, is shown below. Which of the following could be the graph of f?
2.
Let f be a function with a second derivative given by f "( x)  x 2  x  4  x  7  . What are the x-coordinates of
the points of inflection of the graph of f?
A.
0 only B.
4 only C.
0 and 7 only
D.
3.
A.
B.
C.
D.
4 and 7 only
E.
0, 4, and 7
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Given f(x) = x3 find a tangent line approximation, T(x), at the point (3,27). Using T(x), approximate
f(x) = x3 at 3.1.
A.
E.
29.08
B.
None of these
29.7
C.
29.8
D.
29.75