154FinalExam-Review-and-study-guide.pdf - Review for Final Exam Functions(1 Know the meanings of the following terms for a function f(x Domain range

154FinalExam-Review-and-study-guide.pdf - Review for Final...

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Review for Final Exam Functions (1) Know the meanings of the following terms for a functionf(x).Domain, range, independent and dependent variables, zero off(i.e.x-intercepts),vertical and horizontal asymptote,y-intercept. .(2) Here some common functions with restricted domains.The domain off(x) =g(x)/h(x) excludes values ofxfor whichh(x)6 (3) Know the meanings of the following types of functions:Linear function, power function, polynomial and rational function, periodic function.(4) For a power functionaxn, theexponentisnand itscoefficientisa.(5) Thedegreeof a polynomialadxd+ad-1xd-1+· · ·+a0isd.(6) A functionfiseveniff(-x) =f(x) for allxandoddiff(-x) =-f(x) for allx.Even functions have a mirror symmetry about they-axis while odd functions have180rotational symmetry about the origin.Definitions(1) Thesecantline fory=f(x) atx=aandx=bhas equationy=L(x) whereL(x) =f(a) +f(b)-f(a)b-a(x-a).(This is the point-point equation of the line.) 1
(3) A functionfiscontinuous ataiflimxaf(x) =f(a).Note:although this statement is compact, it asserts three things:(a)f(a) exists (i.e.ais in the domain off)(b) limxaf(x) exists.(c) The numbers from the two previous parts are equal.(4) A functionfiscontinuous on(a, b) iffis continuous at every point in the openinterval (a, b).(5) Here are three types ofdiscontinuitiesthatf(x) might have atx=a.removable discontinuity:Hereais not in the domain offbut limxaf(x) exists(i.e. equals some number).jump discontinuity:Here limxa+f(x) andlimxa-f(x) exist but are not equal.blow-up discontinuity:Here limxa+f(x) =±∞or limxa-f(x) =±∞.Example:Find and classify the discontinuities of the functionf(x) =x2-1x2+x-2.(6) Definition of derivative off(x) atx=a:f0(a) = limh0f(a+h)-f(a)hf(a+h)-f(a)his the slope of the secant line meeting the graph ataanda+h.f0(a) is the slope of the tangent line of the graph ofy=f(x) atx=a.If the limit exists, thenfisdifferentiableata.Iffis differentiable at ever point in an open interval (a, b), then we say thatfisdifferentiable on(a, b).Example:Use the definition of derivative to compute the derivative of the functiony= 2x2 2

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