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MATHS 102 – Functioning in Mathematics

Department of Mathematics

2022 Semester One Assignment 3 Due: 11:59 pm, Friday 3rd Jun

Show all working and note that late assignments will not be marked.

1. Find all the critical points of the functions below on the interval 0 ≤ x ≤ 2π.

(a)

1

sin x

(b)

1

cos x

[6 marks]

2. Solve the following equations leaving your answers in exact form – no decimal solutions (you should

check your solutions work).

(a) 7x = 5 (b) 2e4p = 8 (c) logx 125 = −3 [6 marks]

3. Give the domain and range of the following functions.

(a)

y = ln

(

1

x− 1

) (b)

y = e2x−6 + 1

[4 marks]

4. A toy car was released from the starting point of a layered race track toy set. The car then

travelled around the circuit before crossing the finish-line. It took the car 3 seconds to travel from

start to finish. The height of the car above the floor, in centimetres, at any given time is given by,

H(t) = t4 − 5t3 + 5t2 + 20 for 0 ≤ t ≤ 3 where H is the height and t is time in seconds.

(a) What is the height of the race-track at the starting point and the finish-line? [2 marks]

(b) Find the following:

(i) All the local maxima or minima [2 marks]

(ii) Interval(s) where H is increasing/decreasing. [2 marks]

(iii) Points of inflection [2 marks]

(iv) Intervals where H is concave downward/upward. [2 marks]

(c) Fully describe the path of the car in terms of its height and time using the information in (b).

[2 marks]

2022 Semester One Assignment 3 Page 1 of 2

5. Tama volunteered to take part in a laboratory caffeine experiment. The experiment wanted to test

how long it took the chemical caffeine found in coffee to remain in the human body, in this case

Tama’s body. Tama was given a standard cup of coffee to drink. The amount of caffeine in his

blood from when it peaked can be modelled by the function C(t) = 2.65e(−1.2t+3.6) where C is the

amount of caffeine in his blood in milligrams and t is time in hours. In the experiment, any reading

below 0.001mg was undetectable and considered to be zero.

(a) What was Tama’s caffeine level when it peaked? [1 marks]

(b) How long did the model predict the caffeine level to remain in Tama’s body after it had peaked?

[3 marks]

6. Differentiate the following simplifying where it is obvious to do so: [6 marks]

(a) y =

1

x

sin(3x) (b) h(t) =

at

√

1 −e2t

,a ∈ R (c) g(x) = ln(x

2ebx), b ∈ R

7. Integrate the following.

(a)

∫

x2 −x

3

√

x

dx (b)

∫

x2 sec2(x3 − 1) dx [4 marks]

8. (a) Write the following sum using sigma notation:

1

3

+

1

12

+

1

27

+

1

48

+ … +

1

300

[2 marks]

(b) Calculate

3∑

n=0

[

sin

(nπ

2

)

+ cos

(nπ

2

)]

[3 marks]

(c) Use the poperties of summation to calculate

130∑

i=50

(3i2 − i). [3 marks]

Total Marks: 50

2022 Semester One Assignment 3 Page 2 of 2

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