1. Consider the function f(x) := 1 − x 1 + x 2 for x ∈ [−2, 4]. (a) Why does f possess both an absolute maximum and an absolute minimum on the interval [−2, 4]?
(b) Please determine the absolute extrema of f on [−2, 4].
2. Consider the equation 4x−5 = cos(3x). Please show (by referring to the correct Theorems and verifying that their assumptions are satsified!) that the equation has (a) at least one real root. (b) at most one real root.
3. Use Newton’s Method to approximate the positive zero of 2 sin x = x − 2 correctly to six decimal places.
4. Find the limit (a) limx→∞ 1 + x + 3x 5/ x 5 − 5x 2 + 1
(b) lim x→−∞ p 4x 2 − 3x + 2x
5. Find the function f whose 2nd derivative is f 00(x) = x 3 + 2x 2 + 4 and which has the boundary values f(0) = 1, and f(1) = 2. 1
6. Please use the guidelines of Section 3.5 to sketch the graph of the function f(x) := x 2 x + 2 7. A rectangular storage container with an open top is to have a volume of 10 m3 . The length of its base is twice the width. Material for the base costs $8 per square meter. Material for the sides costs $5 per square meter. Find the cost of materials for the cheapest such container.
