By Matthew Boelkins, David Austin, Steven Schlicker

Lively Calculus isn't like so much current texts in at the least the next methods: the fashion of the textual content calls for scholars to be energetic freshmen; there are only a few labored examples within the textual content, with there in its place being three or four actions in line with part that have interaction scholars in connecting rules, fixing difficulties, and constructing knowing of key calculus principles. every one part starts off with motivating questions, a short creation, and a preview task, all of that are designed to be learn and accomplished sooner than category. The routines are few in quantity and not easy in nature. The publication is open resource and will be used as a prime or supplemental textual content.

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Recall that for a moving object with position function s, its average velocity on the time interval t = a to t = a + h is given by the quotient AV[a,a+h] = s(a + h) − s(a) . 3. THE DERIVATIVE OF A FUNCTION AT A POINT 23 In a similar way, we make the following deﬁnition for an arbitrary function y = f (x). 2. For a function f , the average rate of change of f on the interval [a, a + h] is given by the value f (a + h) − f (a) . AV[a,a+h] = h Equivalently, if we want to consider the average rate of change of f on [a, b], we compute AV[a,b] = f (b) − f (a) .

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.

Provide your discussion in practical terms regarding the value of the car and the rate at which that value is changing. 6. 6 The second derivative Motivating Questions In this section, we strive to understand the ideas generated by the following important questions: • How does the derivative of a function tell us whether the function is increasing or decreasing at a point or on an interval? • What can we learn by taking the derivative of the derivative (to achieve the second derivative) of a function f ?

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

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