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# MATH 125 FINAL EXAM STUDY GUIDE MATH 125 010

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This 10 page Study Guide was uploaded by Rachel Klein on Thursday April 28, 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 44 views. For similar materials see Basic Calculus 125 in Math at University of Tennessee - Knoxville.

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Date Created: 04/28/16

Math 125 Final Exam Spring 2016 HIGHLIGHT = formula HIGHLIGHT = key phrase HIGHLIGHT = rules 7.1 Limits •! A limit looks at how a function is behaving as a function approaches a certain point •! Rules for basic limits o! lim b = 0 x!c o! lim x!cx = c n n o! lim x!cx = c " " o! lim x!c ! = # If n is even, c is positive. If n is odd, c is negative. o! lim x cf(x) = L and limx c g(x) = K ! ! "! lim x c (b(f(x)) = bL ! "! lim x c (f(x) + g(x)) = L + K ! "! lim (f(x)g(x)) = LK x!c $(&) ) "! lim x!c = ((&) * n n "! lim x!c (f(x) ) = L " " "! lim x!c +(!) = , •! The limit does not exist if there are different 2 ways to approach the limit o! However, it can have a limit from the left and a limit from the right 7.2 Continuity •! A function is continuous at a point c when the following are true o! F(c) is defined o! lim x!cf(x) exists o! lim x cf(x) = f(c) ! •! Polynomials are always continuous •! A rational function is continuous at every point on its domain (when x is defined) •! Removable discontinuity is possible o! You can “make” a function continuous if it can be simplified to a form in which it has an unlimited domain •! You can have a closed interval [a,b] in which a function is continuous just in this range ! ! ! 1! 7.3 Derivatives •! The formula to find a derivative: -. 0 /12 30(/) = lim x h -/ ! 2 •! How to find tangent line of a point (a, b) 45 1.! Find 4& 2.! Plug a into 45; this value is the slope the tangent line, m 4& 3.! Use point slope formula: y-b = m(x-a) 4.! Simplify •! Relationship between differentiable and continuous 1.! Differentiable ! continuous 2.! NOT continuous ! NOT differentiable 7.4 Differentiation Rules •! The Constant Rule: f(x) = c, f’(x) = 0 •! The Simple Power Rule: f(x) = x , f;(x) = nx n-1 •! The Constant Multiple Rule: f(x) = c(f(x)), f’(x) = c(f’(x)) •! The Sum (and difference) Rule: f(x) = f(x) + g(x), f’(x) = f’(x) + g’(x) 7.5 Rates of Change ∆. 0 7 30(8) •! Average Rate of Change between points a and b: ∆/ = 738 •! Average Velocity = 9:;<=>?@<?A@BC;<9> 9:;<=>?@<?C@D> •! Instantaneous Rate of Change is the same thing as finding the derivative •! Marginals are rates of change in economics o! Profit = revenue – cost o! Marginals show an estimate of how much profit/revenue/cost would increase or decrease if x increases by 1 i.! The derivative of profit is marginal profit ii.! The derivative of revenue is marginal revenue iii.! The derivative of cost is marginal cost ! ! ! 2! •! The Demand Function o! Demand- the number of goods that customers are willing to buy o! Demand function- the relationship between demand and price, P=f(x) o! Revenue: R(x) = xp = xf(x) 7.6 Product and Quotient Rules •! The Product Rule 0 / ∙ F / = 0 / F / + ?0 / F′(/) o! Can also be solved by multiplying out the equation and solving as a polynomial o! When there are 3+ functions + ! ∙ K ! ∙ ℎ ! = + ! K ! ℎ ! + ?+ ! K ! ℎ ! + + ! K ! ℎ′(!) •! The Quotient Rule 0 / H 0 / F / − 0 / F′(/) = F / (F / ) N 77, Chain Rule •! The Chain Rule OP OP OQ O! =?OQ O! can also be written as -. H -/ 0 F / =?0 F / ∙ F′(/) •! Used for composite functions o! Example of when to use chain rule: + ! =? 2! + 1 "! + ! =? ! and K ! = ?2! + 1 •! The General Power Rule -. T3V -/ = T(U / ) ∙ U′(/) o! Example of when to use chain rule: + ! = ! + 2!X Y 8.1 Higher Order Derivatives ! ! ! 3! Y X •! Original function: + ! =?3! + 4! + 2! H W ^ •! First order derivative: + ! = ?15! + 8! + 2 HH ^ X •! Second order derivative: + ! = ?60! + 24! •! Third order derivative: + HHH! = ?180! + 48! •! Fourth order derivative: + W ! = 360! + 48 Y •! Fifth order derivative: + ! = ?360 a •! Sixth order derivative: + ! = 0 •! The position of any free falling object is given by b c =?−Vdc + e c + 2 f "! h = height of the object when it starts to fall "! v 0 initial velocity "! t = time in seconds o! g(h) represents distance as a function of time o! g′(h) represents velocity with respect to time o! g′′(h) represents acceleration with respect to time 10.3 Derivatives of Exponential Functions •! Review of Exponential Functions o! An exponential function is when the variable is in the exponent: K ! =?2 x o! When the constant (in this case the constant is 2) is positive, then the function will have exponential decay to left of the y-axis and exponential growth to the right o! Properties of Exponents (not calculus) "! a = 1 x y x+y "! a a = a i j "! k =?l &35 i & 5 xy "! l ) = a x x x "! (lm) = a b n "! a =-x ij •! Natural Number e x o! Natural exponential function is f(x) = e x x o! The Derivative of f(x) = e is f’(x) = e •! Sometimes this function can be more complicated ! ! ! 4! - oU / =?o U(/)∙ U′(/) -/ 10.5 Logarithms and their Derivatives •! Quick Review of Logarithms (not calculus) o! Basically the opposite of an exponent … o! ln! =… ! e = x s ln! = m???????? =??????????r = ! o! No such thing as a logarithm of a negative number or a logarithm of 0 •! Basic Properties of Logarithms (not calculus) o! lnr = ! o! rt<&= ! o! ln!P = ln! + lnP & o! ln5= ln! − lnP o! ln! = vln! •! Derivatives of Logarithms - V -/ wx/ = / OR - V wx(U / ) = ∙ U′(/) -/ U(/) ! ! ! 5! 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. 3.! F’(x) is undefined when the denominator = 0. 4.! These values are called critical numbers 5.! A critical number is the x value for which the function changes from increasing to decreasing 6.! Use the critical numbers to create intervals. 7.! 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 ! ! ! 6! 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 + HH ! . 2.! Find + HH ! = 0 and where + HH ! is undefined. HH 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 + HH ! is concave up for these values. If the sign is negative, then + HH! ?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 + HH ! . 2.! Plug in the critical value for x. 3.! If + HH! is negative, then the critical value is a relative maximum. 4.! If + HH! 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. ! ! ! 7! 4.! Find either the maximum or the minimum. 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 primary 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. 11.1 Antiderivatives and Integrals •! + ! O! = z ! + { Integral! sign! Integrand! Differential! Antiderivative! •! An integral is the opposite function of the derivative + ! O! = + ! + ?{ •! Integration Rules 1.! |O! = |! + { 2.! |+(!)O! = | + ! O! 3.! [+ ! + K ! ]O! =? +(!)O!? +? K ! O! 4.! [+ ! − K ! ]O! =? +(!)O!? −? K ! O! /TV 5.! / -/ =? + Ä, T ≠ V T1V ! ! ! 8! 11.2 Integration by Substitution and General Power Rule -U U TV •! U T -/ =? U -U =? + Ä -/ T1V 1.! Identify a factor of u of the integrand that is raised to a power. Write down this factor as u = __________. 4É 2.! Then find and write do4&. 3.! Show that this derivative is also a factor of the integrand. 4.! If it is, then continue to the final step of the equation. 5.! If it is not, then manipulate the equation so that it is. ** Remember that if you multiply something inside the integral, then you must divide by the same factor outside the integral. 11.3 Exponential and Logarithmic Integrals •! o -U =?o + ÄU , when u is a function of x. V -U 3V •! -U =? =? U =?wx U + Ä U U 11.4 The Fundamental Theorem •! If f(x) is non-negative and continuous in the interval [a,b], then 7 0 / -/ = Ñ / 7 = ?Ñ 7 − ?Ñ(8) 8 8 o! Used to find the area under a curve •! Positive, Negative, and Nearly Zero s o! Positive:i + ! O! > 0 o! Negative: s + ! O! < 0 i s o! Nearly Zero: i+ ! O! = nearly zero •! The Average Value of a Function V 7 o! 0 / -/ 738 8 ! ! ! 9! 11.5 Area of a Region Bounded by 2 Graphs 7 7 7 •! 0 / − F(/) -/? =?? 0 / -/ −? F / -/ 8 8 8 o! In order to determine which function should be subtracted from which function, it is helpful to graph the functions in order to determine which function is located above the other. The one that is on top is the one that is used as f(x). •! Consumer Surplus and Producer Surplus o! The demand function: consumer’s behavior; shows what quantity of goods is bought at various prices o! The supply function: producer’s behavior; shows what quantity of the item producers will supply at different price levels o! Equilibrium: where supply = demand /ë o! Consumer surplus: ÄÖ =? (ÜáàâxÜ?äãxåçéèx − ê )-/ ë , where the ë equilibrium is (o ,0p ). "! The amount gained by buying an item at the current price rather than than the price they are willing to pay o! Producer surplus:íÖ =? /ë ê ìîãïïwñ?äãxåçéèx -/ , where the ë ë equilibrium is (0 ,op ). "! The amount gained by selling at the current price rather than the price they are willing to accept o! Total gains = consumer surplus + producer surplus ! ! ! 10!

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