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Calculus Part 3

This document contains multiple calculus problems involving finding points where tangents are parallel, finding dimensions and volume of a tray, sketching graphs, identifying concavity, and finding equations of tangents. The problems require calculating derivatives, integrals, areas under curves, and applying calculus concepts like maxima and minima.

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Farah
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47 views10 pages

Calculus Part 3

This document contains multiple calculus problems involving finding points where tangents are parallel, finding dimensions and volume of a tray, sketching graphs, identifying concavity, and finding equations of tangents. The problems require calculating derivatives, integrals, areas under curves, and applying calculus concepts like maxima and minima.

Uploaded by

Farah
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Calculus Part 3 [37 marks]

1. Find the x-coordinates of all the points on the curve [7 marks]


y = 2x4 + 6x3 + 72 x2 − 5x + 32 at which
the tangent to the curve is parallel to the tangent at (−1, 6).

16 cm 10 cm
2a. Hugo is given a rectangular piece of thin cardboard, 16 cm by 10 cm . He [2 marks]
decides to design a tray with it.
He removes from each corner the shaded squares of side x cm , as shown in the
following diagram.

The remainder of the cardboard is folded up to form the tray as shown in the
following diagram.

Write down, in terms of x , the length and the width of the tray.

5
2b. (i) State whether x can have a value of 5. Give a reason for your [4 marks]
answer.
(ii) Write down the interval for the possible values of x .

3
2c. Show that the volume, V cm3 , of this tray is given by [2 marks]

V = 4x3 − 52x2 + 160x.

2d. Find dV . [3 marks]


dx

3 2
2e. Sketch the graph of V = 4x3 − 51x2 + 160x , for the possible values of [4 marks]
x found in part (b)(ii), and 0 ⩽ V ⩽ 200 . Clearly label the maximum point.

= ( ) −0.5 ≤ ≤ 6.5 ′
Let y = f(x), for −0.5 ≤ x ≤ 6.5. The following diagram shows the graph of f ′ ,
the derivative of f .

The graph of f ′ has a local maximum when x = 2, a local minimum when x = 4,


and it crosses the x-axis at the point (5, 0).

3a. Explain why the graph of f has a local minimum when x = 5. [2 marks]
3b. Find the set of values of x for which the graph of f is concave down. [2 marks]
3c. The following diagram shows the shaded regions A, B and C. [5 marks]

The regions are enclosed by the graph of f ′ , the x-axis, the y-axis, and the line
x = 6.
The area of region A is 12, the area of region B is 6.75 and the area of region C is
6.75.
Given that f(0) = 14, find f(6).
3d. The following diagram shows the shaded regions A, B and C. [6 marks]

The regions are enclosed by the graph of f ′ , the x-axis, the y-axis, and the line
x = 6.
The area of region A is 12, the area of region B is 6.75 and the area of region C is
6.75.
2
Let g(x) = (f(x)) . Given that f ′ (6) = 16, find the equation of the tangent to the
graph of g at the point where x = 6.

© International Baccalaureate Organization 2021


International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®
Printed for NEFERTARI INTL SCH

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