AHL 5.9—Differentiating standard functions and derivative rules Questionbank

Practice AHL 5.9—Differentiating standard functions and derivative rules 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 2

The velocity of a particle moving in a straight line is given by $v(t)=t^{2} e^{-0.5 t}$ , for $t \geq 0$ , where $t$ is time in seconds and $v$ is in meters per second.

Find the acceleration $a(t)$ .

[3]

Find the time $t$ when the acceleration is zero, for $0<t<10$ .

[3]

Sketch the graph of $a(t)$ for $0 \leq t \leq 10$ , indicating the point where $a(t)=0$ . marks]

[3]

Question 2

HLPaper 1

A curve is defined by $y=x^{3} \ln (x)$ , for $x>0$ .

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ .

[5]

Find the x -coordinate of the point of inflection.

[3]

Determine the intervals where the curve is concave up.

[2]

Question 3

HLPaper 2

A tank in the shape of a cone with height 4 meters and base radius 2 meters is filled with water. The water drains through a small valve at the vertex, and the height $h(t)$ meters of water at time $t$ minutes satisfies the differential equation $\frac{d h}{d t}=-0.01 \sqrt{h}$ .

Solve the differential equation to find $h(t)$ .

[5]

Given that $h(0)=4$ , find the time $T$ when the tank is empty ( $h=0$ ).

[3]

Find the rate of change of the volume of water when $h=1$ .

[4]

Question 4

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 5

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3{t^2} - 14t + 8$ , for $0 \leqslant t \leqslant 5$ .

When $t = 0$ , the velocity of P is $3{\text{ m}}\,{{\text{s}}^{ - 1}}$ .

Write down the values of $t$ when $a = 0$ .

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$ .

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 6

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 7

HLPaper 1

The position vector of a particle, P, relative to a fixed origin O at time t is given by $\overrightarrow{OP} = \begin{pmatrix} \sin(t^2) \cos(t^2) \end{pmatrix}$

Find the velocity vector of P.

[2]

Show that the acceleration vector of P is never parallel to the position vector of P.

[5]

Question 8

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 9

HLPaper 1

A researcher is analyzing the oscillatory behavior of a specific biological signal, modeled by the function $f(x) = e^{2x} \sin(x)$ . This function represents the interaction between exponential growth and periodic oscillations.

Find the first derivative of $f(x)$ using the product rule to understand how the signal changes over time.

[3]

Determine the second derivative $f^{\prime \prime}(x)$ to analyze the acceleration of the signal's behavior.

[3]

Question 10

HLPaper 1

A particle moves along a straight line. Its displacement, $s$ metres, at time $t$ seconds is given by $s = t + \cos 2t,{\text{ }}t \geqslant 0$ . The first two times when the particle is at rest are denoted by ${t_1}$ and ${t_2}$ , where ${t_1} < {t_2}$ .

Find ${t_1}$ and ${t_2}$ .

[5]

Find the displacement of the particle when $t = {t_1}$

[2]