The closer x is to a, the better the slope of the secant line will approximate the slope of the tangent line. y=f(x)

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1 Math 1710 Topics for ærst exam Chapter 1: Limits an Continuity x1: Rates of change an its Calculus = Precalculus + èitsè Limit of a function f at a point x 0 = the value the function `shoul' take at the point = the value that the points `near' x 0 tell you f shoul have at x 0 fèxè =L means fèxè isclose to L when x is close to èbut not equal toè x 0 Iea: slopes of tangent lines secant line y=f(x) (a,f(a)) tangent line (x,f(x)) The closer x is to a, the better the slope of the secant line will approximate the slope of the tangent line. The slope of the tangent line = it of slopes of the secant lines ( through (a,f(a)) ) fèxè =L oes not care what fèx 0 è is; itignores it f èxè nee not exist! èfunction can't make upit's min?è x2: Rules for æning its If two functions fèxè an gèxè agree èare equalè for every x near a èbut maybe not at aè, then fèxè = gèxè x 2, 3x +2 Ex.: x!2 x 2, 4 = x!2 x, 1 x +2 If fèxè! L an gèxè! M as x! x 0 èan c is a constantè, then fèxè+gèxè! L + M ; fèxè,gèxè! L, M ; cfèxè! cl ; f èxègèxè! LM ; an f èxèègèxè! L=M provie M 6= 0 If f èxè is a polynomial, then Basic principle: to solve fèxè= fèx 0 è, plug in x = x 0! If èan whenè you get 0è0, try something else! èfactor èx, x 0 è out of top an bottom...è If a function has something like p x, p a in it, try multiplying ètop an bottomè with p x + p a Sanwich Theorem: If fèxèç gèxèç hèxè, for all x near a èbut not at aè, an

2 fèxè = hèxè = L, then gèxè =L. x4: Extensions of the it concept Motivation: the Heavisie function 1 y=h(x) The Heavisie function has no it at 0; it can't make up its min whether to be 0 or 1. But if we just look to either sie of 0, everything is fine; on the left, H(0) `wants' to be 0, while on the right, H(0) `wants' to be 1. It's because these numbers are ifferent that the it as we approach 0 oes not exist; but the `one-sie' its DO exist. Limit from the right: +fèxè =L means fèxè isclose to L when x is close to, an bigger than, a Limit from the left: fèxè =M means fèxè isclose to M, when x is close to, an smaller than, a f èxè = L then means +fèxè =, fèxè = L Inænite its: 1 represents something bigger than any number we can think of fèxè =1 means fèxè gets really large as x gets close to a Also have fèxè =,1 ; +fèxè =1 ; fèxè =1 ; etc..., Typically, aninænite it occurs where the enomenator of f èxè iszero èalthough not alwaysè x5: Continuity A function f is continuous èctsè at a if This means: è1è fèxè =fèaè fèxè exists ; è2è fèaè exists ; an è3è they're equal. Limit theorems say èsum, iæerence, prouct, quotientè of cts functions are cts. Polynomials are continuous at every point; rational functions are continuous except where enom=0. Points where a function is not continuous are calle iscontinuities Four æavors: removable: both one-sie its are the same jump: one-sie ts exist, not the same inænite: one or both one-sie its is 1 or,1

3 oscillating: one or both one-sie its DNE Intermeiate Value Theorem: If fèxè iscts at every point in an interval ëa; bë, an M is between fèaè an fèbè, then there is èat least oneè c between a an b so that fècè =M. Application: æning roots of polynomials x6: Tangent lines Slope of tangent line = it of slopes of secant lines; at èx 0 ;fèx 0 è : fèxè, fèx 0 è Notation: call this it f 0 èx 0 è = erivative of f at x 0 x, x 0 Diæerent formulation: h = x, x 0, x = x 0 + h f 0 èx 0 è = h!0 Chapter 2: Derivatives fèx 0 + hè, fèx 0 è h x1: The erivative of a function erivative = it of iæerence quotient ètwo æavorsè f 0 èx 0 è exists, say f is iæerentiable at x 0 Fact: f iæerentiable èiæ'bleè at x 0, then f cts at x 0 h! 0 notation: replace x 0 with x è= variableè, get f 0 èxè = new function f 0 èxè =the erivative of f = function whose vaules are the slopes of the tangent lines to the graph of y=fèxè.domain = every point where the it exists Notation: f 0 èxè = y x = f èfèxèè = x x = y0 = D x f = Df = èfèxèè 0 x2: Diæerentiation rules x èconstantè = 0 x èxè = 1 èfèxè+gèxèè 0 = èfèxèè 0 + ègèxèè 0 èfèxè-gèxèè 0 = èfèxèè 0 - ègèxèè 0 ècfèxèè 0 = cèfèxèè 0 èfèxègèxèè 0 = èfèxèè 0 gèxè+ fèxèègèxèè 0 è fèxè gèxè è0 = f 0 èxègèxè, fèxèg 0 èxè g 2 èxè èx n è 0 = nx n,1, for n=0,1,,1,2,,2,3,... èè è1ègèxèè 0 = -ggpèègèxèè 2 èè f 0 èxè is`just' a function, so we can take its erivative! èf 0 èxèè 0 = f 00 èxè è= y 00 = 2 y x = 2 f 2 x è 2 = secon erivative off è=rate of change of rate of change of f!è Keep going! f 000 èxè =3r erivative, f ènè èxè = nth erivative

4 x3: Rates of change Physical interpretation: fètè= position at time t f 0 ètè= rate of change of position = velocity f 00 ètè= rate of change of velocity =acceleration jf 0 ètèj = spee Basic principle: for object to change irection èvelocity changes signè, f 0 ètè= 0 somewhere èivt!è Examples: Free-fall: object falling near earth; sètè = s 0 + v 0 t, g 2 t2 s 0 = sè0è = initial position; v 0 = initial velocity; g= acceleration ue to gravity Economics: Cèxè = cost of making x objects; Rèxè = revenue from selling x objects; P = R, C = proæt C 0 èxè =marginal cost = cost of making `one more' object R 0 èxè =marginal revenue ; proæt is maximize when P 0 èxè = 0 ; i.e., R 0 èxè =C 0 èxè x4: Derivatives of trigonometric functions sin x Basic it: = 1 ; everything else comes from this! x!0 x sinèbxè Note: this uses raian measure! = b x!0 x Then we get: èsin xè 0 = cos x ècos xè 0 =, sin x ètan xè 0 = sec 2 x ècot xè 0 = = csc 2 x èsec xè 0 = sec x tan x ècsc xè 0 = = csc x cot x x5: The Chain Rule Composition èg æ fèèx 0 è = gèfèx 0 èè ; ènote: we on't know what gèx 0 è is.è èg æ fè 0 ought to have something to o with g 0 èxè an f 0 èxè in particular, èg æ fè 0 èx 0 è shoul epen on f 0 èx 0 è an g 0 èfèx 0 èè Chain Rule: èg æ fè 0 èx 0 è = g 0 èfèx 0 èèf 0 èx 0 è = èèoutsieè eval' at insie fcnèæèèinsieèè Ex: èèx 3 + x, 1è 4 è 0 = è4èx 3 +1, 1è 3 èè3x 2 +1è Diæerent notiation: y = gèfèxèè = gèuè, where u = fèxè, then y x = y u u x x6: Implicit iæerentiation We can iæerentiate functions; what about equations? èe.g., x 2 + y 2 =1è graph looks like it has tangent lines

5 tangent line? (a,b) Iea: Preten equation eænes y as a function of x : x 2 +èfèxèè 2 = 1 an iæerentiate! 2x +2fèxèf 0 èxè =0; so f 0 èxè =,x fèxè =,x y Diæerent notation: x 2 + xy 2, y 3 =6; then 2x +èy 2 + xè2y y y è, 3y2 x x =0 y,2x, y2 = x 2xy, 3y 2 Application: exten the power rule è=rx r,1 works for any x èxr rational number r x7: Relate Rates Iea: If two èor moreè quantities are relate èa change in one value means a change in othersè, then their rates of change are relate, too. xyz = 3; preten each is a function of t, an iæerentiate èimplicitlyè. General proceure: Draw a picture, escribing the situation; label things with variables. Which variables, rates of change o you know, or want to know? Fin an equation relating the variables whose rates of change you know or want to know. Diæerentiate! Plug in the values that you know.