Question

1)         Prove (with an ε- δ proof) limx→22x3-x2-3x=6

2)         fx= x5-5x3      

a)   Find the first derivative.

b)   Find all critical numbers.

c)   Make a single line graph showing where the function is increasing and where it is decreasing.

d) Find the coordinates of all stationary points, maxima, and minima.

e)   Find the second derivative. Find any numbers where the concavity of the function may change.

f) Make a single line graph showing the concavity of the function. Find the coordinates of all inflection points.

g) What are the x-intercepts?

h) Does this graph have any type of symmetry? If so, what leads you to this conclusion?

i) Graph the function. Label all intercepts, max, min, stationary points, and inflection points and give their coordinates. Use the full page of graph paper which has been provided.

3)         For the function fx= x3-5x+1, use Newton’s method to find an approximation of x such that f(x) = 0. Start with x0=-3 and find x2. Answer with 6 decimal places.

4)         A right circular cylinder is placed inside a cone whose radius is R and height H so that the base of the cylinder lies on the base of the cone. Find the maximum lateral surface area of the cylinder (the area of the curved part of the cylinder, not the top or bottom).

5)         Find the antiderivative (also called indefinite integral) of    fx=3x5-4x3+7x2-12

6)         If fx= x3-8x-5 and x∈ 1,4, find all numbers c∈(1,4) satisfying the conclusion of the Mean Value Theorem.

Answer 1