Answer all problems showing all working.
1) Find the equation of the tangent line to the curve y=2x-x^{3} at the point (1,1).
2) Evaluate the derivative of the function below from first principles.
f(x)=(x^{2}-2x/x^{2}-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)= x^{4}-2x^{2}
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) x^{2}=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 z_{1} and z_{2} in polar form. Find the product z_{1}z_{2} and the quotients z_{1}/z_{2} and 1/z_{1}. Leave your answer in polar form.
a) z_{1}= 4√3 - 4i z_{2}=8i
b) z_{1}=3+4i z_{2}=2-2i
11) Utilising De Moivre’s theorem to evaluate the following
a) (1-√3i)^{5}
b) (-1/2 -√3/2)^{15}