IB Maths AA HL Topic 5: Calculus questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 1, no calculatorMedium6 marks

In Figure 1, the solid curve is $y=g(x)$ The dashed line is its tangent at $P$ where $x=0$

(a)

Using Figure 1, write down the values of $g(0)$ and $g'(0)$

[2]

(b)

The function $F$ is defined by

F(x)=\frac{e^{g(x)}}{x+2}.

Determine $F'(0)$ giving your answer exactly.

[4]

Question 3

Paper 2, calculatorEasy6 marks

Consider the functions $f(x)=x^3-3x$ and $g(x)=x^3$

The stationary points of $f$ are $(-1,2)$ and $(1,-2)$

(a)

Use the second derivative test to classify each stationary point of $f$

[2]

(b)

Determine whether $(0,0)$ is a point of inflexion of $f$ and state the gradient there.

[2]

(c)

Determine whether $(0,0)$ is a point of inflexion of $g$ and state the gradient there.

[2]

Question 4

Paper 2, calculatorHard6 marks

A manufacturer models the permitted side lengths $x$ and $y$ in metres, of a rectangular mounting panel by

x^2+xy+2y^2=36.

The positive value of $y$ is used, and installation requirements restrict $x$ to $4\leq x\leq 5$ The area of the panel is $A=xy$

(a)

Determine $\dfrac{dy}{dx}$ in terms of $x$ and $y$

[2]

(b)

Hence, or otherwise, determine the maximum possible area of the panel on the allowed interval.

[4]

Question 5

Paper 2, calculatorHard6 marks

For a real parameter $k>0$ let

f_k(x)=\frac{x^2-k}{x^2+1},\qquad -3\leq x\leq 3.

It is known that

\int_{-3}^{3} f_k(x)\,dx=0.

(a)

Determine the exact value of $k$

[2]

(b)

Hence determine the total area between the graph of $y=f_k(x)$ and the $x$ -axis for $-3\leq x\leq 3$ Show a definite-integral expression before evaluating. Do not use numerical integration. Give your answer to 3 significant figures.

[4]

Question 6

Paper 1, no calculatorHard6 marks

Figure 1 shows the graph of $f$ on the interval shown.

(a)

Determine an antiderivative of $f$

[4]

(b)

Hence determine, in exact form, the area of the shaded region $R$

[2]

Question 7

Paper 1, no calculatorMedium6 marks

The positive solution $y$ of

\frac{dy}{dx}=\frac{2x}{y}

satisfies $y(1)=1$ for $1\leq x\leq 2$ An Euler estimate is made using a step length of $\frac12$

(a)

Write down the Euler estimate for $y\left(\frac32\right)$

[1]

(b)

Hence, write down the Euler estimate for $y(2)$

[1]

(c)

Determine the exact value of $y(2)$ by solving the differential equation.

[3]

(d)

Hence, state whether the estimate in (b) is an overestimate or an underestimate of the exact value. Justify your answer.

[1]

Question 8

Paper 2, calculatorMedium12 marks

A liquid is cooling in a room where the temperature remains at $20\,^\circ\mathrm C$ The measured temperatures are shown in Figure 1. Temperatures were recorded to the nearest $1\,^\circ\mathrm C$

For $0\leq t\leq 15$ where $t$ is measured in minutes and $T$ is measured in degrees Celsius, the cooling is modelled by

\frac{dT}{dt}=-k(T-20), \qquad k>0,

where $k$ is constant.

(a)

Write down the measured temperatures at $t=0$ and $t=10$ minutes, as shown in Figure 1.

[2]

(b)

Solve the differential equation and use the measurements in Figure 1 to determine $k$ giving your answer to four significant figures with units.

[4]

(c)

Determine estimates for $T$ at $t=5$ and $t=10$ minutes using Euler’s method with step length $h=5$ minutes. Give both estimates to three significant figures.

[3]

(d)

Hence, use the solution from part (b) to determine the model’s predicted temperature at $t=15$ minutes. Give your answer to the nearest $0.1\,^\circ\mathrm C$ and compare it with the measurement in Figure 1.

[3]

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