By Stephen R. Munzer

ISBN-10: 0521372844

ISBN-13: 9780521372848

ISBN-10: 0521378869

ISBN-13: 9780521378864

This ebook represents an immense new assertion at the factor of estate rights. It argues for the justification of a few rights of non-public estate whereas displaying why unequal distributions of personal estate are indefensible.

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The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. It states the following: If a function f (x) is continuous on a closed interval [a,b], then f (x) has both a maximum and minimum value on [a,b]. 46 CliffsQuickReview Calculus The procedure for applying the Extreme Value Theorem is to first establish that the function is continuous on the closed interval. The next step is to determine all critical points in the given interval and evaluate the function at these critical points and at the endpoints of the interval.

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. A function f (x) is said to be concave up on an interval I if f'(x) is increasing on I; f (x) is concave down on I if f'(x) is decreasing on I.

F l(x) = 2x - 8 f l(3) = (2)( 3) - 8 =- 2 Example 3-8: If y = x +4 2 , find y lat (2, 1). (x + 2) ^ 0h - 4 (1) (x + 2) 2 = -4 2 (x + 2) At (2, 1), y l= - 4 2 (2 + 2) = -4 16 =- 1 4 y l= 33 34 CliffsQuickReview Calculus Example 3-9: Find the slope of the tangent line to the curve y = 12 – 3x2 at the point (–1,9). Because the slope of the tangent line to a curve is the derivative, you find that y' = –6x; hence, at (–1,9), y' = 6, and the tangent line has slope 6 at the point (–1,9). Trigonometric Function Differentiation The six trigonometric functions also have differentiation formulas that can be used in application problems of the derivative.

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