Each of the following limits represents a derivative f'(a). (4 + h)3 – 64 (a) lim h→0 h Choose the correct expression for f(x). 1 х — 4 4 + x x3 x3 – 64 х — 4 O x' + x Determine a. (Give your answer as a whole or exact number.) = D

17. Sketch the graph of f(x): = 2x + 2, 3 x³ = x, x ≤ -1 −1 and find the derivative. Determine whether f satisfies the conditions of the mean-value theorem on the interval [-3,2] and, if so, find the numbers c that satisfy the conclusion of the theorem.

Consider the following. 2x f(x) x + 2 Use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. O y y 2- 0 --2 0 -1- 1 1 2 f(x) f(x) 2 3 3 X 4 O f' is never zero. O f approaches √2 when f' is zero. Of approaches -√2 when f' is zero. 4 5 f'(x) f'(x) 5 0 -1- Describe the behavior of the function when the derivative is zero. Of approaches to when f' is zero. O 3¹¹f'(x) 0.5 3 f'(x) 0 0.5 1 to X 2 1.5 2 X 2.5 f(x) f(x) 2.5 3

Apply the limit definition to compute the derivative of each function. Use proper limit notation and show all details. Simplify your final answers as much as possible. a) g(x) = 4 + 3x b) h(x) = 5 - x c) p(x) = /5x? – 3 2.