IB Maths AI SL 5.1 Calculus Sl Content Questions

Practise IB Mathematics AI SL 5.1 by interpreting derivatives, integrals, rates of change, area and optimisation in contextual models with technology.

Exam points

Interpret limits, derivatives, gradient/rate notation and derivative signs to analyse function behaviour.
Differentiate integer-power functions, construct tangents/normals and locate/classify stationary points.
Use antiderivatives, definite integrals and the trapezoidal rule to calculate accumulation, area and approximations.
Set up and solve optimisation problems, then interpret feasible extrema, units and assumptions.

Covered syllabus points

SL 5.1—Limits and derivative concept SL 5.2—Increasing and decreasing functions SL 5.3—Derivative of powers SL 5.4—Tangents and normals SL 5.5—Integration as anti-differentiation SL 5.6—Stationary points SL 5.7—Optimization SL 5.8—Trapezoidal rule

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Study 5.1 Calculus - SL content conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 4]

Consider the graph of the cubic function $f(x)=x^{3}+2 x^{2}-4 x-2$ . Part of the graph of y=f(x) is shown in the following diagram.

(a)

Write down

[ 4 ]

(i)

the gradient of the tangent.

[ 2 ]

(ii)

the equation of the tangent.

[ 2 ]

$\quad y+3=3(x-1)$ OR y=3 x-6

[Maximum number: 5]

Consider the graph of the following function:

g(x)=\frac{8}{x}+\frac{x^{2}}{2}, \text { for } x \neq 0

(a)

Find $g^{\prime}(x)$ .

[ 3 ]

$\quad\left(g^{\prime}(x)=\right)-8 x^{-2}+x$
Note: Award A1 for -8 seen, A1 for $x^{-2}$ (or $\frac{1}{x^{2}}$ ) and A1 for second term being x. Award at most A1A1AO if additional terms are seen.

(b)

Write down the interval in which g(x) is increasing.

[ 2 ]

$x>2 \mathbf{OR}(2, \infty) \mathbf{OR} \quad 2<x<\infty$
Note: Award A1 for 2 seen and award A1 for correct inequality.

[Maximum number: 7]

A hollow chocolate box is manufactured in the form of a right prism with a regular hexagonal base. The height of the prism is $h \mathrm{~cm}$ , and the top and base of the prism have sides of length $x \mathrm{~cm}$ .

(a)

Find an expression for $\frac{\mathrm{d} V}{\mathrm{~d} x}$ .

[ 2 ]

$\frac{\mathrm{d} V}{\mathrm{~d} x}=300 \sqrt{3}-\frac{27}{4} x^{2}$

Note: Award A1 for a correct term.

(b)

Find the value of x which maximizes the volume of the box.

[ 2 ]

from the graph of V or $\frac{\mathrm{d} V}{\mathrm{~d} x}$ OR solving $\frac{\mathrm{d} V}{\mathrm{~d} x}=0$

x=8.77(8.77382 \ldots)

(c)

Hence, or otherwise, find the maximum possible volume of the box.

[ 2 ]

from the graph of V OR substituting their value for x into V
$V_{\text {max }}=3040 \mathrm{~cm}^{3}(3039.34 \ldots)$

(d)

The box will contain spherical chocolates. The production manager assumes that they can calculate the exact number of chocolates in each box by dividing the volume of the box by the volume of a single chocolate and then rounding down to the nearest integer.

Explain why the production manager is incorrect.

[ 1 ]

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Covered syllabus points

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