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 2

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 3

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]

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