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.
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?
a) y= (1+sinx)/(x+cosx)
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.
8) Find all solutions of the equation and express them in the form a + bi.
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
b) (-1/2 -√3/2)15