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# Calculus I MATH 1351

TTU

GPA 3.62

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This 37 page Class Notes was uploaded by Ms. Ally Koelpin on Thursday October 22, 2015. The Class Notes belongs to MATH 1351 at Texas Tech University taught by Victoria Ellen Howle in Fall. Since its upload, it has received 28 views. For similar materials see /class/226461/math-1351-texas-tech-university in Mathematics (M) at Texas Tech University.

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Date Created: 10/22/15

Math 1351 011 August 29 2007 1 Announcements 0 Homework 1 due today 0 Homework 2 due next Friday in class 0 In class quizzes during discussion sections begin next week TTU Department of Mathematics amp Statistics Math 1351 011 August 29 2007 2 Section 13 functions and graphs 0 De nitions A function f is a rule that assigns to each element x of a set X a unique element y of a set Y The element y is called the image of 16 under f and is denoted f The set X is called the domain of f and the set of all images of elements of X is called the range of the function 0 A function f can be thought of as the set of ordered pairs 16 y Where each member 16 of the domain is associated With exactly one member y x of the range 0 A function assigns a unique output to each legitimate input TTU Department of Mathematics amp Statistics Math 1351 011 August 29 2007 3 o If the range of f consists of all of Y then f is said to map X onto Y o If each element in the range is the image of one and only one element in the domain then f is said to be a onetoone function 0 A function f is bounded on a b if there exists a number B such that g B for all x in a b o In a function represented as y f x x and y are called variables Since y is determined by x y is called the dependent variable and IE is called the independent variable 0 Example f x2 l 476 l 5 is the set of all ordered pairs 16 y satisfying y x2 4x 5 TTU Department of Mathematics amp Statistics Math 1351 011 August 29 2007 4 0 Evaluating a function means to nd the value of f for a particular value in the domain 0 Example fr 2x2 16 Find fr h Find JCiIChL f957 Where x and h are real numbers and h 7E 0 o W is called a difference quotient It Will be important in chapter 3 derivatives TTU Department of Mathematics amp Statistics Math 1351 011 August 29 2007 5 Piecewise de ned functions 0 Some functions are de ned differently on different parts of their domain 0 Example xsinx ifzrlt 2 x 2 316 1 1 1fx22 c Find and f2 TTU Department of Mathematics amp Statistics Math 1351 011 August 29 2007 6 Application An object dropped from a height in a vacuum Will fall a distance of 8 feet in t seconds according to the formula 8t 16t2t2 0 1 How far With the object fall in the rst second In the next 2 seconds 2 How far With it fall during the time interval t 1 sec to t 1 l h sec 3 What is the average rate of change of distance during the time t1sectot3sec 4 What is the average rate of change of distance during the time txsectotxhsec TTU Department of Mathematics amp Statistics Math 1351 011 August 29 2007 7 0 Domain of f is assumed to be the set of real numbers for Which the function is de ned unless otherwise speci ed 0 Examples nd the domain x 2x 1 gx 2ZE 1 x7 3 hm Zx xljrngrZB m GJc 5 Cosac 0 Function equality Two functions f and g are equal if and only if 1 f and g have the same domain 2 g1c for all SE in the domain TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 1 Announcements 0 Homework 6 due next Monday 1082007 Will be posted on course web page today TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 2 Using derivatives in graphing We already know how to nd 16 and y intercepts NOW we can also nd places Where the graph turns 3 0 Example graph y x2 416 7 Example graph y IE 2ZE2 416 7 TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 3 Derivatives of trigonometric functions Theorem The functions sinx and cosx are differentiable for all x and sinxcosx cosxsinx dx dx TTU Department of Mathematics amp Statistics Math 1351 011 Recall October 1 2007 More Trig review sinac cosac tanx cow cotx Sm 1 1 SEC IE CSC IE cosac sinx COS2ZE l sin2 25 1 1tan2x sec2 16 cot2 16 1 csc2 x sina l sinozcos l cosozsin cosa cos oz cos sina sin TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 5 Derivatives of trigonometric functions The trigonometric functions sin cos tan csc sec cot are all differentiable Wherever they are de ned and isinxcosx icosx sinx dac dac itanxsec2x icotx csc2x dac dac sccxsecxtanx cscx cscxcotx TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 6 Derivatives of exponentials and logarithms The natural exponential ex is differentiable for all 16 With derivative d il39 il39 e dx 6 The slope of the graph of the natural expontial at any point C is ea The natural logarithm lnx is differentiable for all E gt 0 With derivative d 1 1 dxnx x TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 7 Rates of Change A A Average rate of change y W A96 Ax A Instantaneous rate of change lim y 11m W Air gt0 ASE Ax gtO Ax Suppose f is differentiable at x 950 Then the instantaneous rate of change of y x With respect to x at 960 is dy 960 x2160 TTU Department of Mathematics amp Statistics Math 1351 011 October 1 2007 8 The relative rate of change of y at E x0 is given by the ratio Instantaneous rate of change f x0 size of the quantity f 960 TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 1 Announcements 0 Homework 6 due today 0 Homework 7 due next Monday 10152007 0 Exam 2 on Friday 10192007 Strauss sections 24 through 38 TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 2 Implicit Differentiation If an equation de nes y implicitly as a differentiable function of 16 nd by 1 Differentiate both sides of equation With respect to 16 Remember that y is really a function of x for part of the curve and use the chain rule When differentiating terms containing 2 Solve the differentiated equation algebraically for TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 3 warning Beware that implicit differentiation is only valid if y is a differentiable function of 16 Can get errors if you apply it carelessly Example 362 y2 1 TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 Differentiation formulas for inverse trig functions i 39 1 1 1 1 1 dxs1n u fa dag dag cos u fag dag d 1 1 du d 1 1 du tan 10 1u2 33 10 1u2 d 1 1 du A 1 1 dmSeC u Mm dac dacltCSC u umdzr TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 5 Derivatives of exponential and logarithmic functions with base b Let u be a differentiable function of x and b be a positive number other than 1 Then d du u 1 u dxb nbb dx 110 ui 16L dx gb 1nb udx TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 6 Derivative of In If x lnixi 25 7E 0 then f x l 139 Also if u is a differentiable function of x then 1du d Inw dx udx TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 7 Logarithmic Differentiation Using logarithms we can trade differentiating products and quotients for differentiating sums and differences Useful to handling complicated product or quotient functions and exponential functions Where variables appear in both the base and the exponent Take logarithm of both side then apply logarithm rules to simplify then differentiate TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 8 Related Rates Many problems involve a functional relationship y f in Which both 16 and y are themselves functions of another variable such as time t We can use implicit differentiation to relate the rate of change to the rate Cell f TTU Department of Mathematics amp Statistics Math 1351 011 October 8 2007 9 General Procedure for Related Rates 1 Draw a gure if appropriate 2 Assign variables to the quantities that vary 3 Find a formula or equation that relates the variables 4 Differentiate the equations often implicitly With respect to time 5 Substitute speci c numerical values Where known 6 Solve algebraically for any required rate TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 1 Announcements 0 Exam 2 this Friday 10192007 Strauss sections 24 through 37 Discussions sections this week are exam review no quiz Extra of ce hours this week Tuesday 1000 to 1130 am Wednesday 1000 to 1130 am Thursday 1000 to 1130 am Same format as exam 1 in class no notes no calculators etc 0 See extra review problem on course web page 0 Homework 8 due Friday 10 262007 will be posted later today TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 2 In the immediate vicinity of P7 the tangent line Closely approximates the shape of the curve at y TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 3 Linearization If an is near a then fZE1 is close to the point on the tangent line to y x at x 961 That is fx1 fa f a1 a This is a linear approaimation of f at x a This process is called linearization of the function at point x a Incremental approaimation formula f961 a f a961 a Ay fCLAZE TTU Department of Mathematics amp Statistics Math 1351 011 October 177 2007 4 Incremental approximation of cosine Let sinaj Then approximates faj cosajz Af i sin90 A93 sin 930 N A9 A9 N f 930 Ax10 Ax05 TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 5 Differentials We give the dy and dx of Leibniz notation Ell Z meaning as separate quantities dx is called the differential of x dy is called the differential of y 0 dx A96 0 But dy 7E Ay dy f xdx or equivalently df f xdx 0 Ag is the rise of f that occurs With a change of A96 But dy is the rise in the tangent line relative to the change in x TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 6 Linearity Product Quotient Power Trig Exp and log Differential rules dfg f dgg df d 9 d2quot d9 9 7A 0 dx men 1 dzr dsin x cosx dzr dcos x sinx dzr dtan x S C2 x dx dcot x csc2 x dx dsec x secxtanx dzr dcsc cscrcotr dzr TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 7 Differentials for inverse trig dsin1 u Xldf dc0s1 u dtan1 u 11 dc0t1 u 1322 dsec u W dcsc1 u W TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 8 Exam 2 Review TTU Department of Mathematics amp Statistics Math 1351 011 October 17 2007 9 Trig Identities 0 De nitions of trig and inverse trig functions SOH CAH TOA Remember that sin1 16 7E sin1 16 arcsin x 1 sinac 39 o Pythagorean identities cos2 x l sin2 25 1 1 tan2 x sec2 16 o Opposite angles odd vs even functions cos Jc cosr ie even sin r sinr ie odd tan Jc tanr ie odd 0 Angle addition sina l sina cos cosa sin cosa cosa cos sina sin TTU Department of Mathematics amp Statistics Math 1351 011 September 10 2007 1 Announcements 0 Homework 3 due Friday 0 Exam 1 on Friday 921 0 One volunteer note taker needed TTU Department of Mathematics amp Statistics Math 1351 011 September 10 2007 2 Informal limit de nition The notation lim f L ac gtc means that the function values f can be made arbitrarily close to a unique number L by choosing x suf ciently close to c but not equal to 0 Notation Also written as fr gt L as x gt c TTU Department of Mathematics amp Statistics Math 1351 011 September 10 2007 3 Limits do not always exist If the limit of the function f fails to exist f is said to diverge asx gtc o The function may grow arbitrarily large or small as x gt c 1 Eg limxno 36 2 A function f that increases or decreases Without bound as x approaches 0 is said to tend to in nity inf as x gt 0 lim f 00 if f increases Without bound gtC lim f oo if f decreases Without bound gtC o The function may oscillate as x gt c 1 Eg limxgt0 sin 5 divergence by oscillation TTU Department of Mathematics amp Statistics

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