Task MTH1010 Assignment 3. This assignment mark will count 5% towards your final MTH1010 mark.

Objectives When you have completed this assignment you should have developed the following skills:

• Understanding and working with circular functions;

• Understanding and working with the derivative by first principle;

• Understanding and working with differentiation of functions;

• Using differential calculus to assist in curve sketching;

• Using differential calculus in applications of minimum-maximum problems and/or curve sketch- ing;

• Consulting in groups but writing up final drafts individually.

Instructions Before you start this assignment

• Read your lecture notes up to and including optimisation and the relevant sections in

Bolster Academy.

• Read through the assessed exercises you have completed as an aid to doing the assignment problems.

• Be aware of Monash University guidelines and plagiarism and cheating, they are available for viewing at http://www.policy.monash.edu/policy-bank/academic/education/conduct/plagiarism- policy.html

Attempting the assignment Keep in mind the following:

• Write neatly.

• Show all steps and working.

• Use words to briefly describe/explain steps in your working and diagrams to illustrate.

• 90% of marks are given to correct working and explanations.

• Approximately 10% of the total assignment marks are given to mathematical communication. Please read the Mathematics Assignment Writing Guide on the MTH1010 Moodle page.

• Final answers are only worth one mark each.

Submission Be aware that:

• Assignment 3is due on Friday 14th October  by 23:55pm .  It should be submitted online in  the submission tool allocated to Assignment3

• Extension to an assignments due date, on reasonable grounds, must be sort from the Unit Coordinator before the due date, if possible.

• Assignments submitted after the due date should be submitted directly to the Unit Coordinator.

• Assignments submitted after the due date, without Unit Coordinator granted extension, will be penalised for lateness by 10% of the assignment total per calendar day.

1. The depth, D, (in metres) of water at the entrance to a harbour at t hours after midnight on a given day is represented by

D(t) = 10 + 3 sin. πt + π Σ for t ∈ [0, 24]

(a) State the amplitude, the equation of the midline, phase shift and period of D(t).

(b) Sketch the graph of D(t) for t ∈ [0, 24].

(c) Algebraically find the intervals of time during the day when the depth is greater than 8.5 metres.

(d) A boat is permitted to enter the harbour only if the depth of water at the entrance is at least d metres for a continuous interval of three hours. Algebraically approximate the largest value of d which satisfies this condition.

2. Use the first principles definition of the derivative, that is,

df

= lim

. f (x + ∆x) − f (x) Σ

to find the first derivative of the function

x

f (x) =

2 .

(x + 1)

3. Find the first derivative of each of the following functions with respect to their independent variable. Your working should clearly show only one differentiation rule per line.

(a) f (y) = log .. 3y − 2 Σ.

5y

(b) f (x) = 100 .

5 + 2e−3x

(c) f (t) = e^−5t sin(πt).

(d) f (θ) = cos.√3 θ2 + 1Σ.

(e) f (z) = .5z loge. sin(3z − 1) ΣΣ10.