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How to Guide for Math Midterm 2.docx

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Richard Blute

Basic Rules and Concepts  If x=c is not in the domain, then there are not critical points  If F’(x) is undefined, there is also a critical point  If local maximum at x=c, there will be a critical point at x=c What if there is a critical point at x=c which is neither a local max or local minimum? Saddle Point More cool rules  Always throw away points not in interval  Absolute extreme occurs at x=c if f > f(x) for every x Concavity  F”(x)<0 concave down Absolute Maximum  F”(x)>0 concave up Absolute Minimum  If there is an inflection point at x=c, then f”(x)=0 or f”(c) is undefined  You must check if function is concave up from one side and concave down from the other Second Derivative Test Suppose f(x) has CP 1) There is a local max at x=c if concave down so f”(x) > 0 2) There is a local minimum if p”(x) > 0 3 methods for Validating the Presence of Absolute Extrema 1) Your function is a simple function like a parabola A<0 (absolute max) A>0 (absolute min) 2) Use extreme value theorem F(x) is continuous [a,b] F(x) has an absolute maximum and minimum 3) Thm not in book F(x) is continuous on an open interval I and has only one CP on interval I, then if that CP is a local max it is an absolute max as well How-to Questions for Midterm How to find elasticity of demand: 1. Find derivative 2. Sub in 2 in original equation given to get p 3. Sub in 2 in derivative to get denominator value 4. Divide value attained from original equation by elasticity of demand (x value) by value attained by subbing 2 in derivative How to find local minimums and maximums: 1. Find the first derivative of the given function 2. Get the critical point of the given function 3. Look at a value to the left and to the right of the critical point 4. Make a table How to determine the concavity of a function 1. Find the first derivative 2. Find the second derivative 3. Find the critical points of the second derivative 4. Sub points in between the critical points 5. Wherever the curve begins to turn up or down is called a point of inflection Optimization Question About Maximizing Profit 1. Isolate x and P 2. X value will be quantity of item sold 3. P value will be price of item 4. Make a chart; Put P on one side and X on the other a. Find the slope (you will typically have two price values and two quantity values) 5. Insert the slope in y=mx+b form 6. Sub in slope value 7. SUB IN ONE OF THE POINTS IN THE ORIGINAL TABLE FOR Y IN Y=MX+B to find B 8. Find the revenue function (R(x)) by multplyng by x
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