IB Maths AA SL 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

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Question 1

Paper 1, no calculatorEasy7 marks

The curve $C$ has equation $y=f(x)=(x+1)e^{x^2}$ where $x\in\mathbb R$ The curve passes through $(0,1)$

(a)

Determine $f'(x)$

[3]

(b)

Hence, determine the equation of the tangent to $C$ at $x=0$

[3]

(c)

The tangent meets the $x$ -axis at $P$ Find the $x$ -coordinate of $P$

[1]

Question 2

Paper 1, no calculatorMedium5 marks

The curve $C$ has equation $y=f(x)=x^3-3x$ for $-2\leq x\leq 2$ Figure 1 shows $C$

(a)

Use Figure 1 to state the $x$ -coordinates of the stationary points of $C$

[1]

(b)

Determine whether each stationary point is a local maximum or a local minimum. Justify your answers.

[2]

(c)

Show that $(0,0)$ is a point of inflexion of $C$ and determine the gradient of $C$ there. State whether this gradient is zero or non-zero.

[2]

Question 3

Paper 1, no calculatorEasy5 marks

The derivative of a function $f$ is

f'(x)=6x+2,\qquad x\in\mathbb{R},

and $f(0)=3$

(a)

Determine $f(x)$

[4]

(b)

Hence write down a definite integral representing the area bounded by the graph $y=f(x)$ the $x$ -axis, $x=0$ and $x=2$ Do not evaluate the integral.

[1]

Question 4

Paper 1, no calculatorEasy7 marks

The curve $y=x^2-1$ the $x$ -axis and the lines $x=-2$ and $x=2$ bound several regions.

(a)

Write down, without evaluating, an expression in terms of definite integrals for the total area of these regions.

[3]

(b)

Hence determine the exact total area of these regions.

[4]

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