Math 165 Review Sheet Note: Remember to use the Chain Rule where appropriate.Name Formula1.1 Intro to Limitslimx cf  x=L if the Left-Hand and Right-Hand Limits = L1.3 Limit Theoremslimx cf  x= f c1.4 Trig Limitslimt 0sin tt=1limt 01−cos tt=01.5 Infinite Limitslimx ∞1x=0limx a1x−a=±∞Use sign analysis.1.6 Continuitylimx cf  xexists, f(c) exists, and limx cf  x= f c2.2 The Derivativef '  x=limh 0f xh− f x h2.3 Derivative RulesDxxn =nxn−1Dx[ f  x g  x]= f  x Dxg  xg  x  Dxf x Dxf x g x =g  x Dxf  x− f x  Dxg x g2x 2.4 Trig DerivativesDxsin x=cos xDxcos x=−sin xDxtan x=sec2xDxsec x=sec x tan xDxcsc x=−csc x cot xDxcot x=−csc2x2.5 Chain RuleDx[ f  g x  ]= f ' g  x g ' x 2.6 Higher-Order Acceleration = Velocity' = Position''2.7 Implicit DifferentiationTaking the derivative without the form y = f(x).Dxy2=2ydydt2.8 Related RatesTake derivative with respect to time. Dxx=dxdt2.9 Differentialsf  x x = f  x f '  x  x3.1 Max and Min Critical points occur at End Points, when f(x) = 0, and when f(x) does not exist.3.2 Monotonicity &Concavityf is increasing whenf '  x0, decreasing whenf '  x0concave up when f ' ' x 0, concave downf ' ' x 03.3 Extrema1st Deriv test: Max whenf '  xgoes from +, 0, -. Min -, 0, +.2nd Deriv test: Max whenf ' ' c0, Min whenf ' ' c03.4 Practical Problems Get the problem in terms of one variable and solve. Check answer with Extrema tests.3.5 GraphingFind wheref  x, f '  x, and f ' ' x are +, -, and 03.6 MVT for Derivativef b− f a b−a= f ' c 3.7 Newton's Methodxn1=xn−f xnf '  xn3.8 Antiderivative∫[ g x ]rg '  x dx=[ g  x]r1r1C3.9 Differential Equations Separate the different variables to opposite sides. Integrate.4.1 Intro to Area∑i=1ni=nn12∑i=1ni2=nn12n16∑i=1ni3=[nn12]2∑i=1ni4=nn12n13n23n−1304.2 Definite Integral∫abf x dx=limn ∞∑i=1nf  xi xi4.3 1st Fundamentalddx∫axf tdt = f x 4.4 2nd Fundamental∫abf x dx=F b−F af  g x  g '  xdx=∫g a g bf u  duwhere u=g  x 4.5 MVT for Integrals1b−a∫abf  x dxEven: ∫−aaf x dx=2∫0af  xdx, Odd: ∫−aaf x dx=04.6 Trapezoidal &Simpson'sh2 f  x02 f  x12 f  x2...2 f xn−1 f  xnh3 f  x04 f x12 f  x2...4 f  xn−1 f xnh=b−a6.1 Logarithmln x=∫1x1tdt6.2 Inverse Functions Switch x and y, then solve. f−1'  y =1f '  x6.3 Exponentialeln x=xln ey= yDxex=ex6.4 Exp and Logax=exln aDxax=axln a∫axdx=1ln aaxC6.5 Exponential Growth/Decayy= y0ekty= y012thT t =TsT 0−Tsekty= y01rnnt6.8 Inverse TrigDxarcsin x =11−x2Dxarccos x =−11−x2Dxarctan  x=11x2Dxarcsec x