M129: Applied Calculus

Tutor Marked Assignment

Cut-Off Date:--------, 2015 Total Marks: 40

ContentsFeedback form …………………..…………………….…...….. 2Question 1 …………………………………………………..……… 3Question 2 …………………..………………..………………………3Question 3 ……………………..………………..……………………4Question 4 ………………..………………………..……………… 4Question 5 …………………………………………………..……… 5Question 6 …………………………………..……………………… 5Question 7 …………………………………..…………………… 6Question 8 ……………………………………..…………………… 6

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M129 TMA Feedback Form

[A] Student Component

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[B] Tutor Component

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QUESTION 1 2 3 4 5 6 7 8

MARK 5 5 5 5 5 5 5 5

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Tutor’s Comments:

This TMA covers only chapters 0, 1, 2, 3 and4. The TMA consists of eight questions for a total of 40 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results.

Q−1: [3+2Marks]

a) Given f ( x )=2 x2−16 x+30 , g ( x )=x (x−3)2(x+1)4

Find the zeros of g(x)and f(x).b) Find the points of intersection, if any, of the pair of curves :

x2+ y2=13, y=x+1

Q−2: [2+2+1Marks] a) Letf ( x )=√x−3 , g ( x )=x2+8.

(I) Find f(g(x)).(II) Find g(f (x)).

b) Find the domains of g(f (x)).

Q−3: [2+3 Marks] Use the definition of the derivative to find f ′(x) ifa) f ( x )=x2+xb) f ( x )= 1

x−1

Q−4: [2 + 3 Marks]

a) Find an equation of the tangent line to the graph of f ( x )= x3+1x2+ x+1

at the point whose x-coordinate is 1.

b) Find the equation of the tangent line to the curve at the point (4,1).

Q−5:[2.5+2.5 Marks] differentiate the following function:

a) f ( x )=(x2+

1

x3 ) ( x3−x2+1 )

b) f ( x )=e(x3+2 x)

Q−6: [5×1Marks] Letf ( x )=23

x3−5x2+12 x−7

a) Find f ′(x) and f ′′(x).b) Find the intervals on which f (x) is increasing or decreasing.c) Find the local maximum and minimum of f (x), if any.d) Find the intervals on which the graph of f (x) is concave up or

concave down.e) Find the inflection points of f(x), if any.

Q−7:[5 Marks]Poster is to be designed with 50 in2 of printed type, 4 inch margins on both the top and the bottom, and 2 inch margins on each side. Find the dimensions of the poster which minimize the amount of paper used.

Q−8:[5 Marks]Use the logarithmic differentiation to differentiate the function

y=(x+2 )2

ex √x2+16