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Post 12 months word: First released January 1st 1978
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Extra info for Calculus (Cliffs Quick Review)
The restrictions stated or implied for such functions will determine the domain from which you must work. 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.
X " 0 Multiplying the numerator and the denominator by 4 produces lim sinx4x = lim 4 sin 4x 4x x " 0 x " 0 = c lim 4 m $ lim sin 4x 4x x " 0 x " 0 =4$1 lim sinx4x = 4 Example 2-20: Evaluate lim sec xx - 1. x " 0 x " 0 Because sec x = 1/cos x, you find that 1 -1 lim sec xx - 1 = lim cos xx x " 0 x " 0 cos x = lim 1 x-cos x x " 0 1 $ 1 - cos x = lim c cos m xm c x x " 0 1 $ lim 1 - cos x = ; lim cos E x E ;x " 0 x x " 0 =1$0 lim sec xx - 1 = 0 x " 0 Continuity A function f (x) is said to be continuous at a point (c,f (c)) if each of the following conditions is satisfied: (1) f (c) exists (c is in the domain of f ), (2) lim f (x) exists, and x " c (3) lim f (x) = f (c).
If f'(x) > 0 at each point in an interval I, then the function is said to be increasing on I. If f'(x) < 0 at each point in an interval I, then the function is said to be decreasing on I. Because the derivative is zero or does not exist only at critical points of the function, it must be positive or negative at all other points where the function exists. In determining intervals where a function is increasing or decreasing, you first find domain values where all critical points will occur; then, test all intervals in the domain of the function to the left and to the right of these values to determine if the derivative is positive or negative.