Popular in Calculus I
Popular in Mathematics (M)
This 7 page Study Guide was uploaded by Alejandra Miranda on Friday October 16, 2015. The Study Guide belongs to Math 251 at University of Oregon taught by Ellis A in Summer 2015. Since its upload, it has received 45 views. For similar materials see Calculus I in Mathematics (M) at University of Oregon.
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Date Created: 10/16/15
Calculus 1 Study Guide Basic things you should know Standard algebraic exponential rules How to find limits eg Solving equations to find limits Limit laws Understand the concept of continuity How to find the derivative long way and short way Understand the concept of differentiable eg techniques of differentiation Understand how to find domain and range How to find equations of tangent lines Understand the concept of conjugate radicals How to find vertical and horizontal asymptotes FX F X F X Understand inflection points Rules for computing derivatives Know instantaneous rateaverage velocity Look back through your HW to get more familiar with the concepts Here we go Exponents Fx aquotx X All real number agt0 Rules 1 aquotbaquotc a quotbc 2 aquotbquotCaquotbC 3 aquot b 1aquotb 4 aquot01 a ex OquotO1 Log Rules bquotyx 9 log bxy 1 Product a Log bquotmnlog bquotmlog b n 2 Quotient a Log b mn log bmlog b n 3 Power a Log b mquotnnlog bm Log b0 undefined Log b10 Log bb1 Log binfinf when x 9 infinity Limit laws 1 Lim fxgxlim fxlim gx x a x a x a Lim cfx cim fx x a x a Lim fXgX Iim fxllllim glxll x a x a x a Lim fxgxlim fx lim gx WHEN lim gx 0 x a x a x a x a If fx is a Rational Function amp a is in the domain of fx THEN lim fx fa x a Lim fxquotn lim fxquotn x a x a nsqrt lim fx lim nsqrt fx UNLESS n is even amp lim fxlt 0 x a x a x a If fxgx for all x near but not necessarily equal to a THEN Lim fxlim gx x a x a If fx gx for all x near a THEN lim fx ltlim gx x a x a 10 Squeeze Theorem If fx ltg x lth x for all x near a amp lim fxlim hx L x a x a THEN lim gx exists amp equals L x a A Function fX is continuous at xa if fa exists Lim fx exists x a amp lim fxfa x a fx is continuous if it exists on all Real numbers Limits involving infinity fliEIi L zza x x ENE flirt ll EJtf 3quot r l m x 4 n I n m x NE ll i iixi m 1 Lim fx inf if we can make fx arbitrarily large amppositive by taking x close to a both Left amp Right x a Variations lim inf Instead of inf x a 2 Lim fx inf Or lim fxinf Then we say fx has a vertical asymptote at xa x9aquot x9aquot 3 Lim fxb if we can make fx arbitrarily close to b by taking x to be large amp positive x9 inf Variation inf 4 Fx has a horizontal asymptote at yb if either lim fxb or lim fx b x inf x inf Differentiability f Iim fafxax exists we say fx is differentiable at xa amp that its derivative at xa is x a faim falfXaX x a Alternate definition of derivative f x Iim fxhfxh h o if fx is differentiable at xa then it s continuous at xa eg III E i 31 I I 3 3 4 JIL f I E HY Notice how only graph 1 is differential because it is smooth and continuous Finding Domain and Range II F a hi in quotquotH quot39 1F J v 1 Domain all x values ZI How to find equation of tangent lines You are given the coordinate You want to find m in the point slope equation yy1mxx1 1 Differentiate fx 2 Put x value into the derivative f x is the gradient of the curve and the tangent line 3 F xm in the point slope equation If going further to make equation ymxc 4 Now find fxy from the equation by substituting the value of x into fx 5 Now find value of c by substituting the values of x and y 6 Put the value of m and c into ymxc Conjugate Radicals in green xe o we o x 4 4 w 16 y 16 x Vertical asym ptotes To find vertical asymptotes set denominator equal to O and solve for x Horizontal asym ptotes How do we find the horizontal asymptotes Compare the degrees of the numerator and denominator Degrees egual It s the ratio of the leading coefficients Degree of numerator is greatest No asymptote The function is unbounded at the ends Goes on forever to infinity Degree of denominator is greatest y O xaxis is horizontal asymptote Facts if fx has a vertical asymptote at xa then it s not differentiable at xa amp if fx has a horizontal asymptote as x9 inf then lim f xO and likewise at inf x9 inf lel f lxlI f xl graph fee39fure ies F39TK Netee rising in m e3 elepe in El felling n m e3 elepe r U r nL derieel39ieeinue net E mneirnum elepe t E m R 39 ll Em meanme er E v mineg quot3 minimum elepe El ellquot nmm v 39urnreture ehengee i tenil39el IHHEETIDH r quot infleetien paint eeneeve up I eeneeve dewn Inflection point An inflection point of fx is a point at which f x goes from gt0 to lt O or vice versa at an inflection point f x O I iFlEL Inflation quotFul l Emmw Immune Emmw I Emmeta I Epitome Emmion Upward Upward wumw rcl Upward Downward Notation ddx the derivative with respect to x of Le dfdx f x Rules for computing derivatives ddx fg dfdx dgdx if c is a constant ddx cf c dfdx Power Rule if n is wreal then ddx xquotnnxquotn1 Product Rule ddx fg dfdx g fdgdx Quotient Rule ddx fxgx gf fg gquot2 91 WN o equotx is the only function that is its own derivative Average rate of change Fbfaba Instantaneous rate of change is the slope of a tangent line Use lim fahfah h O
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