AHL 5.12—First principles, higher derivatives Questionbank

Practice AHL 5.12—First principles, higher derivatives 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 the introduction of differential calculus and focuses on finding the derivative of a function using first principles and applying basic differentiation rules.

Consider the function $f(x) = 3x^2 - 2x + 4$ . Using the definition of the derivative, find $f'(x)$ .

[5]

Using the result from part 1, find the slope of the tangent to the curve $y = f(x)$ at the point where $x = 1$ .

[2]

Find the equation of the tangent line to the curve $y = f(x)$ at the point where $x = 1$ .

[3]

Determine the x-coordinate where the tangent to the curve $y = f(x)$ is horizontal.

[2]

Question 2

HLPaper 1

Given the function $g(x)=e^{2x}$ :

Find the derivative of $g(x)$ from first principles.

[5]

Determine the intervals where $g(x)$ is increasing or decreasing.

[3]

Question 3

HLPaper 1

The power rule used for differentiation is $\frac{d}{dx}x^n=nx^{n-1}$ .

Prove the power rule for a real exponent $n$ using first principles.

[6]

Question 4

HLPaper 2

In physics, the displacement of a particle in a damped oscillatory system is modeled by $h(x)=e^{-x} \sin (2 x)$ , where $x$ is time in seconds.

Find $h^{\prime}(x)$ using first principles.

[8]

Show that $h(x)$ satisfies the differential equation $h^{\prime \prime}(x)+2 h^{\prime}(x)+5 h(x)=0$ .

[6]

Find the Maclaurin series for $h(x)$ up to the $x^{5}$ term.

[6]

Determine the coordinates of the first positive maximum of $h(x)$ . Sketch the graph, labeling this point.

[7]

Question 5

HLPaper 2

In financial modeling, the function $g(x)=\frac{\ln (1+x)}{1+x}, x>-1$ , models the adjusted return rate of an investment over time $x$ (in years).

Find $g^{\prime}(x)$ using first principles.

[7]

Find the first three derivatives of $g(x)$ .

[6]

Find the Maclaurin series for $g(x)$ up to the $x^{3}$ term.

[6]

Estimate the adjusted return rate at $x=0.1$ using the series from (c). Discuss the accuracy of this approximation in the financial context.

[4]

Question 6

HLPaper 1

In modeling the growth of a bacterial colony under varying nutrient conditions, the function $f(x)=\sqrt{x^{2}+4}, x \geq 0$ , represents the effective nutrient absorption rate, where $x$ is time in hours.

Derive $f^{\prime}(x)$ using first principles.

Find the first four derivatives of $f(x)$ . Establish a general formula for $f^{(n)}(x)$ .

Determine the equation of the tangent line to $f(x)$ at $x=2$ . Sketch the curve and the tangent line, labeling the point of tangency and the x-intercept of the tangent.

In the context of the bacterial colony, interpret $f^{\prime \prime}(2)$ and compute its value. Use this to predict whether the absorption rate is accelerating or decelerating at $t=2$ hours.

Question 7

HLPaper 1

In financial modeling, the function $g(x)=\frac{\ln (1+x)}{1+x}, x>-1$ , models the adjusted return rate of an investment over time $x$ (in years).

Find $g^{\prime}(x)$ using first principles.

[8]

Prove by induction that $g^{(n)}(x)=(-1)^{n+1} \frac{n!}{(1+x)^{n+1}} \sum_{k=0}^{n-1} \frac{1}{k!(n-1-k)!}$ .

[10]

Find the Maclaurin series for $g(x)$ up to the $x^{4}$ term.

[6]

Estimate the adjusted return rate at $x=0.1$ using the series from (c). Discuss the accuracy of this approximation in the financial context.

[4]

Question 8

HLPaper 2

Let $f(x)=\tan x$ , for $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

Find $f^{\prime}(x)$ using first principles.

[6]

Compute the first three derivatives of $f(x)$ .

[5]

Find the Taylor polynomial of degree 3 for $f(x)$ centered at $x=0$ .

[5]

Question 9

HLPaper 1

In engineering, the stress distribution in a material is modeled by $k(x)=\frac{x^{2}}{1+x^{3}}, x \geq 0$ , where $x$ is the distance from a reference point.

Find $k^{\prime}(x)$ using first principles.

Find the first four derivatives of $k(x)$ . Derive a general expression for the numerator of $k^{(n)}(x)$ .

Find the Taylor series for $k(x)$ centered at $x=1$ up to the $(x-1)^{3}$ term.

Estimate $k$ (1.1) using the series from (c). Discuss the reliability of this approximation in the engineering context.

Question 10

HLPaper 2

Let $p(x)=x^{2} \cos x$ .

Find $p^{\prime}(x)$ using first principles.

[6]

Find $p^{\prime \prime}(x)$ . Determine the $x$ -coordinates where $p^{\prime \prime}(x)=0$ in $[0, \pi]$ , and sketch the graph of $p(x)$ , labeling these points.

[7]

Find the equation of the normal line to $p(x)$ at $x=\frac{\pi}{2}$ .

[4]