By Matthew Boelkins, David Austin, Steven Schlicker

Energetic Calculus isn't like so much present texts in a minimum of the subsequent methods: the fashion of the textual content calls for scholars to be lively inexperienced persons; there are only a few labored examples within the textual content, with there as an alternative being three or four actions in keeping with part that have interaction scholars in connecting rules, fixing difficulties, and constructing knowing of key calculus rules. each one part starts with motivating questions, a short creation, and a preview job, all of that are designed to be learn and accomplished sooner than type. The workouts are few in quantity and hard in nature. The booklet is open resource and will be used as a chief or supplemental textual content.

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Summary In this section, we encountered the following important ideas: f (b) − f (a) • The average rate of change of a function f on the interval [a, b] is . The b−a units on the average rate of change are units of f per unit of x, and the numerical value of the average rate of change represents the slope of the secant line between the points (a, f (a)) and (b, f (b)) on the graph of y = f (x). If we view the interval as being [a, a + h] instead of [a, b], the meaning is still the same, but the average rate of change f (a + h) − f (a) is now computed by .

Slopes on the graph of f correspond to heights on the graph of f . same as the height at the corresponding point on the righthand graph. That is, at each respective value of x, the slope of the tangent line to the original function at that x-value is the same as the height of the derivative function at that x-value. Do note, however, that the units on the vertical axes are diﬀerent: in the left graph, the vertical units are simply the output units of f . On the righthand graph of y = f (x), the units on the vertical axis are units of f per unit of x.

5. 13. A company manufactures rope, and the total cost of producing r feet of rope is C(r) dollars. (a) What does it mean to say that C(2000) = 800? (b) What are the units of C (r)? 35. Estimate C(2100), and justify your estimate by writing at least one sentence that explains your thinking. (d) Which of the following statements do you think is true, and why? 1. What would the practical meaning of this derivative value tell you about the approximate cost of the next foot of rope? Is this possible?

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