Calculus (Cliffs Quick Review) by Bernard V. Zandy, Jonathan J. White

By Bernard V. Zandy, Jonathan J. White

By way of pinpointing the things you actually need to grasp, no one does it greater than CliffsNotes. This quickly, powerful instructional is helping you grasp middle Calculus options -- from capabilities, limits, and derivatives to differentials, integration, and yes integrals -- and get the very best grade.
At CliffsNotes, we're devoted to assisting you do your top, regardless of how difficult the topic. Our authors are veteran academics and gifted writers who understand how to chop to the chase -- and nil in at the crucial details you want to be successful.

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The function, together with its domain, will suggest which technique is appropriate to use in determining a maximum or minimum value—the Extreme Value Theorem, the First Derivative Test, or the Second Derivative Test. Example 4-16: A rectangular box with a square base and no top is to have a volume of 108 cubic inches. Find the dimensions for the box that require the least amount of material. The function that is to be minimized is the surface area (S) while the volume (V ) remains fixed at 108 cubic inches (Figure 4-2).

Example 2-24: Discuss the continuity of f (x) = x at x = 0. When the definition of continuity is applied to f (x) at x = 0, you find that (1) f (0) = 0 (2) lim f (x) = lim x DNE because lim x " 0 x " 0 but lim x DNE x " 0- x " 0+ x = 0, (3) lim f (x) = f (0) x " 0+ hence, f is continuous at x = 0 from the right only. F 4/25/01 8:53 AM Page 27 Chapter 2: Limits Example 2-25: Discuss the continuity of f (x) = * 5 - 2x, x < - 3 x 2 + 2, x $ - 3 27 at x = –3. When the definition of continuity is applied to f (x) at x = –3, you find that (1) f (- 3) = (- 3) 2 + 2 = 11 (2) lim f (x) = lim (5 - 2x) = 11 x " - 3- x " - 3- lim f (x) = lim (x 2 + 2) = 11 x " - 3+ x " - 3+ hence, lim f (x) = 11 because lim f (x) = lim f (x) x " 3 (3) x " - 3- x " - 3+ lim f (x) = f (- 3) x "-3 hence, f is continuous at x = –3.

X " c Geometrically, this means that there is no gap, split, or missing point for f (x) at c and that a pencil could be moved along the graph of f (x) through (c,f (c)) without lifting it off the graph. F 4/25/01 8:53 AM Page 25 Chapter 2: Limits 25 at (c,f (c)) from the right if lim f (x) = f (c) and continuous at (c,f (c)) x " c+ from the left if lim f (x) = f (c). Many of our familiar functions such as x " c linear, quadratic and other polynomial functions, rational functions, and the trigonometric functions are continuous at each point in their domain.

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