Elementary Calculus - mecmath

1 ElementaryCalculusMichael Corral0v20gv202gElementary CalculusMichael CorralSchoolcraft CollegeAbout the author:Michael Corral is an Adjunct Faculty member of the Department ofMathematics at School-craft College. He received a in Mathematics from the University of California at Berkeley,and received an in Mathematics and an in Industrial &Operations Engineeringfrom the University of text was typeset in LATEX with theKOMA-Scriptbundle, using the GNU Emacstext editor on a Fedora Linux system. The graphics were created using TikZ and 2015 Michael is granted to copy, distribute and/or modify this document under the terms of theGNU Free Documentation License, Version or any later version published by the FreeSoftware Foundation; with no Invariant Sections, no Front-Cover Texts, and no Introduction.

2 The Derivative: Limit Approach .. The Derivative: Infinitesimal Approach .. Derivatives of Sums, Products and Quotients .. The Chain Rule .. Higher Order Derivatives .. 312 Derivatives of Common Inverse Functions .. Trigonometric Functions .. The Exponential Function and Natural Logarithm Function.. General Exponential and Logarithmic Functions .. 523 Topics in Differential Tangent Lines .. Limits .. Continuity .. Implicit Differentiation .. Related Rates.

3 Differentials .. 814 Applications of Optimization and the Second Derivative Test .. Curve Sketching .. Numerical Approximation of Roots of Functions .. The Mean Value Theorem .. 1135 The The Indefinite Integral .. The Definite Integral .. The Fundamental Theorem of Calculus .. Integration by Substitution .. Average Value of a Function .. 145 Appendix AAnswers and Hints to Selected Exercises149 GNU Free Documentation License152 History161 Index162 The Greek AlphabetLetters NameLetters NameLetters NameA alphaI iotaP rhoB betaK kappa sigma gamma lambdaT tau deltaM mu upsilonE epsilonN nu phiZ zeta xiX chiH etaOoomicron psi theta pi omegaMathematical NotationSymbolMeaningExample ; implies|x|>1 x2>1 if and only if; two-way implication|x|>1 x2>1iffif and only if; two-way implication|x|>1 iffx2>1;does not imply|x|>1.

4 X>1 there exists a numberc>0 there does not exist xsuch thatx2<0 !there exists a unique !xsuch that 2x 1=3 for every x 0,pxis a real number is identically equal tof 0 f(x)=0 for allx is proportional toy x2 y=kx2for somek is a subset of{0,1} {0,1,2} is an element of1 {1,2,3} is not an element of1 {2,3} union of sets{0,1} {2,3}={0,1,2,3} intersection of sets{0,1} {1,2}={1} empty set{0,1} {2,3}= therefore nmust existCHAPTER 1 The IntroductionCalculus can be thought of as the analysis of curved example, suppose that anobject at rest 100 ft above the ground is dropped.

5 Ignoring air resistance and wind, the objectwill fall straight down until it hits the ground (see Figure (a)). As will be proved later,tseconds after being dropped the object will bes=s(t)= 16t2+100 ft above the that the object will hit the ground after seconds (why?). While the object s path isa straight line, the graph of its heightsabove the ground aftertseconds is curved, part of adownward-pointing parabola (see Figure (b)).100 ftt=0 sect= secobject falling(a)Path of (b)Graph of object s heights= 16t2+100 Figure dropped from 100 ft above the groundHow fast is the object moving before it hits the ground?

6 This is where Calculus comes is more than that, of course. But we are in good company in usingthat definition: the first European textbookon Calculus , written by the French mathematician Guillaumede l H pital in 1696, was titledAnalyse des InfinimentPetits pour l Intelligence des Lignes Courbes(which translates asAnalysis of the Infinitely Small for UnderstandingCurved Lines). That book (in French) can be obtained freely in electronic form 1 The Derivative this is the type of question that will motivate the rest ofthis chapter, the answer will bedetermined , the object travels 100 ft in seconds, so itsaverage speedin that time isdistance traveledtime elapsed=100 seconds=40 ft/s,and itsaverage velocityin that time ischange in positionchange in time=final position initial positionend time start time=0 ft 100 sec 0 sec= 40 velocity takes direction into account.

7 The object s downward motion means it has negativevelocity. Positive velocity would mean upward the idea of average velocity, there is a natural way to define the object sinstantaneousvelocity, that is, its velocity at a particularinstantof timet, not over anintervalof time,namely:1. Find the average velocity over an interval of Let the interval become smaller and smaller indefinitely, shrinking to a pointt. If theaverage velocity over that smaller and smaller interval approaches some value, call thatvalue the instantaneous velocity at below shows how to choose the interval: for any timetbetween 0 and , use theinterval [t,t+ t], where tis a small positive number.

8 So tis the change in time over theinterval, and denote by sthe change in the heightsover that + t ss(t+ t)s(t)average velocity= s t=s(t+ t) s(t) tFigure velocity s tover the interval [t,t+ t]The average velocity of the object over the interval [t,t+ t] is s t, so sinces(t)= 16t2+100:Introduction Section s t=s(t+ t) s(t) t= 16(t+ t)2+100 ( 16t2+100) t= 16t2 32t t 16( t)2+100+16t2 100 t= 32t t 16( t)2 t= t( 32t 16 t) t= 32t 16 t,since t>0, and hence tcan be canceled from the numerator and denominator. Now let theinterval [t,t+ t] get smaller and smaller indefinitely, that is, let tget closer and closer to the average velocity s t= 32t 16 tgets closer and closer to 32t 0= 32t.

9 So theobject has instantaneous velocity 32tat timet. This calculation can be interpreted as takingthelimitof s tas tapproaches0, written as follows:instantaneous velocity att=limit of average velocity over [t,t+ t] as tapproaches to 0=lim t 0 s t=lim t 0( 32t 16 t)= 32t 16(0)= 32tNotice that tis not replaced by 0 ( take the limit as tapproaches 0) in the ratio s tuntilafterdoing as much cancellation as possible. Notice also that the instantaneous velocityof the object varies witht, as it should (why?). In particular, at the instant when the objecthits the ground at timet= sec, the instantaneous velocity is 32( )= 80 this makes sense so far, then you understand the crux of the idea of what a limit is andhow to calculate a limit.

10 The instantaneous velocity 32tis called thederivativeof the functions(t)= 16t2+100. Calculating derivatives, analyzing their properties, and using them to solvevarious problems are part ofdifferential does this have to do with curved shapes? Instantaneous velocity is a special case of aninstantaneous rate of changeof a function; in this case the instantaneous rate of change of theposition (height above the ground) of the object. Similar to how the rate of change of a line isits slope, the instantaneous rate of change of a general curverepresents theslope of the example, the parabolas(t)= 16t2+100 has slope 32tfor type of problem which Calculus was created to solve is tofind the area inside curvedregions ( the area bounded by curved lines which intersect).