Math 103, week 3 notes
Math 103, week 3 notes MATH 103 001
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MATH 103 001
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This 0 page Class Notes was uploaded by Cambria Revsine on Sunday February 7, 2016. The Class Notes belongs to MATH 103 001 at University of Pennsylvania taught by William Simmons in Spring 2016. Since its upload, it has received 27 views. For similar materials see Intermediate Algebra Part III in Mathematics (M) at University of Pennsylvania.
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Date Created: 02/07/16
Math lO3 Week 3 Notes 2461 24 One Sided Limits Limits as X approaches C from the right or left x gtCZ fxL lim 2 X gtC fxM lim 2 Z I Right hand limit I Left hand limit A function has a limit as X approaches C if it has left and right handed limits and if those two limits are equal Sin lim 1 r gt0 lim 60w 1 0 a gt0 25 Continuity A function is continuous if its outputs vary consistently with its inputs gt kCan be drawn without lifting the pencil fX is continuous at point Xc if 1 fX eXists 2 lim x eXists X C 3 1imfxf6 X C xec fxfcgt lim 2 warfltxgtfltcgt 11mg I function is right continuous I function is left continuous Properties of Continuous Functions If functions f and g are continuous at X c then the following are continuous at X c f g I Sums f g I Differences k f I Constant multiples f o g Products fg gc i O l Quotients fn n is a positive integer l Powers 1 n is a positive integer I Roots Composite of Continuous Functions If f is continuous at C and g is continuous at fc then the composite g 0 f is continuous at c Limits of Continuous Functions If g is continuous at the point b and 11in f x 19 f ix lim 2 X C 9 Then limgifixiigibig X C Ex l x l x2 limZ xgtl 1X 1 lim sm 1 2 s1n xgtl 1 X l lx limamp xgtl 1 sm 6 11 H s1n 2 6 Intermediate Value Theorem If f is continuous on a closed interval ab and if yo is any value between fa and fb then yo fc for some c in ab ll a v a I I Q Show that there is a solution to X3 X l 0 between 1 and 2 fX X3 X l fll3 l l 1 f223 2 15 Since 1 lt O and 5 gt 0 there must be a solution in between 1 and 2 gt Try values in between 1 and 2 to get a more accurate solution 26 Limits Involving Infinity lim fx xgtioo gt Plug in a really big number into X to represent infinity Limits at Infinity of Rational Functions i 00 depending on signs of coefs I if highest degree of num gt highest degree of denom Leading coe icient of mm Leading coe iciem 0f denom I if highest degree of num highest degree of denom O I if highest degree of num lt highest degree of denom Horizontal Asymptotes y b of function yfX if lim f x xgtioo gt HA the limit Oblique Slant Asymptotes In a rational function if the highest degree of the numerator is one more than the highest degree of the denominator the function has an oblique asymptote gt kDivide the numerator by the denominator to find the equation of the asymptote Vertical Asymptotes X a of function yfX if xgtaz fxioo xgtaz fxioo 11m 2 0r 11m 2 L L gt VA of a rational function is where the limit does not eXist denominator O Dominance For limits log lt polynomials lt eXponents EX lim m0 x OO lim Inf 0 xgtoo X 31 Tangent of a Function Slope of a curve yfX at the point P X0 fX0 fxohfxO m 1m hgt0 h I tangent line at P has this slope recap of 21 gt Plug in point P and m into ymX b to find b to complete the tangent equation Derivatives Same as this slope of the tangent line I fx0hfltx0gt f x0 231 h I aka derivative of function f at point X0
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