By Bernard V Zandy; Jonathan J White

Calculus : CliffsQuickReview Calculus, Anton/Bivens/Davis model via Cliffs quickly evaluation Publishing employees. Cliffs Notes, Inc.,2003

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As noted in Example 4–4, f'(x) = 0 at x = π/4 and 5π/4. Because f"(x) = – sin x – cos x, you find that f" (π/4) = - 2 and f has a relative maximum at (r/4, 2 ). Also, f" (5π/4)= 2, and f has a relative minimum at (5r/4, - 2 ). These results agree with the relative extrema determined in Example 4-11 using the First Derivative Test on f (x) = – sin x – cos x on [0,2π]. 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.

F (w) - f (2) (w 2 - 5) - (2 2 - 5) w-2 = w-2 2 = ww --24 ^w + 2 h ^w - 2 h = w-2 f (w) - f (2) =2 w - 2 = w + 2, w Y f l^ 2 h = lim ^w + 2 h = 2 + 2 = 4 w " 2 Hence, the derivative of f (x) = x2 – 5 at the point (2,–1) is 4. One interpretation of the derivative of a function at a point is the slope of the tangent line at this point. The derivative may be thought of as the limit of the slopes of the secant lines passing through a fixed point on a curve and other points on the curve that get closer and closer to the fixed point.

F ^ x h = * xx +-21, x Y 2, x =- 1 Answers: 1. (a) 6 (b) –6 (c) DNE 2. + ∞ 3. 0 4. 5/3 5. f is not continuous at x = –1. Chapter 3 THE DERIVATIVE Chapter Check-In ❑ Understanding derivatives ❑ Computing basic derivatives ❑ Finding derivatives of more complicated functions ne of the most important applications of limits is the concept of the O derivative of a function. In calculus, the derivative of a function is used in a wide variety of problems, and understanding it is essential to applying it to such problems.