By Matthew Boelkins, David Austin, Steven Schlicker

Energetic Calculus isn't like such a lot current texts in at the very least the subsequent methods: the fashion of the textual content calls for scholars to be lively freshmen; there are only a few labored examples within the textual content, with there as an alternative being three or four actions in step with part that have interaction scholars in connecting principles, fixing difficulties, and constructing realizing of key calculus principles. every one part starts off with motivating questions, a quick advent, and a preview task, all of that are designed to be learn and accomplished sooner than category. The workouts are few in quantity and difficult in nature. The ebook is open resource and will be used as a first-rate or supplemental textual content.

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The central diﬀerence approximation to the value of the ﬁrst derivative is given by f (a) ≈ f (a + h) − f (a − h) , 2h and this quantity measures the slope of the secant line to y = f (x) through the points (a − h, f (a − h)) and (a + h, f (a + h)). The central diﬀerence generates a good approximation of the derivative’s value any time we have symmetric data surrounding a point of interest. • Knowing the derivative and function values at a single point enables us to estimate other function values nearby.

For now, it suﬃces to understand this as a consequence of the jump that must occur in the derivative function at a sharp corner on the graph of the original function. Exercises 1. Let f be a function with the following properties: f is diﬀerentiable at every value of x (that is, f has a derivative at every point), f (−2) = 1, and f (−2) = −2, f (−1) = −1, f (0) = 0, f (1) = 1, and f (2) = 2. 19, sketch a possible graph of y = f (x). Explain why your graph meets the stated criteria. 19, sketch a possible graph of y = f (x).

Include units on your answer and write one sentence to explain the meaning (in everyday language) of the value you found. (c) Use the limit deﬁnition to write an expression for the instantaneous rate of change of P with respect to time, t, at the instant a = 2. Explain why this limit is diﬃcult to evaluate exactly. (d) Estimate the limit in (c) for the instantaneous rate of change of P at the instant a = 2 by using several small h values. 3. THE DERIVATIVE OF A FUNCTION AT A POINT estimate of P (2), include units on your answer, and write one sentence (using everyday language) to explain the meaning of the value you found.