CliffsQuickReview calculus : Anton/Bivens/Davis version by Bernard V Zandy; Jonathan J White

By Bernard V Zandy; Jonathan J White

Calculus : CliffsQuickReview Calculus, Anton/Bivens/Davis model through Cliffs speedy assessment Publishing employees. Cliffs Notes, Inc.,2003

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Example 4-12: Find any relative extrema of f (x) = x4 – 8x2 using the Second Derivative Test. As noted in Example 4-3, f'(x) = 0 at x = –2, 0, and 2. Because f"(x) = 12x2 – 16, you find that f"(–2) = 32 > 0, and f has a relative minimum at (–2,–16); f"(0) = –16 < 0, and f has relative maximum at (0,0); and f"(2) = 32 > 0, and f has a relative minimum (2,–16). These results agree with the relative extrema determined in Example 4-10 using the First Derivative Test on f (x) = x4 – 8x2. Example 4-13: Find any relative extrema of f (x) = sin x + cos x on [0,2π] using the Second Derivative Test.

Chapter 3: The Derivative 39 Higher Order Derivatives Because the derivative of a function y = f (x) is itself a function y' = f'(x), you can take the derivative of f'(x), which is generally referred to as the second derivative of f(x) and written f"(x) or f (2)(x). This differentiation process can be continued to find the third, fourth, and successive derivatives of f (x), which are called higher order derivatives of f (x). Because the “prime” notation for derivatives would eventually become somewhat messy, it is preferable to use the numerical notation f (n)(x) = y(n) to denote the nth derivative of f (x).

Because y = arctan(x 3/2 ) 1 $ 3 x 1/2 1 + (x 3/2 ) 2 2 = 1 3 $ 32 x 1/2 1+x y l= y l= 3 x 2 (1 + x 3 ) Differentiation of Exponential and Logarithmic Functions Exponential functions and their corresponding inverse functions, called logarithmic functions, have the following differentiation formulas: (1) If f (x) = e x , then f l(x) = e x . (2) If f (x) = a x , a > 0, a ! 1, then f l(x) = ( ln a) $ a x . (3) If f (x) = ln x, then f l(x) = 1x . 1 . (4) If f (x) = log a x, a > 0, a ! 1, then f l(x) = ( ln a) $ x Note that the exponential function f (x) = ex has the special property that its derivative is the function itself, f '(x) = ex = f (x).

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