RevisionDojo

SL 5.4—Tangents and normals - IB Questionbank | RevisionDojo

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 $°C$ , 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 $(km h^{-1})$ 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 $5$ $\text{km h}^{-1} \text{ minute}^{-1}$ .

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

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

Question 4

HLPaper 2

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

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

[4]

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

[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 where the curve intersects the $x$ -axis. [1]

[1]

Calculate the gradient (slope) of the curve at point $P$ .

[2]

Find the equation of the normal line to the curve at point $P$ in the form $y=ax+b$ .

[3]

Question 6

SLPaper 1

The surface area of an open box with a volume of 32 cm³ and a square base with sides of length x cm is given by S(x) = x² + 128/x where x > 0.

Find S'(x).

[2]

Solve S'(x) = 0.

[1]

Interpret your answer to (i) in context.

[3]

Question 7

SLPaper 1

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

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 ${g^{\prime}(x)}$ 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 $3x - 2{y^2}{{\text{e}}^{x - 1}} = 2$ .

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]

SL 5.4—Tangents and normals Questionbank