math homework on ( calculus + number system )

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show the working out.

ASAP 🙂 today 

 

5 to 10$ maximum 

MAT

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CNS and MAT1CPE ASSIGNMENT 4,

2

013

Place your assignment solutions in the appropriate pigeonhole in the boxes on the third level of the
Mathematics Building before 12.00noon on Thursday 2ndndnd May.

Each page of your solutions must carry (1) your name, (2) your demonstrator’s name,
and (3) the day and time of your first Practice Class of the week.

In submitting your work, you are consenting that it may be copied and transmitted by the
University for the detection of plagiarism. You must start your assignment solutions with
the following Statement of Originality, signed and dated by you:

“This is my own work. I have not copied any of it from anyone else.”

The assignments are designed to help you master the concepts in this subject and also for you to
develop your mathematical communication skills. Please note that often it is not the final answer
that is important but your mastery of the required techniques and the way you communicate your
ideas and your approach to the problems. Note that your use of language and mathematical symbols
are worth marks in the assignment and that these marks will also be used to assess the written
communication aspect of your faculty graduate capability score.

For Question 1 refer to Chapter 5 of Survival Skills for First-Year Mathematics.

1. Find the following by adding or subtracting multiples of 2π, using the basic triangles, using the
fact that cos is an even function (i.e., cos(−x) = cos(x)) while sin and tan are odd functions
(i.e., sin(−x) = −sin(x) and tan(−x) = −tan(x)), and using the unit circle.

(a) sin( 17π
3

), (b) cos( 13π
4

), (c) tan(−19π
3

), (d) sin(−11π
2

), (e) cos(9π).

2. In each of the following, find all stationary points, in the form (x,y), and then use the second
derivative test to classify them. You need only evaluate the second derivative to the point
where its sign is clear.

Recall that a point a such that f ′(a) = 0 is the domain co-ordinate for a stationary
point on the graph of f. The second derivative test tells us that:

If f ′′(a) > 0, then the stationary point is a local minimum.

If f ′′(a) < 0, then the stationary point is a local maximum.

If f ′′(a) = 0, then the test is inconclusive.

(a) g(x) = x +

1

x
(b) y =

x− 2
x2 + 5

In Questions 3 and 4 you should refer to Chapter 7 of the Notes on Number
Systems and you should use the setting out given in the model answers to Number
Systems Practice Classes 6 and 7 as a guide.

3. (a) Express each of the following numbers in the form a + bi where a,b ∈ R.

You must show your working.

(i) (2 − 3i)(−4 + i)

(ii)
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2 − 3i
(b) Express each of the following numbers in the form r cis(θ) with r > 0 and −π < θ 6 π.

You must show your working.

(i) 3 − 3i, (ii) −5 cis(5π/6), (iii) z9 where z = 10 cis(2π/5).

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4. (a) (i) Let w = cis(π
3
). Use deM to calculate w2, w3 and w4 in polar form.

(ii) Plot w, w2, w3 and w4 on an Argand diagram

(iii) Use your diagram or the fact that cis(θ) := cos(θ) + i sin(θ) to convert w, w2, w3 and
w4 to the form x + yi, with x,y ∈ R.

(iv) Use your answer to (iii) to write down a solution to the equation z4 = −1
2
−
√

3
2
i.

(v) Use the kth Roots Theorem (see Number Systems Lecture 7 and page 97 of

the Number Systems Notes) to find all solutions to the equation z4 = −1
2
−
√

3
2
i.

Give answers in the polar form r cis(θ) with −π < θ 6 π. [Base your answer on the model answers to Questions 2, 3 and 4 on Number Systems Practice Class 7.]

(b) [See Number Systems Practice Class 6 and Workshop 7 on the LMS.] Sketch
the following regions in the complex plane. [Give a separate diagram for each part.]

(i) A := {z ∈ C : |z + 2 + 2i| 6 2
√

2}.
(ii) B := {z ∈ C : −π

2

6 arg(z) 6 0}.

(iii) A∩B = {z ∈ C : |z + 2 + 2i| 6 2
√

2 and − π
2

6 arg(z) 6 0}.

5. (a) Draw a neat sketch of the graph of y = 3x − 7 and clearly shade and label the areas

involved in calculating

∫ 4
1

(3x− 7) dx. Also mark in all appropriate values on the x and
y axes.

(b) Calculate

∫ 4
1

(3x− 7) dx in terms of area as in Question 3 of Calculus Practice Class 6.

(Do not use rules of integration to do this!)

6. Consider the definite integral

∫ 2
0

1

2x + 1
dx.

(a) Using your calculator, or otherwise, sketch the graph of f(x) =
1

2x + 1
, for 0 6 x 6 2.

On the graph you have drawn, sketch the graph of the function that approximates f(x)
according to the Riemann approximation with four subintervals. Label the height of each
step on the co-domain axis.

(b) In each of the following, use the Riemann formula to write down the expression for and
then, correct to 4 decimal places, obtain a numerical approximation to the integral using,

(i) the Riemann approximation with four subintervals.

(ii) the Riemann approximation with eight subintervals.

(c) Carry out the integration to calculate the exact answer and evaluate this correct to 4
decimal places. (Check the table of common integrals in the appendix).

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