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Answer all problems showing all working.

1) Find the equation of the tangent line to the curve y=2x-x3 at the point (1,1).
2) Evaluate the derivative of the function below from first principles.

              f(x)=(x2-2x/x2-x-2)

3) In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area.In 1999, the population of this area was 961, 400, and the population was increasing at roughly 9200 people per year. The average annual income was $30, 593 per capita, and this average was increasing at about $1400 per year. Use the Product Rule and these figures to estimate he rate at which total personal income was rising in the Richmond-Petersburg area in 1999. describe the meaning of each term in the Product Rule.

4) Let f(x)= x4-2x2

a) Use the definition of a derivative to find f'(x) and f''(x)

b) On what intervals is f(x) increasing or decreasing?

c) On what intervals is f(x) concave upward or concave downward?

5) Differentiate

a) y= (1+sinx)/(x+cosx)

b) √x+√y=1

6) Sketch the graph of a function that satisfies all of the given conditions:

a) f’(x) > 0 if x < 2; f’(x) > 0 if x > 2; f’(2) = 0

b) f’’(x) < 0 if x < 2; f’’(x) < 0 if x >2; f not differentiable at x = 2

7) Evaluate the radical expression and express the result in the form a + bi.

a) √(-9/4)

b) √1/3(√-27)

8) Find all solutions of the equation and express them in the form a + bi.

a) x2=1/2x+1=0

b) z+4+12/z=0

9) prepare the complex number in polar form with argument θ between 0 and 2π

a) 3i(1 + i)

b)√2 + √2i

10) prepare z1 and z2 in polar form. Find the product z1z2 and the quotients z1/z2 and 1/z1. Leave your answer in polar form.

a) z1= 4√3 - 4i       z2=8i

b) z1=3+4i             z2=2-2i

11) Utilising De Moivre’s theorem to evaluate the following

a) (1-√3i)5

b) (-1/2 -√3/2)15

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  • Category:- Math
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