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AHL 5.12—First principles, higher derivatives - IB Questionbank | RevisionDojo

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-coordinates 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 $g(x)$ 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 using first principles.

[6]

Question 4

HLPaper 1

An art gallery hangs a painting on a vertical wall. The bottom edge of the painting is $a$ metres above the eye level of an observer, and the top edge is $b$ metres above eye level, where $b > a > 0$ . The observer stands $x$ metres horizontally from the wall ( $x>0$ ).

Let $\theta(x)$ be the vertical angle subtended by the painting at the observer's eye.

Show that $\theta(x) = \arctan\left(\frac{b}{x}\right) - \arctan\left(\frac{a}{x}\right)$ .

[2]

Find $\theta'(x)$ .

[4]

Hence, find the exact distance $x$ that maximizes the viewing angle $\theta$ .

[4]

Question 5

HLPaper 1

Consider the function $g( y ) = y\,{{\text{e}}^{2y}}$ , where $y \in \mathbb{R}$ . The $m^{{\text{th}}}$ derivative of $g( y )$ is denoted by ${g^{\left( m \right)}}( y )$ .

Prove, by mathematical induction, that ${g^{\left( m \right)}}( y ) = \left( {{2^m}y + m{2^{m - 1}}} \right){{\text{e}}^{2y}}$ , $m \in {\mathbb{Z}^ + }$ .

[7]

Question 6

HLPaper 1

Given the function $f(x)=x^2+3x$ :

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

[5]

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

[3]

Question 7

HLPaper 1

Consider the trignometric function $f(x)=\cos(x)\sin(x)$

Find the derivative of $f(x)$ using the definition of the derivative.

[6]

Question 8

SL & HLPaper 1

Let $f(x) = \cos x$ , $g(x) = x^k$ , where $k \in \mathbb{Z}^+$ and $h(x) = \left( {{f^{(19)}}(x) \times {g^{(19)}}(x)} \right)$ .

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

(ii) Given that $g^{(19)}(x) = \frac{{k!}}{{(k - p)!}}(x^{k - 19})$ , find $p$ .

[5]

(i) Find the first four derivatives of $f(x)$ . (ii) Find ${f^{(19)}}(x)$ .

[4]

(i) Find $h'(x)$ .

(ii) For $k=21$ show that $h(\pi) = \frac{-21!}{2}\pi^2$ .

[7]

Question 9

HLPaper 2

Differentiate from first principles the function $g(y) = 3y^3 - y$ .

[5]

Question 10

HLPaper 2

Consider the function $y^{e^{pz}}$ where $p$ is a rational number.

Use mathematical induction to prove that ${\frac{d^n}{dx^n}\left(y^{e^{pz}}\right) = p^{n-1}(pz+n)e^{pz}}$ for $n \in \mathbb{Z}^+$ , $p \in \mathbb{Q}$ .

[7]

AHL 5.12—First principles, higher derivatives Questionbank