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 2

A model for the population $P(t)$ of a species in a habitat is given by $P(t)=\frac{1000 t}{e^{0.2 t}+2}$ , where $t$ is time in years, $t \geq 0$ .

Find the rate of change of the population, $\frac{d P}{d t}$ .

[4]

Find the time $t$ when the population is growing at its maximum rate.

[4]

Question 5

HLPaper 1

Consider the function $f(x)=\frac{4 x^{2}}{x^{2}+1}$ , for $x \in \mathbb{R}$ .

Show that $f^{\prime}(x)=\frac{8 x}{\left(x^{2}+1\right)^{2}}$ .

[4]

Find $f^{\prime \prime}(x)$ and hence determine the x -coordinate of the point of inflection of the graph of $f$ .

[4]

Sketch the graph of $f^{\prime}(x)$ , indicating the behavior as $x \rightarrow \pm \infty$ .

[3]

Question 6

HLPaper 1

Consider the function $g(x)=x e^{-2 x}$ , for $x \in \mathbb{R}$ .

Find $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ .

[4]

Find the coordinates of the local maximum of $g(x)$ .

[3]

Sketch the graph of $g(x)$ , showing the local maximum and behavior as $x \rightarrow \pm \infty$ .

[3]

Question 7

HLPaper 2

A particle moves along a straight line with velocity given by $v(t)=3 t \sin \left(t^{2}\right)$ , for $t \geq 0$ , where $t$ is time in seconds.

Find the acceleration $a(t)$ .

[3]

Determine the values of $t$ in the interval $0 \leq t \leq \sqrt{\pi}$ where the particle is speeding up.

[4]

Find the total distance traveled by the particle from $t=0$ to $t=\sqrt{\pi}$ .

[5]

Question 8

HLPaper 2

A curve is defined by $y=\ln \left(x^{2}+3 x+4\right)$ , for $x>-2$ .

Show that $y^{\prime}=\frac{2 x+3}{x^{2}+3 x+4}$ .

[3]

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

[3]

The region bounded by the curve, the x-axis, and the lines $x=-1$ and $x=1$ has area $A$ . Find $A$ .

[5]

Question 9

HLPaper 1

A function is defined as $f(x)=\frac{\ln (2 x+3)}{x+1}$ , for $x>-1$ .

Show that $f^{\prime}(x)=\frac{-\ln (2 x+3)+2}{(x+1)^{2}}$ .

[4]

Find the x -coordinate of the stationary point of $f(x)$ .

[3]

Determine the nature of the stationary point found in part (b).

[3]

Question 10

HLPaper 1

The height of a wave is modeled by $h(t)=3 \sin (t) e^{-0.1 t}$ , where $t$ is time in seconds, $t \geq 0$ , and $h$ is in meters.

Find $\frac{d h}{d t}$ .

[3]

Show that the first time the wave reaches a local maximum is at $t=5$ .

[4]

Find the rate of change of the height at $t=5$ .

[2]