MATH 1077 UO Essential Mathematics 2: Calculus

Problem Solving Exercise 1

Total Marks Available: 70+5=75.

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� Submission: Please submit a good quality digital copy of your assignment as a single
pdf �le on the course learnonline website, by the due date above.

• Presentation:
Please start the solution of each new question on a new page.

You may write your solutions out by hand or type them (my preference is for hand-written
work). Typed mathematical expressions must be notationally correct. Marks will be
deducted for poorly-presented typed mathematical expressions. Marks will not be deducted
for neatly handwritten assignments.

Show all necessary steps so that the reader can follow your solution procedure. Write
out your solutions clearly so that they are well organized and easy to follow. Use words.
When you answer a mathematical problem you are telling a story, and that story should
make sense, and be logical.

Generally, use exact values: use any necessary expressions such as π,
√
3, etc.; approx-

imate a �nal answer if it makes sense to do so; for example if it is a measured quantity.

• Style points are awarded for good mathematical writing.

Here, there are �ve (5) marks available for presentation and communication:

Excellent Notation is pro�cient and accurate. Layout is clear and easy to follow.
(5 marks) Diagrams are appropriate and very well presented. Presentation requirements

have been met.

Good Notation is generally appropriate, with some inaccuracies. Layout is mostly
(3 marks) clear and easy to follow. Diagrams are mostly appropriate and well presented.

Presentation requirements have been mostly met.

Satisfactory Notation has several inaccuracies. Layout is adequate. Some attempt has
(2 marks) been made to produce appropriate diagrams. Presentation is satisfactory.

Poor Limited accuracy of notation. Layout is poor. Limited attempt has been
(1 mark) made to produce appropriate diagrams. Presentation is adequate.

None Inadequate
(0 marks)

• Plagiarism: Even though you may discuss the exercises with others, the solutions you
present here should be the result of your separate and individual work. The University’s
policy on plagiarism will be applied strictly, and hence any �joint work� must be indicated
for cross-checking purposes.

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1. Rectilinear motion

The evil Professor Mayhem is planning to drop a time-bomb from the top of a 180 m tall
building. If the bomb hits the ground it will explode and destroy all of the new Adelaide

University City. Even if it doesn’t hit the ground, the bomb is set to explode 24 s after
its release.

The superhero Mercurious is 864 m from the base of the building, at (x, y) = (0, 0), when
he sees Professor Mayhem release the bomb. In an instant Mercurious works out that,
assuming that t = 0 is when the bomb is released, he needs to run with super speed along
a path described by the mathematical equation,

x(t) = t3 − 36t2 + Ct+D,

in order to catch the bomb before it hits the ground, turn around and deposit it a safe
distance from the city, and then turn around again and return before the bomb explodes.

(a) Draw an appropriate sketch of the situation with Mercurious’s position (along the
horizonal) at any time t identi�ed by the function x(t), assuming all the action takes
place to the left of the building (i.e., x < 0), and the bomb’s vertical position at any
time t during its fall is described by the function y(t) ≥ 0.

(b) Determine the time it would take for the bomb to hit the ground if it falls under
gravity with an acceleration of 10 ms−2.

(c) Determine the parameters C and D if Mercurious runs and catches the bomb at the
base of the building at the exact moment it would have hit the ground (at which point
he also reverses direction for the �rst time).

(d) Determine where Mercurious leaves the bomb (at which point he simultaneously re-
verses direction a second time).

(e) Determine Mercurious’s position when the bomb �nally explodes.

(f) Determine Mercurious’s maximum speed over the 24 s period. At what position(s)
is he when this occurs?

[2 + 3 + 10 + 3 + 2 + 5 = 25 marks]

2. Trigonometric functions

(a) Determine the period, frequency and amplitude of

y = 7 cos 5x− 24 sin 5x.

(b) Determine the slope-intercept form of the equation of the tangent line to

y = 2
√
2 x+ π cosx,

at the point x = π/4.

(c) Determine the absolute maximum and absolute minimum of

y = x− cosx

on the interval [0, 2π].

[5 + 5 + 5 = 15 marks]

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3. Inde�nite integration

Evaluate the following antiderivatives, i.e., inde�nite integrals. Show all steps of your
calculations. Marks will be deducted for lack of detail.

(a)

∫ (
8×3 − 5×1/3 − 1

x2/3
+ e2x

)
dx.

(b)

∫
(x− 3)1/4 x dx.

(c)

∫
cos
(
x2 + 1

)
x dx.

(d)

∫
sin
√
t√

t
dt.

[4 + 4 + 4 + 4 = 16 marks]

4. Application of integration

Given functions f(x) = x+ 1 and g(x) = (x+ 1)3,

(a) The graphs of functions f and g intersect each other at three points. Find the (x, y)
coordinates of those points.

(b) Sketch the graphs of functions f and g on the same set of axes. You may use technology
to help you.

(c) Find the area of the region enclosed by the graphs of f and g.

[4 + 4 + 6 = 14 marks]

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