Calculus & Analytic Geometry I
Calculus & Analytic Geometry I MATH 131
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This 11 page Class Notes was uploaded by Dr. Cleo Wunsch on Thursday October 15, 2015. The Class Notes belongs to MATH 131 at New Mexico Institute of Mining and Technology taught by Lynda Ballou in Fall. Since its upload, it has received 37 views. For similar materials see /class/223624/math-131-new-mexico-institute-of-mining-and-technology in Mathematics (M) at New Mexico Institute of Mining and Technology.
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Date Created: 10/15/15
Chapter 2 Limits and Continuity Section 21 Rates of Change and T angents t0 Curves For the function y fx let s look at the slope ofthe secant line from x a to x b and the slope ofthe tangent line at x a Now the slope of the secant line tells us the average rate change of the function from x a to x b a h average rate of change M W Ax b a h For example suppose s f I where s is the displacement directed distance from origin at time t and f is the position function describes the motion of the object In the time interval from t1 ato t2 a h the average rate of change would tell us the average velocity over this time interval is idisplacement ft2ft1 fahfa average vuluvit tlme At t2 t1 h Let s look at the tangent line to the curve y f x at x a again Notice how the tangent line just touches the curve but it also goes in the same direction that the curve is traveling at that point The slope of the tangent line tells us the instantaneous rate of change of the function at a point What happens to the slope of the secant line as h gt 0 Section 22 Limit of a Function and Limit Laws Informal De nition We write lim f x L and say the limit of f x as x approaches a is equal to L provided that we can make values of f x arbitrarily close to L by taking x suf ciently close to a on either side but not equal to a Example 1 limfx Xgt0 2 13321 fx Limit Laws Theorem 1 Math 131 Suppose that c is a constant and the limits 1im f x and 1im gx eXist Then 1 N E 4 UI ON 1 9 0 14 Quotient rule 1im Constant rule 1imc c X l Obvious rule 1imx a 1im cfx c 1im fx Sum rule 1imfxi gx1im fxi1imgx Product rule lim fx gx 1im fx1imgx 1im f x xgta 1imgx f x gx if 1imgx 0 X l xgta 11 Power rule 1imfxp 1imfxp Substitution rule If 1im gx L and lin Li f x f L then 13 fgx fg gx fL If fx gx for all x at a then 1imfx1imgx X l X l Squeeze Theorem 4 If fx S gx S hx when x is near a except possibly at a and 1imfx limhx L then limgx L 1im fx does not eXist if 1im fx b at 0 and 1imgx 0 xgta xgta xgta Trig rule Ema 1 Xgt0 1im i 0 Hoe x Let u ux and 1imux b then 1imfx linl71fu Theorem 2 Limits of Polynomials If Px anxquot anilxm1 a0 then lingPx Pc ancquot ch391391 a0 Theorem 3 Limits of Rational Functions IfPx and Qx are polynomials and Qc 0 then m Pltxgt z Pltcgt H Qx Q c Theorem 5 If fx S gx for all x in some open interval containing c except possible at x c itself and the limits of bothfand g both exist as x approaches 0 then lim fx S limg x Example Evaluate 1 lim2x 5 xgt77 2 xlim33x2x l 3 lim H5 x7 5 5 39 f H5x 25 2 6 II W 8 From example 11 lim sinx 0 and lim cos x 1 nd limtanx xgt0 xgt0 r 0 9 If 5 2x2 SfxS5 x2 for 1sxsl nd 1in01fx Section 23 Precise De nition of a Limit If we want fx to be within any prescribe error from L how close does a have to be to x De nition Let fx be defined on an open interval about a except possibly at a itself We say that the limit of f x as x approaches a is the number L and we write M L if for every number 8 gt 0 there is a corresponding number 5 gt 0 such that for all x 0ltx alt5 3 fx Llte How to Find Algebraically a 5 for a Given f L a and e gt 0 Solve the inequality I f x L lt e to nd an open interval x1 x2 containing a on which the inequality holds for all x at a Find a value of 5 gt 0 that places the open interval a 5 a 5 centered at a inside the N intervalx x The inequality fx L lt 8 will hold for all xi a in this 5interval 1 2 Section 24 OneSided Limits and Limits at Infinity Right Hand Limit Look at the behavior of the function f x as x approaches a from the right That is what happens as to the function for values of x slightly larger than 61 For the graph above what is lim fx What is lim fx XgtZ Xgt72 Left Hand Limit Look at the behavior of the function f x as x approaches a from the right That is what happens as to the function for values of x slightly larger than 61 For the graph above what is lim fx What is lim fx XgtZT rarl39 Theorem 6 A function f x has a limit as x approaches a if and only if it has a lefthand limit and right hand limits there and these onesided limits are equal limfxLcgt linifxL and linrfxL Note The limit laws in theorem 1 hold for right and lefthanded limits l x2 0 S x lt 1 Example Suppose fx x 2 1S x lt 2 at what point a does limfx exist At what 1 x 2 point does only the lefthand limit exist Righthand limit Example 1 lim x3 x2 H4 x2 x2 2 139 3 24 gtx2 Finite Limits as x gt ioo What happens to f x as x approaches positive or negative in nity Note the Sum Product Quotient and Power Rule hold in theorem 1 hold if x gt ioo Basics lim 0 0 lim 1 0 ratoo 16 th Example 2x3 7 wa x x7 2 lim 3H7 Xgtroo x2 2 x2 2 11m Hm 3x7 Horizontal Asymptotes De nition A line y b is a horizontal asymptote of the graph of a function y f x if either limfxb or mp fxb 2 3 7 Not1ce from the examples 1 and 2 above the functlons f x and x x x 7 3x 7 x 2 2 both have horlzontal asymptotes x Example 26 1 11m H024 2 lim ex xgt7oo Z Z x 2 the function fx x 3x 7 3x horizontal asymptote however it has an oblique or slant asymptote This occurs when you have a rational function where the degree of the numerator is one more than the degree of the denominator We can see that as x goes to in nity that the function goes to plus or minus in nity as well But it will go to in nity along a straight line 2 Oblique Asymptotes Look at 11m 7 does not have a Section 25 In nite Limits and Vertical Asymptotes Infinite Limits These occur when the limit of the function becomes arbitrarily large and positive or negative as x approaches a These are both cases where the limit does not exist however we can still describe the behavior of the function near a l l Cons1der 11m 00 and 11m oo xgt0 x xgt x Example 1 lim 2 xgt7 x 7 2 lim sec x xgt77r2 Vertical Asymptotes De nition A line x a is a vertical asymptote of the graph of a function y f x if either lim fx ice or lim fx ioo Example Find the asymptotes of 1 fltxgtm 2 fltxgtxxt1
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