Calculus & Analytic Geometry I
Calculus & Analytic Geometry I MATH 131
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This 23 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 49 views. For similar materials see /class/223625/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 3 Differentiation Section 32 Differentiation Rules for Polynomials Exponentials Products and Qnotients Rule 1 If f has the constant value f x c then i 1a 0 dx dx Power Rule for Positive Integers If n is a positive integer then xn 70Can Proof xquot 1im fZ fxlimz x x Hz Zx Hz Zx 1 z xquot391zquot 2xzxquot391 xquot 1 1m Hz Zx r171 r172 r171 r171 11m 2 2 xzx x Hz mnil Rule 2 If n is any real number then ixn 70Can dx for all x where the powers of xquot and x quot1 are de ned Example Find the derivative for each of the following 1 fxx2 2 fxx13 3 fltxgtiz Rule 3 If u is a differentiable function of x and c is a constant then d du cu c dx dx Example Find the derivative for each of the following 1 fx3x4 Rule 4 If u and v are differentiable functions of x then their sum u v is differentiable at every point where u and v are both differentiable At such points d du dv u v dx dx dx Example Find the derivative of fx x2 x g 7239 Derivative 0f the Natural Exponential Function 61 x x ae 6 Example Find the derivative for each of the following 1 fx78 4e 2 fx 6M 6 Rule 5 Product Rule for Derivatives If u and v are differentiable at x the so is their product uv and d dv du uv u v dx dx dx Function notation fxgx fxg39xgxf39x Example Find the derivative for each of the following 1 fx2x2lx3 2 fx10x36 Rule 6 Quotient Rule for Derivatives If u and v are differentiable at x and if vx 0 then the quotient uv is differentiable at x and du dv v 7 u 7 d a dx dx d Functlon notatlon dx One more way ihigh J W ow dx low Example Find the derivative for each of the following x23x4 1 2 x 2 6x6 2 x21 Second and Higher Order Derivatives If y f xis a differentiable function then its derivative f x is also a function If f 39 is also differentiable then we can differentiate it to get the second derivative off fquotx gjwl fltxgtgt Example Find the following 1 fx 3x2 6x4 a f x b fquotx c fwltxgt 2 yxe dy dx dzy b abc2 d3y CI de Section 33 The Derivative as a Rate of Change De nition The instantaneous rate of change of f with respect to x at x0 is the derivative f xo1igg fx hh fx provided the limit exists De nition Velocity instantaneous velocity is the derivative of the position function with respect to time Ifa body s position at time t is s ft then the body s velocity at time tis d t At t vt S11m f f dt Atao At Example At time t the position ofa body moving along the saXis is s t3 6t2 9t m 1 Find the body s velocity function 2 When is the velocity zero De nition Speed is the absolute value of velocity ds Speed vt dt 1 mm 1 1 AGAIN l pro 1 quotgt 1 Speeds quotP 4 Movasmnwmn ugt0 l 1 Spwds Sleady Slows 9k Ale ltgt P lvconsl 4 w w w w w l l x l w 1 6 7 FIGURE 3 13 The velocity graph fur Example 2 De nition Acceleration is the derivative of the velocity with respect to time If a body s position at time t is s ft then the body s acceleration at time t is dv dis a dt dtz Jerk is the derivative of acceleration with respect to time da 6133 39 t dt d Example Continued 1 Find the body s acceleration function 2 What is the body s acceleration each time the velocity is zero 3 Find the body s speed each time acceleration is zero 4 Find the total distance traveled by the body from t 0 to t 2 Example 13 page 154 Had Galileo dropped a cannonball from the Tower of Pisa 179 ft above the ground the ball s height above the ground at time tsec into the fall would have been 3 179 16t2 1 What would have been the ball s velocity speed and acceleration at time t 2 About how long would it have taken the ball to hit the ground 3 What would have been the ball s velocity at the moment of impact Section 34 Derivatives of T rigtmometric Functions Derivatives of Trigonometric Functions d d 3smx COSX 5008 x smx d d alttanxgtseczx 3cotx csczx d d secx anxsecx cscx cscxco x dx dx Example Show isin x cos x dx Example Show itan x secx dx Example Find the derivative for each of the following 1 y 10x3cos x 2 fx sinxcosxsecx COS x x x COS x 4 yx2 cosx szinx Zcosx sint 1 cost Example Find y dAyaly4 ify 9005 x Example Find the horizontal tangents for y x 2 cos x on 0 S x S 27239 Section 35 The Chain Rule and Parametric Equations Theorem 3 The Chain Rule If f is differentiable at the point it gx and g x is differentiable at x then the composite function f0 gx fgx is differentiable at x and d l l 51 gxf gxg x In Leibniz s notation if y and u g x then dy it dx du dx where dy du is evaluated at u g Note we can translate the following in words differentiate the outside function and evaluate it at the inside function then multiple by the derivative of the inside function fltgltxgtgtrltgltxgtgtgltxgt Example Find the derivatives of the following functions 1 fxx216 2 fx sin3 x 5 fx xx2 1712 6 fx xtan27 Parametric Equations De nition If x and y are given as functions xfta ygt over an interval of t values then the set of points xy ft g de ned by these equations is a parametric curve The equations are parametric equations Comments The variable tis a parameter for the curve 2 If we have a closed interval for our parameter a S t S b then fa g 61 is the initial point of the curve and fb g is the terminal point 3 The parametric equations and the parameter interval is called a parameterization of the curve Example Consider the curve de ned by the parametric equations x 3 3t y 2t 0 S t S l 1 Identify the particle s path 2 Find the Cartesian equation for the particles path Example Consider the curve de ned by the parametric equations x 4co st y 2sin t 0 S t S 27239 1 Identify the particle s path 2 Find the Cartesian equation for the particles path Parametric Formula for dy dx If all three derivatives exist and dxdt 0 dydt dx dxdt Example Consider the curve de ned by the parametric equations x 4co st y 2sin t 0 S t S 27239 Find 37239 an equation for the line tangent to the curve at t Parametric Formula for ally611x2 If the equations x f I y g I de ne y as a twicedifferentiable function of x then at any point where dxdt 0 and y39 dydx flail dy39dt dt dx abc2 dxdt dxdt Example Consider the curve de ned by the parametric equations x 4co st y 2sin t 0 S t S 27239 Find allynix2 at t3 4 Section 3 6 Implicit Differentiation In the previous sections we have mostly dealt with equations of the form y f x that expresses y explicitly in terms of the variable x In other words y is the dependent variable and is a function of only the independent variable x In section 35 we also looked at parametric curves But for both cases we have learned how to nd the derivative dydx However what happens if we have an implicit relation between the variables y and x Look at the following Examples x3y318xy xsinyxy exzy2x2y Implicit Differentiation l Differentiate both sides of the equation with respect to x treating y as a differentiable function of x Collect the terms with dydx on one side of the equation Solve for dydx 9 Example Find dydx for x2 y2 r2 Example Find dydx for x3 y3 18xy Example Find dydx for xsiny xy Example Find dydx for 8ny 2x2y Example Use implicit differentiation to find dydx and then find ally611x2 for 26 x y Example For the equation xzy2 9 nd the lines that are tangent and normal to the curve at 13 Example For the equation y 2sin7rx y nd the lines that are tangent and normal to the curve atl 0 Section 3 7Derivatives 0f Inverse Functions and Logarithms Theorem 4 The Derivative Rule for Inverses If f has an interval I as domain and f x exists and is never zero on I then f lis differentiable at every point in its domain The value of d f 1dx at point b in the domain of f 1 is the reciprocal ofthe value of dfdx at the point a f391b 1 f 1 39 b 1 W m w or alf 1 dx in dxwwb Example Let fxl3x 2 for x25 Findthe value of df ldx at x5f9 Derivative of the Natural Logarithm Function Example Find the derivative of y In x Example Find the derivative of y 1n XS2 The derivative of y In x with respect to x is iltlnxgtl xgt 0 dx x If u is a differentiable function of x with u gt 0 then iltlnugt ld u u gt 0 dx u dx Example Find the derivative of a y 1n 3x b y 1n kx k constant c Suppose x lt 0 k lt 0 then kx gt 0 and the equation in part b still applies So ln x 1 forxlt 0 x d ionlxl xi 0 The Derivatives of a and loga u Example Find the derivative of y 2 Example Find the derivative of y log2 x If a gt 0 and u is differentiable function of x then a is a differentiable function of x and d u u du a a In a dx dx For agt0 andail d l du loga u dx ulna dx Logarithmic Differentiation 1 Take the natural logarithm of both sides of the equation 2 Simplify using the laws of logarithms 3 Take the derivatives of both sides with respect to x 4 Solve for dydx Example Find dydx for the following 1 y x2 lx l2 2 y sin x Laws of Logarithms If x and y are positive numbers 0 loga xy log xloga y 10ga 10ga x loga y y o loga x39 r loga x for any real number r Theorem 5 The Number e as a Limit The number 6 can be calculated as the limit 11 6 1x1301lx Proof page 191 oftext If fx lnx then f x lx so f l 1 But by the de nition of derivative f1h f1 f1x f1 flt1gt1 a h i quot3 0 1imlnlt1x ln11iml1n1x xgt0 x xgt0x lx lx l1irglnlx h1l111lx Since f39ll then lx lnl1 13lx l exp In linoll xm exp 1 lx lirrgltlxgt 6 Section 38 Inverse T rigonometric Functions Inverse Trigonometric Functions see page 194 for graphs 0 sin391xycgtsinyxforIr2SySIL392 o cos391xycgtcosyxfor0SySIr o tan39lxycgttanyxfor7r2ltylt7r2 o cot391xycgtcotyxfor0ltylt7r o sec391xycgtsecyxfor0SyS7ryak o csc 1xycgtcscyxforSy y 0 Example Find the angles in the following 1 a arccos 2 Inverse Function 7 Inverse Cofunction Identities arccos x 7r2 arcsin x are cot x 7r2 arctan x arccscx 7r2 arcsecx Example Find the derivative of y arcsin x Example Find the derivative of y arcsinsE Example Find the derivative of y arctan x Example Find the derivative of y arctan In x Table 31 Derivative of Inverse Trigonometric Functions d 1 du 1 arcs1nu ult1 dx 1u2 dx 2 iltarccosu 1 ult1 dx 1u2 dx d 1 du 3 altarctanu 1u2 a d 1 du 4 altarccotugt W d 1 du 5 altarcsecugt mg gt1 6 iltarccscugt d u gt1 dx u2 1 dx Section 3 9 Related Rates Related Rates Problem Strategy 1 Draw a picture and select symbols for the variables use tfor time Write down the information given using the symbols you chose Determine what you are asked to find usually a rate so express it as a derivative Find an equation that relates the variables you have Differentiate both sides of the equation with respect to t Plug in the known information and solve 9959 Example 21 page 207 A spherical balloon is in ated with helium at the rate of 1007239 ft3 min How fast is the balloon s radius increasing at the instant the radius is 5 ft How fast is the surface area increasing Example 18 page 207 Draining a conical tank Water is owing at the rate of 50 m3 min from a shallow concrete reservoir vertex down of base radius 45m and height 6 m a How fast is the water level falling when the water is 5m deep b How fast is the radius of the water s surface changing then Example 23 page 207 A balloon is rising vertically above a level straight road at a constant rate of 1 sec Just when the balloon is 65 ft above the ground a bicycle moving at a constant rate of 17 ftsec passes under it How fast is the distance between the bicycle and balloon increasing 3 seconds later Example 22 page 207 A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow The rope is hauled in at the rate of 2 ft sec a How fast is the boat approaching the dock when 10 ft of rope are out b At what rate is the angle 6 changing then Section 31 0 Linearization and Differentials Recall that the tangent line of y fx at the point a fa is yy0 mm xx0 yfa f axa Now solVing for y y fa f39ax a Notice if you plot a function at its tangent line at a point as you zoomin then fx m fa f39ax a for x near a Example Suppose y x4 then the tangent line at x 6 is y 864x 3886 4000 3000 game mm The tangent line is called the linear approximation or tangent line approximation of f at a 20 De nitions If f is differentiable at x a then the approximating function L06 faf axa is the linearization of f at a The approximation fx m Lx for x near a of f byL is the standard linear approximation of f at a The point x a is the center of the approximation Example Find the linearization of the following functions 1 ysinx at x0 2 ylnlx at x0 Example 1 Suppose you want to estimate 4 82 what linearization would work here 2 Suppose you want to estimate sin 1 what linearization would work here 21 De nition Let y fgtc be a differentiablefunction The dinerentjal m is an independent variable The amerential dy is dyf39rdr Example Find dy 1 y x 5x L yxhx Consider the function y at and its tangent line at y 5 4000 3000 2000 than Estimating with Differentials Suppose we know fa and we want to predict fa h fa Ax fa d 1 then fad x faAy where Ay faAx fa if we use differential approximation then fadxm fady since Ay m dy where dy f adx Approx Error Ay dy faAx a f39aAx fltacgt ltagtfam 8AX As Ax gt 0 then the difference quotient approaches f a by definition of the derivative so the quantity in the parentheses approaches zero Change in y fx near x a If y fx is differentiable at x a and x changes for a to 61 Ax then the change Ay infis given by an equation of the form Ay f xAxsAx inwhich g gt0 asx gt0 True Estimated Absolute change Ay fad Pfa dy f adx A d Relative change f f J A d Percentage change X 100 fa X 100 Example Find20015 using differentials 23
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