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These results agree with the local extrema determined in Example 4-11 using the First Derivative Test on f (x) = – sin x – cos x on [0,2π]. F 4/25/01 8:59 AM Page 51 Chapter 4: Applications of the Derivative 51 Concavity and Points of Inflection The second derivative of a function may also be used to determine the general shape of its graph on selected intervals. A function is said to be concave upward on an interval if f"(x) > 0 at each point in the interval and concave downward on an interval if f"(x) < 0 at each point in the interval.

E5-x – tan x 4. 3 5. F 4/25/01 8:58 AM Page 43 Chapter 4 APPLICATIONS OF THE DERIVATIVE Chapter Check-In ❑ Using the derivative to understand the graph of a function ❑ Locating maximum and minimum values of a function ❑ Finding velocity and acceleration ❑ Relating rates of change ❑ Approximating quantities by using derivatives he derivative of a function has many applications to problems in calculus. It may be used in curve sketching; solving maximum and minT imum problems; solving distance; velocity, and acceleration problems; solving related rate problems; and approximating function values.

The technique of implicit differentiation allows you to find the derivative of y with respect to x without having to solve the given equation for y. The chain rule must be used whenever the function y is being differentiated because of our assumption that y may be expressed as a function of x. dy Example 3-19: Find dx if x 2 y 3 - xy = 10. F 38 4/25/01 8:57 AM Page 38 CliffsQuickReview Calculus Example 3-20: Find y' if y = sin x + cos y. Differentiating implicitly with respect to x, you find that 1 $ y l= cos x - sin y $ y l 1 $ y l+ sin y $ y l= cos x y l(1 + sin y) = cos x cos x y l= 1 + sin y Example 3-21: Find y' at (–1,1) if x2 + 3xy +y2 = –1.