By Matthew Boelkins, David Austin, Steven Schlicker

Lively Calculus isn't like such a lot current texts in not less than 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 consistent with part that interact scholars in connecting rules, fixing difficulties, and constructing knowing of key calculus rules. each one part starts off with motivating questions, a quick creation, and a preview job, all of that are designed to be learn and accomplished sooner than classification. The routines are few in quantity and not easy in nature. The booklet is open resource and will be used as a main or supplemental textual content.

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**Example text**

What does it mean to say that a function is concave up or concave down? How are these characteristics connected to certain properties of the derivative of the function? • What are the units of the second derivative? How do they help us understand the rate of change of the rate of change? Introduction Given a diﬀerentiable function y = f (x), we know that its derivative, y = f (x), is a related function whose output at a value x = a tells us the slope of the tangent line to y = f (x) at the point (a, f (a)).

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) . b−a It is essential to understand how the average rate of change of f on an interval is connected to its graph. 3. Suppose that f is the function given by the graph below and that a and a + h are the input values as labeled on the x-axis. 10 to answer the following questions.

Suppose that f is the function given by the graph below and that a and a + h are the input values as labeled on the x-axis. 10 to answer the following questions. 3. (a) Locate and label the points (a, f (a)) and (a + h, f (a + h)) on the graph. (b) Construct a right triangle whose hypotenuse is the line segment from (a, f (a)) to (a + h, f (a + h)). What are the lengths of the respective legs of this triangle? (c) What is the slope of the line that connects the points (a, f (a)) and (a + h, f (a + h))?

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

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