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IB Maths AA: Calculus — Practice Questions & Answers
Calculus is the study of change: how fast quantities vary (differentiation) and how to accumulate them (integration). This topic builds from the idea of a limit to the derivative, then covers the differentiation rules, derivatives of standard functions, tangents and normals, stationary points, optimization, integration, definite integrals, area, and kinematics. HL extends to the chain, product and quotient rules and to integration by substitution.
50 practice questions available for this topic — here are 10 with full answers and explanations.
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Practice questions with answers
1
Multiple choice · Easy
Differentiate $y = x^{5}$ with respect to $x$.
$5x^{4}$
$5x^{5}$
$x^{4}$
$\frac{1}{6}x^{6}$
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Why By the power rule, $\frac{dy}{dx} = n x^{n-1}$, so the derivative of $x^{5}$ is $5x^{4}$.
2
Multiple choice · Easy
Find $\frac{dy}{dx}$ for $y = 3x^{2} - 4x + 7$.
$6x - 4$
$6x - 4 + 7$
$6x^{2} - 4$
$3x - 4$
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Why Differentiate term by term: $6x - 4 + 0 = 6x - 4$.
3
Multiple choice · Easy
Find $\int x^{3}\,dx$.
$\frac{x^{4}}{4} + C$
$3x^{2} + C$
$\frac{x^{4}}{4}$
$4x^{4} + C$
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Why Add one to the power and divide: $\frac{x^{4}}{4} + C$. The constant of integration must be included.
4
Multiple choice · Easy
What is the derivative of $y = \sin x$?
$\cos x$
$-\cos x$
$-\sin x$
$\tan x$
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Why The derivative of $\sin x$ is $\cos x$.
5
Multiple choice · Easy
What is the derivative of $y = e^{x}$?
$e^{x}$
$x e^{x-1}$
$\frac{e^{x}}{x}$
$e^{x-1}$
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Why The exponential function $e^{x}$ is its own derivative.
6
Multiple choice · Easy
Find the gradient of the curve $y = x^{2}$ at the point where $x = 3$.
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Why $\frac{dy}{dx} = 2x$, so at $x = 3$ the gradient is $2(3) = 6$.
7
Multiple choice · Easy
At a stationary point of a function, the value of $\frac{dy}{dx}$ is:
$0$
$1$
undefined
a maximum value
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Why A stationary point occurs where the gradient is zero, so $\frac{dy}{dx} = 0$.
8
Multiple choice · Easy
Find $\int (2x + 3)\,dx$.
$x^{2} + 3x + C$
$2x^{2} + 3x + C$
$2 + C$
$x^{2} + 3 + C$
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Why Integrate term by term: $\frac{2x^{2}}{2} + 3x + C = x^{2} + 3x + C$.
9
Multiple choice · Easy
The derivative of $y = \cos x$ is:
$-\sin x$
$\sin x$
$\cos x$
$-\cos x$
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Why The derivative of $\cos x$ is $-\sin x$.
10
Multiple choice · Medium
Differentiate $y = \frac{4}{x^{2}}$ with respect to $x$.
$-\frac{8}{x^{3}}$
$\frac{8}{x^{3}}$
$-\frac{4}{x^{3}}$
$\frac{2}{x}$
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Why Write as $4x^{-2}$, then $\frac{dy}{dx} = -8x^{-3} = -\frac{8}{x^{3}}$.
Key terms in Calculus
Limit: The value a function approaches as its input approaches a particular value.
Limit notation: Writing a limit as $\lim_{x \to a} f(x)$.
Continuity: A property of a function with no breaks, jumps or holes in its graph.
Tangent line: A line that touches a curve at a point with the same gradient as the curve there.
Secant line: A line passing through two points on a curve.
Rate of change: How quickly one quantity changes with respect to another.
Average rate of change: The change in a function over an interval divided by the length of that interval.
Instantaneous rate of change: The rate of change of a function at a single point, given by the derivative.
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