AHL 5.19—Maclaurin series Questionbank

Practice AHL 5.19—Maclaurin series with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

This question involves finding the Maclaurin series expansion of given functions and using them to approximate integrals and evaluate limits.

Find the Maclaurin series expansion of the function $f(x) = \cos(x^2)$ up to and including the term in $x^4$ .

[4]

Using the result from part 1, find an approximate value for $\int_0^1 \cos(x^2) \, dx$ .

[5]

Find the Maclaurin series for the function $g(x) = e^{x^2} \sin(x^2)$ up to and including the term in $x^6$ .

[5]

Using the Maclaurin series found in part 3, evaluate $\lim_{x \to 0} \frac{e^{x^2} \sin(x^2) - x^2 - x^4}{x^6}$ .

[3]

Question 2

HLPaper 1

This question explores the Maclaurin series expansion of functions and their applications.

Find the Maclaurin series for the function $f(x) = \sin(x)$ up to and including the term in $x^5$ .

[5]

Using the result from part 1, find an approximate value for $\int_0^{\frac{\pi}{2}} \sin(x) \, dx$ .

[5]

Consider the function $g(x) = e^x \sin(x)$ . Using the maclaurin series of $sin(x)$ and $e^x$ , find the Maclaurin series for $g(x)$ up to and including the term in $x^5$ .

[5]

Using the result from part 3, find an approximate value for $\int_0^1 e^x \sin(x) \, dx$ .

[5]

Consider the function $h(x) = \ln(1 + x)$ . Find the Maclaurin series for $h(x)$ up to and including the term in $x^4$ .

[5]

Question 3

HLPaper 1

This question explores the Maclaurin series expansion of functions and their applications.

Find the Maclaurin series for the function $f(x) = \ln(1 + x)$ up to and including the term in $x^4$ .

[2]

Using the result from part 1, find an approximate value for $\int_0^1 \ln(1 + x) \, dx$ .

[5]

Question 4

HLPaper 1

Prove by mathematical induction that ${d^n \over dy^n}\left(y^2e^y\right) = \left[y^2 + 2ny + n(n-1)\right]e^y$ for ${n \in \mathbb{Z}^+}$ .

[7]

Hence or otherwise, determine the Maclaurin series of $f(y) = y^2e^y$ in ascending powers of $y$ , up to and including the term in $y^4.$

[3]

Question 5

HLPaper 1

Consider the function $g(x) = \ln(1+x) \cos x$ .

Find the Maclaurin series expansion of $g(x)$ using the series of $\ln(1+x)$ and $\cos(x)$ up till $x^4$

[3]

Question 6

HLPaper 1

Consider the function $f(x) = e^{2x}$ .

Find the Maclaurin series expansion for $f(x)$ up to the $x^4$ term.

[6]

Question 7

HLPaper 1

Consider the function $f(x) = \ln(1 + e^x)$ .

Find the first two derivatives of $f(x)$ , $f'(x)$ and $f''(x)$ .

[4]

Hence, find the Maclaurin series for $f(x)$ up to and including the term in $x^2$ .

[3]

By substituting $x=1$ into your series, find an approximation for $\ln(1+e)$ .

[1]

The Maclaurin series for $\ln(1+x)$ is $x - \frac{x^2}{2} + \dots$ . Deduce the Maclaurin series for $\ln(1+e^x)$ by substituting the series expansion for $e^x$ , and verify your answer to part (b).

[6]

Question 8

HLPaper 2

Consider the function $f(x) = \frac{1}{1+x}$ . We are interested in finding the first three non-zero terms of its Maclaurin series expansion.

Find the first derivative of $f(x) = \frac{1}{1+x}$ and evaluate it at $x = 0$ .

[1]

Find the second derivative of $f(x) = \frac{1}{1+x}$ and evaluate it at $x = 0$ .

[2]

Find the third derivative of $f(x) = \frac{1}{1+x}$ and evaluate it at $x = 0$ .

[2]

Using the derivatives found, write the first three non-zero terms of the Maclaurin series for $f(x) = \frac{1}{1+x}$ .

[2]

Question 9

HLPaper 1

This question involves finding the Maclaurin series expansion of a given function and using it to approximate integrals.

Find the Maclaurin series expansion of the function $f(x) = \cos(x^2)$ up to and including the term in $x^8$ .

[4]

Using the result from part 1, find an approximate value for $\int_0^1 \cos(x^2) \, dx$ .

[4]

The function $g(x) = e^{x^2} \cos(x^2)$ is given. Find the Maclaurin series for $g(x)$ up to and including the term in $x^6$ .

[5]

Using the Maclaurin series found in part 3, evaluate $\lim_{x \to 0} \frac{e^{x^2} \cos(x^2) - 1 - x^2}{x^4}$ .

[3]

Question 10

HLPaper 1

Consider the function $f(x) = \frac{e^x - 1}{x}$ for $x \neq 0$ and $f(0) = 1$ .

Show that $\lim_{x \to 0} f(x)$ exists and find its value.

[3]

Evaluate $\lim_{x \to 0} \frac{\ln(1+x)}{x}$ using L'Hôpital's Rule.

[3]

Find the limit $\lim_{x \to 0} \frac{e^{2x} - 1}{x}$ using L'Hôpital's Rule.

[3]

Consider the limit $\lim_{x \to \infty} \frac{x}{e^x}$ . Show that this limit is equal to 0 using L'Hôpital's Rule.

[3]

Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ without using L'Hôpital's Rule.

[3]