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Lecture

7.4-Optimization.pdf

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Department
Mathematics
Course
MATH 157
Professor
Chiaka Drake
Semester
Fall

Description
1 Chapter 7.4 Optimization I ______________ Maximum (y-value): Given a function f defined on its domain D f, iff (x) f (c)for all x in D , then f (c)is the ______________ maximum value. f ______________ Minimum (y-value): Given a function f defined on its domain D f, iff (x) f (c)for all x in D f, then f (c)is the ______________ minimum value. Extreme Value Theorem: Existence Theorem A ____________ function f defined on some closed interval [a,b] will have both an ______________ maximum and an ______________ minimum on the interval. Copy From Text: p. 386 2 Examples Q1: Find the absolute extrema, if they exist, for the following functions defined on the given interval. 3 2 a) f (x) x 6x 1 , 1,5] b) g(x) 3 x 2 , [1,4] 3 2 2 c) h(x) (x 1) (x 1) ,[ 2,3] 3 3 d) k(x) 5x x ,[ 1,4] 4 Q2: Indicate all absolute and relative extrema on the graph of the function f below for the interval [ 8,6] . Q3: Indicate all absolute and relative extrema on the graph of the function f below for the interval [ 1,3] . 5 Q4: The demand equation for a manufacturer’s product is 80 x p 4 with 0 x 80 , where x is the number of units and p is price per unit in dollars. a) At what value of x will there be maximum revenue? b) What is the maximum revenue? 6 1 3 Q5: A bicycle manufacturer has costs C(x) 60x 15x 10 and demands p 90 x for 0 x 90 bicycles per week, where p is the price in dollars per bicycle. a) Find the maximum weekly revenue. b) Find the maximum weekly profit. 7 c) Show that marginal revenue and marginal cost are the same at the point where the maximum profit occurs. d) Find the price the manufacturer should charge to realize maximum profit. 8 Chapter 7.5 Optimization II Optimization Solving Guidelines: I suggest you do the following steps in ______________ in order to solve an optimization problem posed on a function f . 1. Read the problem at least _________. 2. If possible, ___________ a diagram and ___________ pertinent parts. 3. _
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