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# MATH 125 CALCULUS STUDY GUIDE MATH 125 010

UT

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This 3 page Study Guide was uploaded by Rachel Klein on Sunday March 27, 2016. The Study Guide belongs to MATH 125 010 at University of Tennessee - Knoxville taught by in Spring 2016. Since its upload, it has received 12 views. For similar materials see Basic Calculus 125 in Math at University of Tennessee - Knoxville.

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Date Created: 03/27/16

Math 125, Test 3 Sections 8.4, 8.5, 8.6, 9.1, and 9.2 8.4 Increasing and Decreasing Functions How to find whether a function is increasing or decreasing 1.! Find f’(x). 2.! Find where f’(x) = 0 and where f’(x) is undefined. a.! F’(x) is undefined when the denominator = 0. b.! These values are called critical numbers i.! A critical number is the x value for which the function changes from increasing to decreasing 3.! Use the critical numbers to create intervals. 4.! Test the intervals using the following table Interval Interval Interval Test value (in interval) Test value (in interval) Test value (in interval) Sign of test value Sign of test value Sign of test value Increasing/decreasing Increasing/decreasing Increasing/decreasing a.! If the sign of the test value is negative, then f(x) is decreasing on this interval b.! If the sign of the test value is positive, then the f(x) is increasing on this interval 8.5 Extrema and First Derivative How to find if a critical number is a relative extrema 1.! Follow the same steps as above to find where the function is increasing or decreasing. 2.! The relative extrema will exist where there is a critical number a.! Just because there is a critical number does not imply that there is an extrema 3.! If the interval before the critical number is increasing and the interval after the critical number is decreasing, then there is a relative maximum. ! ! ! 1! 4.! If the interval before the critical number is decreasing and the interval after the critical number is increasing, then there is a relative minimum. 8.6 Concavity and Second Derivative How to find whether a function is concave up or concave down. 1.! Find ! "" # . 2.! Find ! "" # = 0 and where ! "" # is undefined. a.! ! "" # 'is undefined where the denominator = 0. 3.! Create test intervals with these values. 4.! Test the signs of the intervals. "" 5.! If the sign is positive, then ! # is concave up for these values. "" If the sign is negative, then ! # 'is concave down for these values. The Second Derivative Test is used to determine if a critical number is a relative maximum or relative minimum; it is a replacement method for the first derivative test, often used because it is easier. 1.! Find ! "" # . 2.! Plug in the critical value for x. "" 3.! If ! # is negative, then the critical value is a relative maximum. "" 4.! If ! # is positive, then the critical value is a relative minimum. 9.1 Optimization How to solve an optimization problem 1.! Write a primary equation that is to be maximized or minimized 2.! If the primary equation has more than one independent variable, write a secondary equation that relates the independent variables. 3.! Determine the domain of primary equation. 4.! Find either the maximum or the minimum. ! ! ! 2! 9.2 Applications Finding maximum profit 1.! Write the primary equation a.! Profit = revenue – cost i.! Revenue = xp 1.! p is price, usually a demand function 2.! Write the secondary equation. 3.! Find the derivative of the profit equation. 4.! Find where P’(x) = 0 and where it is undefined. 5.! Follow normal directions for a maximization problem. 6.! Make sure that your solution answers the question that was originally asked. ! ! ! 3!

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