AP Calculus Testbank
(Chapter 9)
(Mr. Surowski)
Part I. Multiple-Choice Questions
√
∞
X
n−1
1. The series
will converge, provided that
1+p + n + 1
n
n=0
(A) p > 1
2. The series
(B) p > 2
∞
X
an =
n=1
(C) p > .5
∞
X
(−1)n 2n
n=1
n2
(D) p < .5
diverges because
I. The terms are not all positive.
II. The terms
a do not tend to 0 as n tends to ∞.
n+1 III. lim > 1.
n→∞ an
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
(E) p > 0
3. Which of the following series converge?
∞ X
2n
I.
.
n
+
1
n=1
II.
∞
X
3
n
n=1
∞ X
III.
n=1
cos 2nπ
.
n2
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
4. The interval of convergence for the series
∞ X
(3x − 2)n+2
n5/2
n=1
is
1
< x ≤ 1.
3
1
(B) ≤ x < 1.
3
1
(C) < x < 1.
3
1
(D) ≤ x ≤ 1.
3
(A)
1
(E) −1 ≤ x ≤ − .
3
5. If
∞
X
an (x − c)n is a Taylor series that converges to f (x) for every
n=0
real number x, then f 00 (c) =
(A) 0
(B) a1
(C) 2a2
(D)
∞
X
n=0
n−2
n(n − 1)an (x − c)
(E)
∞
X
n=0
an
6. The graph of the function represented by the Taylor series
∞
X
(−1)n (x − 1)n intersects the graph of y = ex at x =
n=0
(A) −9.425
(B) 0.567
(C) 0.703
(D) 0.773
(E) 1.763
7. The graph of the function represented by the Taylor series
∞
X
(−1)n n(x − 1)n−1 intersects the graph of y = ex
n=1
(A) at no values of x
(B) at x = 0.567
(C) at x = 0.703
(D) at x = 0.773
(E) at x = 1.763
8. Using the fifth-degree Maclaurin polynomial y = ex to estimate
e2 , this estimate is
(A) 7.000
(B) 7.267
(C) 7.356
(D) 7.389
∞
X
(x + 2)n
√ n is
9. The interval of convergence of the series
n
n3
n=1
(A) −3 < x < 3
(B) −3 ≤ x ≤ 3
(C) −5 < x < 1
(D) −5 ≤ x ≤ 1
(E) −5 ≤ x < 1
(E) 7.667
10. What is the approximation of the value of cos 2 obtained by using the sixth-degree Taylor polynomial about x = 0 for cos x?
2
4
−
3 45
16
64
(B) 1 + 2 +
+
.
24 720
1
1
1
−
(C) 1 − +
2 24 720
4
4
8
(D) 2 − +
−
3 15 315
8
32
128
(E) 2 + +
+
6 120 5040
(A) 1 − 2 +
∞
X
(2x + 3)n
√
11. The set of all values of x for which
converges is
n
n=1
(A) −2 < x < −1
(B) −2 ≤ x ≤ −1
(C) −2 < x ≤ −1
(D) −2 ≤ x < −1
(E) −2 ≤ x < 1
12. The interval of convergence of the series
∞
X
n=1
an
; a > 0 is
(x + 2)n
(A) (a − 2) ≤ x ≤ (a + 2)
(B) (a − 2) < x < (a + 2)
(C) x > (a − 2) or x < −a − 2.
(D) (a − 2) > x > (−a − 2)
(E) (a − 2) ≤ x ≤ (−a − 2)
13. What is the sum of the Maclaurin series
2n+1
π3 π5
n π
+
+ · · · + (−1)
+ ···?
π−
3!
5!
(2n + 1)!
(A) 1
(B) 0
(C) −1
(D) e
(E) This is divergent
14. If f (x) =
∞
X
2
k
(cos x) , then f
k=0
(A) −2
(B) −1
π
4
=
(C) 0
15. The Maclaurin series expansion of
(D) 1
(E) 2
1
is
1 + x2
x2 x4 x6
+
−
+ ···
(A) 1 −
2!
4!
6!
(B) 1 − x2 + x4 − x6 + · · ·
x2 x4 x6
(C) 1 +
+
+
+ ···
2!
4!
6!
(D) 1 + x2 + x4 + x6 + · · ·
x2 x4
x6
+
−
+ ···
(E) 1 −
4!
8! 12!
16. Which of the following series converge(s)?
I.
∞
X
(−1)n
n=1
∞
X
II.
n
1
√
n3
n=1
∞
X
1
√
III.
3
n2
n=1
(A) I only
(B) II only
(C) I and II
(D) I and III
(E) I, II, and III
17. Which of the following gives a Taylor polynomial approximation about x = 0 for sin 0.4, correct to four decimal places?
(0.4)3 (0.4)5
(A) 0.4 +
+
3!
5!
(0.4)3 (0.4)5
(B) 0.4 −
+
3!
5!
3
(0.4)5
(0.4)
+
(C) 0.4 −
3
5
2
(0.4)
(0.4)3 (0.4)4 (0.4)5
(D) 0.4 +
+
+
+
2!
3!
4!
5!
2
3
4
(0.4)
(0.4)
(0.4)
(0.4)5
(E) 0.4 −
+
−
+
2!
3!
4!
5!
Part II. Free-Response Questions
1. The function f has derivatives of all orders for all real numbers
x. Assume f (2) = −3, f 0 (2) = 5, f 00 (2) = 3, and f 000 (2) = −8.
(a) Write the third-degree Taylor polynomial for f about x = 2
and use it to approximate f (1.5).
(b) The fourth derivative of f satisfies the inequality |f (4) (x)| ≤
3 for all x in the closed interval [1.5, 2]. Use the Lagrange
error bound on the approximation to f (1.5) found in part
(a) to explain why f (1.5) 6= −5.
(c) Write the fourth-degree Taylor polynomial, P (x), for g(x) =
f (x2 + 2) about x = 0. Use P to explain why g must have a
relative minimum at x = 0.
2. The Taylor series about x = 5 for a certain function f converges
to f (x) for all x in the interval of convergence. The nth deriva(−1)n n!
1
tive of f at x = 5 is given by f (n) (5) = n
, and f (5) = .
2 (n + 2)
2
(a) Write the third-degree Taylor polynomial for f about x = 5.
(b) Find the radius of convergence of the Taylor series for f
about x = 5.
(c) Show that the sixth-degree Taylor polynomial for f about
1
x = 5 approximates f (6) with error less than
.
1000
3. A function f is defined by
f (x) =
1
2
3
n+1
+ 2 x + 3 x2 + · · · + n+1 xn + · · ·
3 3
3
3
for all x in the interval of convergence of the given power series.
(a) Find the interval of convergence for this power series. Show
the work that leads to your answer.
f (x) − 31
.
(b) Find lim
x→0
x
(c) Write the first three nonzero terms and the general term for
Z 2
an infinite series that represents f (x) dx.
0
(d) Find the sum of the series determined in part (c).
4. The Maclaurin series for the function f is given by
f (x) =
∞
X
(2x)n+1
n=0
n+1
4x2 8x3 16x4
(2x)n+1
= 2x+
+
+
+· · ·+
+· · ·
2
3
4
n+1
on its interval of convergence.
(a) Find the interval of convergence of the Maclaurin series for
f . Justify your answer.
(b) Find the first four terms and the general term for the Maclaurin series for f 0 (x).
(c) Use the Maclaurin
series you found in part (b) to find the
1
value of f 0 −
.
3
5. The function f is defined by the power series
f (x) =
∞
X
(−1)n x2n
n=0
x2 x4 x6
(−1)2n x2n
= 1− + − +···+
+···
(2n + 1)!
3! 5! 7!
(2n + 1)!
for all real numbers x.
(a) Find f 0 (0) and f 00 (0). Determine whether f has a local maximum, a local minimum, or neither at x = 0. Give a reason
for your answer.
1
(b) Show that 1 − approxmates f (1) with error less than 10−2 .
6
(c) Show that y = f (x) is a solution to the differential equation
xy 0 + y = cos x.
6. The function f has a Taylor series about x = 2 that converges to
f (x) for all x in the interval of convergence. The nth derivative
(n + 1)!
of f at x = 2 is given by f (n) (2) =
for n ≥ 1, and f (2) =
3n
1.
(a) Write the first four terms and the general term of the Taylor
series for f about x = 2.
(b) Find the radius of convergence for the Taylor series for f
about x = 2. Show the work that leads to your answer.
(c) Let g be a function satisfying g(2) = 3 and g 0 (x) = f (x) for
all x. Write the first four terms and the general term of the
Taylor series for g about x = 2.
(d) Does the Taylor series for g as defined in part (c) converge
at x = −2? Give a reason for your answer.
π
7. Let f be the function given by f (x) = sin 5x +
, and let P (x)
4
be the third-degree Taylor polynomial for f about x = 0.
(a) Find P (x).
(b) Find the coefficient of x22 in the Taylor series for f about
x = 0.
1
1 (c) Use the Lagrange error bound to show that f
<
−P
10
10 1
.
100
Z
x
(d) Let G be the function given by G(x) =
f (t) dt. Write the
0
third-degree Taylor polynomial for G about x = 0.
8. Let f be a function having derivatives of all orders for all real
numbers. The third-degree Taylor polynomial for f about x = 2
is given by
T (x) = 7 − 9(x − 2)2 − 3(x − 2)3 .
(a) Find f (2) and f 00 (2).
(b) Is there enough information given to determine whether f
has a critical point at x = 2? If not, explain why not. If so,
determine whether f (2) is a relative maximum, a relative
minimum, or neither, and justify your answer.
(c) Use T (x) to find an approximation for f (0). Is there enough
information given to determine whether f has a critical point
at x = 0? If not, explain why not. If so, determine whether
f (0) is a relative maximum, a relative minimum, or neither,
and justify your answer.
(d) The fourth derivative of f satisfies the inequality f (4) (x) ≤
6 for all x in the closed interval [0, 2]. Use the Lagrange error bound on the approximation to f (0) found in part (c) to
explain why f (0) is negative.
9. The Taylor series about x = 0 for a certain function f converges
to f (x) for all x in the interval of convergence. The nth derivative of f at x = 0 is given by
f (n) (0) =
(−1)n+1 (n + 1)!
for n ≥ 2.
5n (n − 1)2
The graph of f has a horizontal tangent line at x = 0, and f (0) =
6.
(a) Determine whether f has a relative maximum, a relative
minimum, or neither at x = 0. Justify your answer.
(b) Write the third-degree Taylor polynomial for f about x = 0.
(c) Find the radius of convergence of the Taylor series for f
about x = 0. Show the work that leads to your answer.
10. The function f is defined by the power series
2
f (x) = 1 + (x + 1) + (x + 1) + · · · + =
∞
X
(x + 1)n
n=0
for all real numbers for which the series converges.
(a) Find the interval of convergence of the power series for f .
Justify your answer.
(b) The power series above is the Taylor series for f about
x = −1. Find the sum of the series for f .
Z x
(c) Let g be the function defined by g(x) =
f (t) dt. Find the
−1
1
1
value of g − , if it exists, or explain why g −
cannot
2
2
be determined.
(d) Let h be the function defined by h(x) = f x2 − 1 . Find the
first three nonzero terms and the general term of
the
Taylor
1
.
series for h about x = 0, and find the value of h
2