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

[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 ]

[Maximum number: 2]

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.

Write down

the gradient of the tangent.

[Maximum number: 2]

Consider the graph of the following function:

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

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

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

[Maximum number: 2]

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

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

Note: Award A1 for a correct term.

[Maximum number: 2]

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.

Write down

the equation of the tangent.

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

[Maximum number: 4]

A type of dog food is sold in cylindrical containers. A new label for the container is being considered. The label will be a rectangle that is 5 cm wide and 7 cm high.
The design on the label is a curve, as shown on the following axes, where one unit represents 1 cm for both axes.

The values in the table approximate points on the curve, correct to one decimal place.

(a)

Write down the integral that represents the area of the shaded region.

[ 2 ]

$\quad \int_{0}^{5}\left(-\frac{x^{3}}{3}+3 x^{2}-7 x+6\right) \mathrm{d} x$

Note: Award A2 for an integral with correct integrand and correct limits.

(b)

Hence, find the area of the shaded region.

The dog food is sold in closed cylindrical containers, with radius r and height h.

Figure for Question (b) — IB Maths AI SL

The whole container is made from one type of material, and it is assumed that the thickness of the material is negligible.

Each container has a volume of $400 \mathrm{~cm}^{3}$ .

[ 2 ]

$\quad 15.4\left(15.4166 \ldots, \frac{185}{12}\right) \quad\left(\mathrm{cm}^{2}\right)$

Note: If their limits are outside the domain, award at most A1AOFT.

[Maximum number: 4]

Zheng models the height of a section of a rollercoaster.
She uses the function

f(x)=0.001 x^{3}-0.12 x^{2}+4.05 x+3, \text { for } 0 \leq x \leq 60

where x is the horizontal distance from the left-hand end of the section and f(x) is the vertical height. Distances are measured in metres.

Zheng's model is shown on the following graph of y=f(x).

Point O has coordinates (0,0).

Find the coordinates of M .

$x=24.2(24.1886 \ldots), y=44.9(44.9056 \ldots)$ or (24.2,44.9)

Note: Award A1A0 if parentheses are omitted.

[Maximum number: 3]

The values in the table approximate points on the curve, correct to one decimal place.

Use the trapezoidal rule with five intervals, and the values given in the table, to estimate the shaded area below the curve.

The curve used in the label design can be modelled by:

y=-\frac{x^{3}}{3}+3 x^{2}-7 x+6, \text { for } 0 \leq x \leq 5

$\frac{1}{2} \times 1 \times((6+4.3)+2(1.7+1.3+3+4.7))$ $=15.9(15.85)\left(\mathrm{cm}^{2}\right)$

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Covered syllabus points

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