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Applications of Derivatives: Section 4.1

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York University
Mathematics and Statistics
MATH 1013
Anthony Anthony

Unit 4: Applications of Differentiation  Now we know differentiation rules, we can pursue better applications.  Ex. How derivative affect the graph of f(x), to locate max and mins.  Ex. Maximize/Minimize a cost or area for optimization. Recap: Intermediate Value Theorem: If a function f is continuous on a closed interval [a, b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x) = c. - If there is 2 points connected by a function, and it has one point above a horizontal line, and another below, then it MUST cross that line! 4.1: Maximum and Minimum Values: Optimization: To find the optimal/best way of doing something through the use of maximum/minimum values. o The shape of a can to minimize manufacturing costs? o Maximum acceleration of a space shuttle? o Angle blood vessels branch to minimize energy of pumping blood?  (Also known as extreme values of f). Absolute Maximum: If f(d) >= f(x) for ALL x in the domain of f, where f(d) is called the maximum value of f on the domain. Absolute Minimum: If f(a) <= f(x) in ALL x in the domain of f, where f(a) is called the minimum value of f on the domain. Local Maximum: The largest values of a certain interval. Ex. For interval (a,c) - f(b) is the local maximum in f(x), where x is sufficiently close to b. Local Minimum: The smallest values of a certain interval. Ex. For interval (b,d) - f(c) is the local minimum in f(x). Since f(c) <= f(x) where x is sufficiently close to c. ------------------------------------------------------------------------------------------------- Ex. For f(x) = cosx: Its absolute and local maximum is 1, and its absolute and local minimum is -1. 2 Ex. If f(x) = x , then f(x) >= f(0) for all x. Therefore f(0) = 0 is the absolute and local minimum of f.  There is no highest point on a parabola, so there is no max. 3 Ex. If f(x) = x , then the function neither has a max or min (abs and loc) f(-1) = Absolute maximum (It occurs at an end point) f(1) = Local maximum f(0) = Local minimum f(3) = Absolute & Local minimum f(4) = Neither local or absolute maximum. ------------------------------------------------------------------------------------------------- Extreme Value Theorem:  If f is continuous on a closed interval [a,b]: o There is at least an absolute max value f(c) and absolute m
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