By Matthew Boelkins, David Austin, Steven Schlicker

Lively Calculus isn't like such a lot present texts in at the least the subsequent methods: the fashion of the textual content calls for scholars to be energetic newcomers; there are only a few labored examples within the textual content, with there as a substitute being three or four actions according to part that interact scholars in connecting principles, fixing difficulties, and constructing knowing of key calculus rules. every one part starts with motivating questions, a short advent, and a preview job, all of that are designed to be learn and accomplished ahead of category. The routines are few in quantity and demanding in nature. The booklet is open resource and will be used as a chief or supplemental textual content.

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Edu/s/5p. 4. THE DERIVATIVE FUNCTION 37 We now may take two different perspectives on thinking about the derivative function: given a graph of y = f (x), how does this graph lead to the graph of the derivative function y = f (x)? and given a formula for y = f (x), how does the limit definition of the derivative generate a formula for y = f (x)? Both of these issues are explored in the following activities. 10. For each given graph of y = f (x), sketch an approximate graph of its derivative function, y = f (x), on the axes immediately below.

For the first two, determine the formula for the derivative by thinking about the nature of the given function and its slope at various points; do not use the limit definition. For the latter four, use the limit definition. Pay careful attention to the function names and independent variables. It is important to be comfortable with using letters other than f and x. For example, given a function p(z), we call its derivative p (z). (a) f (x) = 1 (b) g(t) = t (c) p(z) = z 2 (d) q(s) = s3 (e) F(t) = 1 t (f) G(y) = √ y Summary In this section, we encountered the following important ideas: f (x) • The limit definition of the derivative, f (x) = limh→0 f (x+h)− , produces a value h for each x at which the derivative is defined, and this leads to a new function whose formula is y = f (x).

If necessary, adjust and label the vertical scale on the axes for f . 4. THE DERIVATIVE FUNCTION s r x x s′ r′ x x z w x x z′ w′ x x When you are finished with all 8 graphs, write several sentences that describe your overall process for sketching the graph of the derivative function, given the graph the original function. What are the values of the derivative function that you tend to identify first? What do you do thereafter? How do key traits of the graph of the derivative function exemplify properties of the graph of the original function?

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