SL 5.4—Tangents and normals Questionbank

Practice SL 5.4—Tangents and normals with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The wind chill index $W$ is a measure of the temperature, in $^\circ\text{C}$ , felt when taking into account the effect of the wind.

When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind $v$ in kilometres per hour $(\text{km\,h}^{-1})$ is given by the equation

W = 19.34 - 7.405v^{0.16}

Find an expression for $\frac{dW}{dv}$ .

[2]

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5$ $\text{km\,h}^{-1}\,\text{min}^{-1}$ .

Find the rate of change of $W$ at this time.

[5]

Question 2

SLPaper 1

A civil engineer is analyzing the behavior of a specific function $f(x) = \frac{x^2 - 1}{x - 2}$ to understand the stress distribution in a beam under load.

Find the derivative $f^{\prime}(x)$ using the quotient rule.

[3]

Find the equation of the tangent to the curve at $x = 3$ .

[4]

Find the slope of the normal to the curve at $x = 3$ .

[2]

Determine whether the function is increasing or decreasing at $x = 3$ .

[2]

Identify any vertical asymptotes and explain their effect on the curve.

[4]

Question 3

SLPaper 1

Calculus: Chain Rule and Tangents/Normals

Consider $f(x)$ , $g(x)$ and $h(x)$ , for $x ∈ \mathbb{R}$ where $h(x) = (f \circ g)(x)$ .

Given that $g(3) = 7$ , $g'(3) = 4$ and $f'(7) = -5$ , find the gradient of the normal to the curve of $h$ at $x = 3$ .

[4]

Question 4

HLPaper 2

The curve $C$ is defined by the equation $xy - \ln y = 1$, $y > 0$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

[4]

Determine the equation of the tangent to $C$ at the point $\left( \frac{2}{\mathrm{e}}, \mathrm{e} \right)$.

[3]

Question 5

SLPaper 2

The function $f(x) = x^3 - 3x - 2$ is shown in the graph below. The curve intersects the $x$ -axis at point $P$ .

Find the $x$ -coordinate of point $P$ .

[1]

Calculate the gradient of the curve at point $P$ .

[2]

Find the equation of the normal line to the curve at point $P$ .

[3]

Question 6

SLPaper 1

The surface area of an open box with a volume of $32 \text{ cm}^3$ and a square base with sides of length $x$ cm is given by $S(x) = x^2 + \frac{128}{x}$ where $x > 0$ .

Find $S'(x)$ .

[3]

Solve $S'(x) = 0$ .

[2]

Interpret your answer to part 2 in context.

[1]

Question 7

SLPaper 1

Consider the function $g(x) = \frac{x^4}{4}$.

Find $g'(x)$.

[1]

Find the $x$-coordinate of the point at which the normal to the graph of $g$ has gradient $-\frac{1}{8}$.

[3]

Find the gradient of the graph of $g$ at $x = -\frac{1}{2}$.

[2]

Question 8

SLPaper 1

A team of researchers is analyzing the oscillatory behavior of a wave function represented by $g(x) = \sin(x) + \cos(x)$ , which models sound waves in a controlled environment.

Calculate the derivative $g'(x)$ to understand the rate of change of the wave function.

[2]

Find the slope of the tangent line at $x = \frac{\pi}{4}$ , which is critical for analyzing the wave's behavior at that point.

[2]

Find the equation of the normal line to the wave function at $x = \frac{\pi}{4}$ , which will help in understanding the wave's behavior at that point.

[5]

Determine the points where the slope of the tangent is $0$ , indicating points of interest in the wave's oscillation.

[4]

State the domain and range of the function $g(x)$ , which is essential for understanding the behavior of the wave.

[5]

Question 9

HLPaper 1

A curve has equation $3x - 2y^2 e^{x - 1} = 2$ .

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ .

[5]

Find the equations of the tangents to this curve at the points where the curve intersects the line $x = 1$ .

[4]

Question 10

SLPaper 1

A technology company is evaluating the efficiency of its software by analyzing the ratio of two functions, where $f(x) = \frac{u(x)}{v(x)}$ . At $x = 0$ , the number of software users is $u(0) = 15$ and the number of software developers is $v(0) = 3$ . The rates of change are given as $u^{\prime}(0) = 2$ and $v^{\prime}(0) = 4$ .

Find the derivative of $f(x) = \frac{u(x)}{v(x)}$ using the quotient rule.

[3]

Determine $f(0)$ and $f^{\prime}(0)$ .

[2]

Find the equation of the normal to the curve at $x = 0$ .

[2]