By Matthew Boelkins, David Austin, Steven Schlicker

Lively Calculus isn't like so much current texts in at the least the subsequent methods: the fashion of the textual content calls for scholars to be energetic newbies; there are only a few labored examples within the textual content, with there in its place being three or four actions according to part that interact scholars in connecting rules, fixing difficulties, and constructing realizing of key calculus principles. each one part starts off with motivating questions, a short advent, and a preview task, all of that are designed to be learn and accomplished ahead of classification. The routines are few in quantity and tough in nature. The booklet is open resource and will be used as a prime or supplemental textual content.

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**Extra resources for Active Calculus**

**Sample text**

3. THE DERIVATIVE OF A FUNCTION AT A POINT resembles the curve near x = a. 12: A sequence of secant lines approaching the tangent line to f at (a, f (a)). At right, we zoom in on the point (a, f (a)). The slope of the tangent line (in green) to f at (a, f (a)) is given by f (a). At this time, it is most important to note that f (a), the instantaneous rate of change of f with respect to x at x = a, also measures the slope of the tangent line to the curve y = f (x) at (a, f (a)). The following example demonstrates several key ideas involving the derivative of a function.

Even more than these individual numbers, if we consider the role of 0, 1, 2, and 3 in the process of computing the value of the derivative through the limit deﬁnition, we observe that the particular number has very little eﬀect on our work. To see this more clearly, we compute f (a), where a represents a number to be named later. Following the now standard process of using the limit deﬁnition of the derivative, f (a + h) − f (a) h 4(a + h) − (a + h)2 − (4a − a2 ) lim h→0 h 4a + 4h − a2 − 2ha − h2 − 4a + a2 lim h→0 h 4h − 2ha − h2 lim h→0 h h(4 − 2a − h) lim h→0 h lim (4 − 2a − h).

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? Why or why not? 14. Researchers at a major car company have found a function that relates gasoline consumption to speed for a particular model of car. In particular, they have determined that the consumption C, in liters per kilometer, at a given speed s, is given by a function C = f (s), where s is the car’s speed in kilometers per hour.

### Active Calculus by Matthew Boelkins, David Austin, Steven Schlicker

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