Taylor and Maclaurin Series Practice Questions
AP (Advanced Placement) · AP Calculus BC · 151 free MCQs with instant results and detailed explanations.
151
Total
44
Easy
78
Medium
29
Hard
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Sample Questions from Taylor and Maclaurin Series
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Q1
Easy
What is the Maclaurin series expansion for the function f(x) = e^x?
A.
1 + x + x^2/2! + x^3/3! + ...
B.
1 - x + x^2/2 - x^3/6 + ...
C.
x + x^2 + x^3 + ...
D.
1 + x^2/2 + x^3/3 + ...
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Correct Answer: A
The Maclaurin series for e^x is the sum of the series 1 + x + x^2/2! + x^3/3! + ..., which includes all derivatives evaluated at 0, resulting in the correct option.
Q2
Easy
If the Taylor series for f(x) converges to f(x) for all x, what can be said about the function?
A.
It is analytic everywhere.
B.
It is continuous everywhere.
C.
It is differentiable everywhere.
D.
It has a finite radius of convergence.
Show Answer & Explanation
Correct Answer: A
If a Taylor series converges to f(x) for all x, the function must be analytic everywhere, meaning it can be represented by a power series in the neighborhood of every point in its domain.
Q3
Easy
What is the Maclaurin series expansion for the function f(x) = e^x?
A.
1 + x + x^2/2! + x^3/3! + ...
B.
1 - x + x^2/2 - x^3/6 + ...
C.
x + x^2 + x^3 + ...
D.
1 + x^2 + x^3 + ...
Show Answer & Explanation
Correct Answer: A
The Maclaurin series of e^x is derived from its derivatives at x=0, resulting in the infinite sum 1 + x + x^2/2! + x^3/3! + ...
Q4
Medium
Which of the following represents the Maclaurin series expansion for the function f(x) = e^x?
A.
1 + x + x^2/2! + x^3/3! + ...
B.
1 - x + x^2/2! - x^3/3! + ...
C.
1 + x - x^2/2 + x^3/6 + ...
D.
x + x^2/2 + x^3/3 + ...
Show Answer & Explanation
Correct Answer: A
The Maclaurin series for e^x is derived from its derivatives at zero, which yield 1 for all orders. Thus, the correct series is the sum of all derivatives evaluated at zero, resulting in 1 + x + x^2/2! + x^3/3! + ...
Q5
Medium
What is the radius of convergence for the series represented by the Taylor series of f(x) = ln(1 + x) centered at x = 0?
A.
1
B.
2
C.
0
D.
∞
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Correct Answer: A
The Taylor series for ln(1 + x) is valid for |x| < 1, giving a radius of convergence of 1. This means the series converges for values within this interval.
Q6
Medium
If the Taylor series for sin(x) is used to approximate sin(0.1), what is the first non-zero term in the approximation?
A.
0.1
B.
0.1^3/6
C.
0.1^5/120
D.
0.1^7/5040
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Correct Answer: A
The Taylor series expansion for sin(x) at x=0 is x - x^3/3! + x^5/5! - ... The first non-zero term for sin(0.1) is simply 0.1, which corresponds to the linear term of the series.
Q7
Medium
Consider the function f(x) = cos(x). What is the fourth-degree Taylor polynomial centered at x = 0?
A.
1 - x^2/2 + x^4/24
B.
1 - x^2/2 - x^4/24
C.
1 + x^2/2 - x^4/24
D.
1 - x^2/2 + x^4/12
Show Answer & Explanation
Correct Answer: A
The fourth-degree Taylor polynomial for cos(x) includes terms up to x^4. The expansion around x=0 is 1 - x^2/2 + x^4/24, which matches option A.
Q8
Hard
What is the Taylor series expansion of f(x) = e^x centered at x = 0 up to the fourth degree term?
A.
1 + x + x^2/2 + x^3/6 + x^4/24
B.
1 + x + x^2 + x^3 + x^4/4!
C.
1 + x + x^2 + x^3 + x^4
D.
1 + x + x^2/2 + x^3/3 + x^4/4
Show Answer & Explanation
Correct Answer: A
The Taylor series expansion of e^x around x=0 is derived from its derivatives at that point, which all evaluate to 1. The fourth-degree term includes x^4/4!. Thus, the correct expression is 1 + x + x^2/2 + x^3/6 + x^4/24.
Q9
Hard
What is the Taylor series expansion of the function f(x) = e^(2x) centered at x = 0 up to the x^4 term?
A.
1 + 2x + 2x^2 + (2/3)x^3 + (1/12)x^4
B.
1 + 2x + 2x^2 + 2x^3 + (4/12)x^4
C.
1 + 2x + 2x^2 + (2/3)x^3 + (1/6)x^4
D.
1 + 2x + 2x^2 + 4x^3 + (8/12)x^4
Show Answer & Explanation
Correct Answer: C
The correct Taylor series for e^(2x) at x=0 is derived from f(x) = e^(2x), and its derivatives evaluated at 0 yield the coefficients. Up to the x^4 term, the series is 1 + 2x + 2x^2 + (2/3)x^3 + (1/6)x^4.
Q10
Hard
Given the Maclaurin series for sin(x), which of the following represents sin(2x) using a polynomial approximation up to the x^5 term?
A.
2x - (4/6)x^3 + (8/120)x^5
B.
2x - (4/3)x^3 + (8/60)x^5
C.
2x - (4/6)x^3 + (4/60)x^5
D.
2x - (4/3)x^3 + (4/12)x^5
Show Answer & Explanation
Correct Answer: B
The Maclaurin series for sin(x) is x - (1/6)x^3 + (1/120)x^5. Substituting 2x gives 2x - (4/3)x^3 + (8/60)x^5, simplifying to 2x - (4/3)x^3 + (2/15)x^5.
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Taylor and Maclaurin Series — AP (Advanced Placement) AP Calculus BC Practice Questions Online
This page contains 151 practice MCQs for the chapter Taylor and Maclaurin Series in AP (Advanced Placement) AP Calculus BC. The questions are organized by difficulty — 44 easy, 78 medium, 29 hard — so you can choose the right level for your preparation.
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