Exercise Set 1
Derivatives

Problem 1

Consider the function f(x) = 3x^0.6.

1. Find the first- and second-order derivatives: f'(x) = ______, f"(x) = _______.

2. Evaluate the expression and the derivatives at x = 10: f(10) = _______, f'(10) = _________, f"(10) = ___________.

3. Plot f vs. x over x = 0 to 15.

4. What is the slope of the function f(x) at x = 2?

5. As x increases, does the slope of the function increase--or decrease?

Answers to Problem 1

a. f'(x) = 1.8/x^0.4, f"(x) = -0.72/x^1.4

b. f(10) = 11.943, f'(10) = 0.717, f"(10) = -0.029.

d. The slope of the function is given by the first derivative, f'(x). At x = 2, the slope is 1.36.

e. Since f" < 0, we conclude that the slope decreases as x increases. Confirm this by looking at the plot: the curvature becomes less steep as x rises.

Problem 2

Consider the function g(x) = 24x - 0.25x².

1. Find the first- and second-order derivatives: g'(x) = ______; g"(x) = _______.
   
   
2. Evaluate the expression and the derivatives at x = 2: g(2) = _______, g'(2) = _________, g"(2) = ___________.
   
   
3. Plot g vs. x over x = 0 to 48.
   
   
4. Plot g' vs. x over x = 0 to 48.
   
   
5. What is the slope of the function g(x) at x = 5? At x = 6?
   
   
6. Is the slope of the function increasing--or decreasing--in x? Explain.