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# Chapter 7 Exam Study Guide MATH 125 010

UT

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This 2 page Study Guide was uploaded by Rachel Klein on Thursday February 4, 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 395 views. For similar materials see Basic Calculus 125 in Math at University of Tennessee - Knoxville.

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Date Created: 02/04/16

Chapter 7 Exam Math 125, Spring 2016 1.! Limits a.! Looks at how a function is behaving as a function approaches a certain point b.! Rules for basic limits !! lim " cb = 0 !! lim " cx = c !! lim x cx = c n " !! lim " ! = " # If n is even, c is positive. If n is odd, c is negative. " c c.! The limit does not exist if there are 2 ways to approach the limit !! However, it can have a limit from the left and a limit from the right d.! Properties of Basic Limits !! lim " cf(x) = L !! lim x cg(x) = K " •! lim (b(f(x)) = bL " c •! lim " c(f(x) + g(x)) = L + K •! lim " c(f(x)g(x)) = LK •! lim $(&)= ) " c((&) * •! lim x c(f(x) ) = L " " " •! lim " c +(!) = , 2.! Continuity a.! A function is continuous at a point c when the following are true !! F(c) is defined !! lim " cf(x) exists !! lim x cf(x) = f(c) " b.! Polynomials are always continuous c.! A rational function is continuous at every point on its domain (when x is defined) d.! Removable discontinuity is possible !! You can “make” a function continuous if it can be simplified to a form in which it has an unlimited domain e.! You can have a closed interval [a,b] in which a function is continuous just in this range 3.! Derivatives a.! -.= lim $ &/0 1$(&) -& x"h 0 b.! How to find tangent line of a point (a,b) -. !! Find -& -. !! Plug a into ; this value is the slope the tangent line, m -& !! Use point slope formula: y-b = m(x-a) !! Simplify c.! Relationship between differentiable and continuous !! Differentiable " continuous !! NOT continuous " NOT differentiable 4.! Differentiation Rules; “Shortcuts” a.! The Constant Rule: f(x) = c, f’(x) = 0 b.! The Simple Power Rule: f(x) = x , f;(x) = nx n-1 c.! The Constant Multiple Rule: f(x) = c(f(x)), f’(x) = c(f’(x)) d.! The Sum (and difference) Rule: f(x) = f(x) + g(x), f’(x) = f’(x) + g’(x) 5.! Rates of Change ∆. $ 3 1$(4) a.! Average Rate of Change between points a and b: ∆& = 314 b.! Average Velocity = 56789:;<8;=<>?785: 56789:;<8;?<@: c.! Instantaneous Rate of Change is the same thing as finding the derivative d.! Marginals are rates of change in economics !! Profit = revenue – cost !! Marginals show an estimate of how much profit/revenue/cost would increase or decrease if x increases by 1 •! The derivative of profit is marginal profit •! The derivative of revenue is marginal revenue •! The derivative of cost is marginal cost e.! The Demand Function !! Demand- the number of goods that customers are willing to buy !! Demand function- the relationship between demand and price, P=f(x) !! Revenue: R(x) = xp = xf(x)

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