2. Intro to Derivatives

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. <MATCH A-D IMAGE>

Analyzing Graphs

Each of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?

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Recovering a function from its derivative

b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).

Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (−π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2