AHL 5.14—Implicit functions, related rates, optimisation Questionbank

Practice AHL 5.14—Implicit functions, related rates, optimisation 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

A simple racetrack can be modelled by the function $y^4+xy^3+x^2y^2+x^3y+x^4=yx$ where $y$ is a function in terms of $x$ . This can be seen on the graph below.

Find $\frac{dy}{dx}$

[4]

If $y=x$ where $x=0$ then what is the value of $\frac{dy}{dx}$

[2]

Question 2

HLPaper 1

Determine the coordinates of the locations on the curve $y^3 + 3xy^2 - x^3 = 27$ where $\frac{{dy}}{{dx}} = 0$ .

[9]

Question 3

HLPaper 1

Consider a water tank in the shape of an inverted cone with a height of 10 meters and a base radius of 5 meters. Water is being pumped into the tank at a rate of 3 cubic meters per minute.

Find the rate at which the water level is rising when the water is 4 meters deep.

[5]

Question 4

HLPaper 2

A rectangular piece of cardboard with dimensions $24$ cm by $36$ cm is to be made into an open box by cutting squares of side length x cm from each corner and folding up the sides.

Find the volume $V$ of the box in terms of $x$ .

[2]

Determine the value of $x$ that maximizes the volume of the box.

[3]

Verify that the value of x found in part 2 is a maximum by using the second derivative test.

[2]

Calculate the maximum volume of the box.

[2]

Question 5

HLPaper 2

A small marble is free to move along a smooth wire in the shape of the curve $y=\frac{10}{3-2e^{-0.5x}}$ , $x \geq 0$ .

At the point on the curve where $x=4$ , it is given that $\frac{dy}{dt}=-0.1\,m\cdot s^{-1}$

Find the value of $\frac{dx}{dt}$ at this exact same instant.

[3]

Find an expression for $\frac{dy}{dx}$ .

[3]

Question 6

HLPaper 2

Consider the differential equation $z^2 \frac{\mathrm{d}w}{\mathrm{d}z} = w^2 - 2z^2$ for $z > 0$ and $w > 2z$ .

It is given that $w = 3$ when $z = 1$ .

Use Euler's technique, with a step length of 0.1, to find an approximate value of $w$ when $z = 1.5$ .

[4]

Use the substitution $w = vz$ to show that $z \frac{\mathrm{d}v}{\mathrm{d}z} = v^2 - v - 2$ .

[3]

By solving the differential equation, show that $w = \frac{8z + z^4}{4 - z^3}$ .

[8]

Find the actual value of $w$ when $z = 1.5$ .

[1]

Using the graph of $w = \frac{8z + z^4}{4 - z^3}$ , suggest a reason why the approximation given by Euler's technique in part is not a good estimate to the actual value of $w$ at $z = 1.5$ .

[1]

Question 7

HLPaper 2

Consider the curve $D$ given by $y=x-xy \ln(x y)$ where $x>0, y>0$ .

Show that ${\frac{dy}{dx}} + (x\frac{dy}{dx} + y) (1 + \ln(xy)) = 1$ .

[3]

Hence find the equation of the tangent to $D$ at the point where $x=1$ .

[5]

Question 8

HLPaper 1

Consider the function defined by $xy^{2}+\frac{5y^{3}}{x}-x^{2}=x$ where $y$ is a function of $x$ .

Implicitly find $\frac{dy}{dx}$ .

[5]

Question 9

SLPaper 1

This question explores the concepts of implicit functions, related rates, and optimization in the context of a geometric problem.

Consider a circle with radius $r$ that is increasing at a constant rate. The area $A$ of the circle is given by $A = \pi r^2$ . Determine the rate of change of the circle's area.

[2]

If the radius is increasing at a rate of $3\, \text{cm/s}$ , find the rate at which the area is increasing when the radius is $10\, \text{cm}$ .

[2]

Now consider a rectangle with length $l$ and width $w$ such that $l = 2w$ . The area $A$ of the rectangle is given by $A = lw$ .

[2]

If the width $w$ is increasing at a rate of $2\, \text{cm/s}$ , find the rate at which the area is increasing when the width is $5\, \text{cm}$ .

[2]

Find the dimensions of the rectangle that maximize the area, given that the length is twice the width and the perimeter is $36\, \text{cm}$ .

[2]

Question 10

HLPaper 2

A curve $D$ is given by the implicit equation

x+y-\cos (x y)=0 .

The curve $x y=-\frac{\pi}{2}$ intersects $\mathrm{D}$ at R and S .

Find the coordinates of $R$ and $S$ .

[4]

Show that ${\frac{{dy}}{{dx}}} = - \left( {\frac{{1 + y\sin(xy)}}{{1 + x\sin(xy)}}} \right)$ for the curve $D$

[5]

Given that the gradients of the tangents to D at R and S are ${n}_1$ and ${n}_2$ respectively, show that ${n}_1 \times {n}_2 = 1$ .

[3]

Find the coordinates of the three points on D, where the tangent is parallel to the line ${y = -x}$ .

[7]