CALCULUS WITH FUNCTIONS I [C3T1G1]
CALCULUS WITH FUNCTIONS I [C3T1G1] MATH 231
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Math 231 Overview The nal exam will be comprehensive However the course has naturally built on earlier material So early concepts are more likely to be applied in a synthesized way with later material Chapter 0 0 Section 01 Numbers and sets 7 Key concepts types of numbers sets intervals unionintersections absolute value and distance 7 Reappearances Intervals occur frequently solution sets to inequalities and then later to describe where functions have different properties UnionsIntersections again in solution sets to inequalities also when we describe a function in curve sketching analysis we are implicitly doing intersections ofintervals Absolute value The most important application is in the de nition of a limit It also appears in inequalities and as an excellent example of a function with a corner so not differentiable Distance and Midpoint usually appear in applications although distance is the generalization of absolute value in later math courses 7 Common errors Only integers some students forget that between any two numbers we always nd more numbers This is particularly problematic when students don t actually solve a problem but just plug in values and hope it tells them enough to answer a problem Confuse union and intersection If a point is in one set or another set we use union If it must be in both at the same time we use intersection Absolute values Not using the de nition Most commonly students move things in and out of absolute values as though they were just parentheses This doesn t work You have to consider if what is inside is positive or negative and then use the de nition 0 Section 02 Equations 7 Key concepts equations solution sets using factoring quadratic formula fractions equations with quotients and systems of equations 7 Comments There is a difference between an expression and an equation An equation says that two different expressions are equal Equations are used to de ne functions eg fxx 3 Equations are also used to nd points of interest 3x 1 2 will be true for the x value when the function fx3x 1 has a height of 2 Similarly 3x 41 7x will be true for the x value where the functions fx3x 4 and gx1 7x intersect Solving an equation is different than simplifying or evaluating an expression We solve an equation when we use algebra to identify the solution set of an equation Simplifying an expression means that we nd new expressions that are equivalent to the original expression 7 Reappearances Equations are used almost everywhere de ning functions by equations nding zeros nding critical points nding intersections Factoring The most important exact method for solving equations we have avail able It is only helpful when we have a factored expression equal to zero Also used for identifying discontinuities cancelling to evaluate indeterminate limits and quickly sketching polynomials and rational functions Fractions lmportant properties for calculating derivatives particularly nding com mon denominators Also essential in understanding rational functions Systems of equations used in identifying unknown functions when trying to model particular graphs 7 Common errors lncomplete factorization miss solutions of equations unless you have all factors Fractions many problems with fractions such as no common denominators or ig noring the rules of algebra Quotients AB0 if and only if A0 and B is not 0 The second part is sometimes forgotten No equation It is very common to see work related to solving an equation but without ever writing an equation This isn t correct You must show the equations otherwise you just have a jumble of symbols on the page 0 Section 03 lnequalities 7 Key concepts rules for inequalities inequalities and factorization inequalities and ab solute values triangle inequality 7 Comments Rules for lnequalities Thm 010 I think this is overlooked on how powerful this is These properties are fundamental for much of the work that we later use when doing delta epsilon proofs of limits Factorization and lnequalities We have two methods of solving inequalities The way we have been doing it recently involves the Intermediate Value Theorem and requires continuity The method in Section 03 is fundamentally different and prob ably more awkward But sometimes it is more direct You need to understand the difference in these two methods 7 Reappearances Solving inequalities nding intervals where f f or f are positivenegative curve sketching analysis nding domains involving even roots Absolute value as inequalities essential to delta epsilon de nition of a limit distance between values Use of inequalities delta epsilon proofs rely on inequalities onesided limits at vertical asymptotes can be analyzed with inequalities 7 Common errors Half of solution inequalities involving absolute values involve two intervals either in a union or in an intersection but many students only nd one interval see Thm 014 Inequalities by factorization many students only list one of the two cases for AB gt 0 or AB lt 0 0 Section 04 Logic 7 Key concepts logical statement quanti ers negation counterexamples andor con junctions implications converse and contrapositive if and only if 7 Comments ln mathematics a logical statement is a statement that can be precisely judged to be true or false For example saying fz 2 is a continuous function77 is a logical statement It is also true Saying fz i is continuous at z 077 is also a logical statement but it is false Quanti ers introduce objects for use in logical statements as well as form part of larger compound statements As an isolated statement m 2 477 is neither true nor false since we haven t been introduced to z Saying for all z E R z 2 477 is a logical statement and is false There exists z E R such that z 2 477 is another logical statement and is true lmplications ifthen7 statements often introduce objects in the hypothesis state ment For example the statement If z 2 4 then x 277 implicitly assumes that z is some real number Formally we might want a quanti er and it would be For all z E R if z 2 4 then x 2 However in practice we skip the quanti er Use of implications Every time we use a theorem we make use of a true implication To use an implication we must always rst verify the hypothesis Once we verify the hypothesis we know the conclusion is true without any further checking Example Use the Intermediate Value Theorem lVT to guarantee that the function fz 3 7 2x has a zero between z 1 and z 2 The lVT states that the following implication is true If f is a function that is continuous on an interval 11 then if y is between at and fb then there exists some 0 E a 1 such that fc To use the theorem we must verify f is continuous on an appropriate interval What should we try 11 1 2 So state f is continuous on 1 2 because 1 is a polynomial which is continuous everywhere We now know that the conclusion is true namely the statement If y is between f1 71 and f2 4 ie y 6 714 then there exists some 0 E 1 2 such that fc y This is another implication To use this implication we must verify that a particular y value is between 71 and 4 So we state Since y 0 6 714 there exists some 0 E 1 2 such that fc O77 7 Reappearances lmplications every theorem can be thought of as an implication and every time we use a theorem we use an implication One key example is that the de nition of a limit involves the use of an implication Contrapositive Many theorems are most easily applied through a contrapositive Key example If a function is differentiable at z c then it is continuous at z c The contrapositive states If a function is not continuous at z c then it is not differentiable at z c Counterexamples Used frequently to demonstrate a statement is false TrueFalse questions Negation Needed for contrapositive and proofs by contradiction 7 Common errors Converse and contrapositive these are frequently confused Negation of an implication is not an implication Use of an implication it is a mistake to assume the conclusion is true You must show the hypothesis is true 0 Section 05 Proofs 7 Key concepts proof chain of implications direct proof proof by contradiction proof by induction inductive axiom inductive hypothesis 7 Reappearances Direct proofs most proofs we used were direct proofs The idea is that we use known implications to reach a nal conclusion Proof by lnduction this can be used prove that power functions with positive integer powers are continuous and it will play an important role in proofs for integration in 2nd semester 7 Common errors See Caution on p 48 If your task is to prove an equation is true or an inequality is true you can not write down the equation or inequality to begin with You must start with only one side and then demonstrate that you can logically reach the other side Do not work both sides of the equation lnduction Must check the case n 1 and also demonstrate that the inductive im plication is true assume the inductive hypothesis and show the inductive conclusion is true Failure to demonstrate that all hypotheses of an implication are true prior to reach ing a conclusion Chapter 1 0 Section 11 What is a function 7 Key concepts function domain target range function notation independentdependent variables tables and diagrams 7 Comments Domain We re asking what are accepted input values Target Where might output values land Range What are all of the output values that actually occur 7 Reappearances The entire course is based on the study of functions Domain Every time we learn about a new type of function we discuss the domain 7 Common errors Function notation still catches some students off guard Example Suppose we de ne a function fz 2 7 3 Some students want to say things like ft ft2 7 3 but this is not good Writing ft means we evaluate the function f with an input oft Instead we should write ft t2 7 3 and writing ft2 7 3 really means 252 7 32 7 3 Another common error is in using the de nition of a derivative Let s say fz x2m 1 Then the derivative is de ned as m 3 rm 1137 M m 2mh ixQz fl 77 Many students want to incorrectly write m lim flmhl flx lim Wwwwm h70 h h70 h This is often done with the above error at the same time What is wrong with the above statement 0 Section 12 Graphs of Functions 7 Key concepts the graph as a set the graph as a gure vertical line test intercepts pos itivenegative increasingdecreasing concavity extrema extremum in ection points vertical and horizontal asymptotes 7 Comments The height of a point on a graph gives the function value We later learn that the slope of the tangent line at a point gives the value of the derivative of the function A graph encodes all of the information about a function without requiring a formula Distinguish between properties at a point single input value at a time and on an interval A maximum occurs at an input value eg f has a maximum at z c A maximum value corresponds to the output value eg f attains a maximum value of y fc Properties on intervals correspond to intervals coming from the domain 7 Reappearances Modeling graphs with functions such as power functions polynomials and rational functions typically show particular points Each point my ab leads to an equation in the unknown coef cients fa b where we use the proposed formula for f on the left and the known height on the right All of the properties are described more precisely later using limits and derivatives 7 Common errors Failure to interpret the height of a point as a function value 9 Using y values to describe intervals such as where a function is increasing decreas ing Concavity Many students go between turning points to identify concavity You really need to decide whether the slope is increasing or decreasing ln ection points are almost always between turning points 9 0 Section 13 Linear Functions 7 Key concepts linear functions slope average rate of change of any function propor tional functions equations of lines parallel and perpendicular 7 Comments 7 Reappearances Average rate of change Motivates the de nition of a derivative key ingredient to mean value theorem Equations of Lines used to nd the equation of a tangent line given a point and a slope Proportional functions Generalized to become power functions 7 Common errors Average rate of change some students have it upside down It should be change in output over change in input 7 Am 0 Section 14 A Basic Library of Functions 7 Key concepts linear power polynomial rational algebraic functions transcendental functions piecewise functions and functions involving absolute value 7 Comments These functions comprise the bulk of the book as we learn how to analyze each of these different types of functions 7 Reappearances Each function type has at least one chapter devoted to it Piecewise functions most popular example for illustrating ideas such as jump dis continuities nondifferentiability at a jump or at a corner left and right continuous left and right differentiable Rewriting functions with absolute values turning each of these above questions about piecewise functions into a function that at rst glance does not look piecewise 7 Common errors Functions involving absolute value Using the de nition to determine the intervals This involves solving an inequality 0 Section 15 Combinations of Functions 7 Key concepts arithmetic combinations sum difference product and quotient com position domain of combination recognizing a composition decomposing 7 Comments 7 Reappearances Arithmetic Combination Used to de ne polynomials rational functions and al gebraic functions Used in the basic limit theorems and Algorithm 21 as well as several rules of differentiation Decomposition Critical idea in understanding transformations of a graph Key to determining limits and continuity of compositions and will later be needed for the chain rule of derivatives 7 Common errors Not using function notation correctly to nd composition Considering both functions to identify the domain of a composition 0 Section 16 Transformations and Symmetry 7 Key concepts Horizontal and vertical translations horizontal and vertical stretchshrinks re ections even and odd functions 7 Comments Transformations are a powerful tool for quickly analyzing a graph Some functions are neither even nor odd You must check the de nition That is you must consider the function re ected across the y axis f7z and see if it is the same as fz even or the same as the re ection across the z axis ie 180 degree rotation odd 7 Reappearances Considering f7z reappears when considering limits as x 7 700 see Example 413 on p 343 EvenOdd functions power functions later we ll deal with trigonometric func tions Vertical stretch shrink leads to the constant multiple rules of limits and derivatives 7 Common errors lncorrect algebraic determination of evenodd You must evaluate f7z and then simplify before checking how it compares to Example fz 3 m2 Then f7z 7m3 7m2 7mg 2 is not even or odd This probably goes back to a weak understanding of functions Calling a polynomial evenodd based on highest order term This will determine the end behavior but the function may not be even or odd Misusing the rules of transformations For example confusing vertical with hori zontal translations or moving the wrong direction Performing transformations in the wrong order 0 Section 17 lnverse Functions 7 Key concepts one to one functions horizontal line test monotonic functions inverse functions graphs and inverses restricted domain 7 Comments Finding the formula for an inverse function is basically the same as nding out if a particular value of y is in the range of a function That is write the equation fz y and now solve for z in terms of y if possible 7 Reappearances lnverse functions power functions involving roots very important in 2nd semester exponential and logarithm trigonometric and inverse trigonometric functions Monotonic related to idea of increasingdecreasing functions 7 Common errors The inverse function f 1m must not be confused with the reciprocal fz 1 1 1 Chapter 2 0 Section 21 Intuitive Notion of Limit 7 Key concepts limits limit notation left and right limits in nite limits limits at in nity horizontal asymptotes made semi precise graphical and table interpretations 7 Comments Relation between limits and onesided limits Thm 21 Must both exist and be equaD We must not consider the actual height fc Later we ll relax if we know 1 is continuous at z c 7 Reappearances Limits are fundamental to calculus Some say this is the feature that de nes calcu lus Continuity and derivatives focus on limits asymptotes and global behavior of functions are characterized by limits One sided limits Analyzing asymptotes left and right derivatives piecewise func tions 7 Common errors A left limit uses the notation z 7 c This does not mean a negative value It means we consider a limit restricted to values where z lt c 0 Section 22 Formal De nition of Limit 7 Key concepts punctured interval delta epsilon implication de nition of limit unique ness nding a delta for given epsilon formalizing special limits leftright in nite limits and limits at in nity 7 Comments Epsilon e gt 0 is given the task is to nd the appropriate delta 6 gt 0 We are trying to nd a local punctured neighborhood of a point z c so that the function remains within some e band about the limit value L Limits involving in nity are harder to grasp Similar idea to e band except now we have some high threshold that we need to cross 7 Reappearances In addition to Delta Epsilon proofs this idea is important in proving that differen tiability implies continuity The general idea of being within a tolerance of a value is important in general including statistical inference estimation and computer computation Common errors 0 Section 23 Delta Epsilon Proofs Key concepts Delta Epsilon proofs bounding 6 and using min special types of limits one sided and in nite Comments Bounds are needed on 6 whenever fz 7 L does not simplify to some simple product involving only z 7 c Reappearances Delta Epsilon proofs reappear when we consider different types of functions and need to prove they are continuous Common errors Confusing the hypothesis and the conclusion of the Delta Epsilon implication We assume that 0 lt lz7cl lt 6 Then we need to nd algebra that proves 7L lt 6 Some students work backwards on this Forgetting to deal with quanti ers We must state that e gt 0 is arbitrary before we declare 6 We must de ne our choice for 6 to show it exists before we assume 0 lt lz 7 C lt 6 We must assume 0 lt lz 7 C lt 6 and state this before we start to consider 7 L 0 Section 24 Limit Rules Key concepts limit of a constant limit of identity function limit of a linear function limit of a positive integer power function limit of sums limit of products limit of quotients Comments All of the limit rules fall into two categories limits we can evaluate directly con stant identity function linear function and positive integer power function and limits we can evaluate if we know how to nd limits of the parts All proofs involving limits using the limit rules require you to identify how to reduce the limit into parts so that we can evaluate limits of known types of functions Reappearances Limits These are key to calculating derivatives The basic rules are essential in proving the basic rules of derivatives Common errors Confusing identity function and constant function Just plugging it in rather than using actual limit rules when asked 0 Section 25 Calculating Limits Key concepts Algorithm 21 plug it in for special functions but not all indeterminate and unde ned limits quotients limits of quotients in nite limits and left and right limits cancellation and limits piecewise functions Comments Until we have continuity algorithm 21 made us explicitly identify a function as a sum difference product or quotient of the basic functions in the limit rules con stants linear functions and positive integer powers of It is continuity that let s us plug it in for more general functions or when we haven t explicitly written the function in a form in which the algorithm applies 7 Reappearances These strategies are used frequently analyzing vertical asymptotes nding deriva tives using a de nition especially for piecewise functions 7 Common errors For piecewise functions many students want to look at the function when x 0 But for a limit we aren t allowed to do this Many students stop evaluating a limit if they nd an indeterminate form But we must continue by factoring and canceling an appropriate factor until we nd it is unde ned something nonzero over zero or a value When we nd an in nite limit we must decide what happens on the left and on the right 0 Section 26 Continuity 7 Key concepts continuity at a point continuous on an interval continuous function left and right continuous removable discontinuities holes in graph jump discontinuities and in nite discontinuities vertical asymptotes Delta Epsilon de nition not punctured interval now special continuous functions linear and positive integer power functions algebraic combinations composition and limits 7 Comments Be sure to distinguish the different types of continuity at a point on an interval and as a function Verifying continuity involves three things function is de ned fc exists the limit is de ned limmnc fz exists and the limit and function agree This is summarized with one equation 3310 W fc 7 Reappearances When we introduce new functions we always decide where they are continuous 7 Common errors With piecewise functions most students think they can stop testing continuity when they show that a limit exists You must also demonstrate that the limit and function agree With compositions you have to be careful with limits To evaluate the limit of a composition f o gz fgm you must rst demonstrate that the function f is continuous at L limmncgz which means you need to evaluate the limit L rst If f is continuous then the limit will be fL Both steps must be shown 0 Section 27 Two Theorems about Continuous Functions 7 Key concepts Extreme Value Theorem EVT lntermediate Value Theorem lVT special uses of lVT to guarantee existence of zeros Thm 224 and to solve inequalities Thm 225 and 226 and Algorithm 22 10 7 Comments We implicitly use the Intermediate Value Theorem every time we use number lines for curve sketching analysis We usually use the Extreme Value Theorem when we look for global extrema on closed intervals But we can t use it if we aren t continuous on the interval 7 Reappearances Extreme Value Theorem is a key ingredient in Rolle s Theorem lntermediate Value Theorem Used primarily to solve inequalities Also allows us to isolate roots of functions 7 Common errors Chapter 3 Biggest problem Confusion between hypotheses and conclusions For example to see if the Extreme Value Theorem applies we don t rst check to see if there is a global maximum and minimum That is the conclusion not the hypothesis Instead we check to see if the function is continuous on a closed interval of interest Once we verify this we can use the theorem to state declare without question that the function must have a maximum and a minimum We don t need to know where they occur we re just happy that they do occur So verify that the function is continuous on a closed interval and state this Then use the conclusion 9 0 Section 31 Tangent Lines and the Derivative at a Point 7 Key concepts tangent line secant line slope derivative at a point 7 Comments Now that we also know how to nd the derivative as a function there is some confusion about nd the derivative at a point by using the de nition which involves evaluating fc as a value and then taking a limit and nding the derivative as a function which involves evaluating the limit with fz using its expression and then evaluating this function f m at z c Finding the equation of a tangent line is a basic skill 7 Reappearances This is a fundamental concept of calculus It is a consistent theme from here on 7 Common errors Major problems when students don t understand function notation For example f1 71 means that we use the formula for fz replacing x by 1 h For piecewise functions To replace fc with a value you must look at the break point and see what value the function uses You do not replace it with the formula for fc h by setting h 0 Why What s the difference When a derivative exists you will be able to cancel a factor If a factor does not cancel and you still have a zero in the denominator the derivative does not exist 0 Section 32 The Derivative as an lnstantaneous Rate of Change 11 7 Key concepts average rate of change instantaneous rate of change velocity and accel eration units of rates 7 Comments A derivative has two essential interpretations Any student of calculus needs to know these A derivative at a point represents the slope of the tangent line of the graph at that point A derivative at a point represents the instantaneous rate of change of the function at that point 7 Reappearances Average rate of change is a key ingredient in the Mean Value Theorem 7 Common errors Change in function value Ay always goes on top Width of interval Am always goes on bottom The order in subtraction needs to be the same on top and bottom Graphs of position velocity and acceleration are often confused 0 Section 33 Differentiability 7 Key concepts di erentiability at a point di erentiability on an interval and differentia bility as a function graphical view algebraic view piecewise functions relation between continuity and di erentiability 7 Comments 7 Reappearances Key ingredient in Rolle s Theorem Local Extrema and Mean Value Theorem We see functions that are continuous but not differentiable with power functions and rational powers 7 Common errors See the remark in Section 31 about piecewise functions 0 Section 34 The Derivative as a Function 7 Key concepts derivative as a function graph of derivative derivative notations second derivative higher order derivatives position velocity and acceleration functions 7 Comments Based on a graph of f you should be able to sketch a graph of f and to some degree you should be able to go in reverse 7 Reappearances Almost everywhere after this These are very critical concepts 7 Common errors Lack of understanding that the height of the graph of f z corresponds to the slope of the graph of The height of the graph of f z corresponds to the slope of the graph of f z Where f z is above the axis fz must have a positive slope increasing A point 2 1 on the graph of f z means that f 2 1 or that the slope of the graph of fz at z 2 is 1 Major problems for some in correctly substituting the correct formula for fz h You may nd yourself more reliable if you use the alternative de nition using ft 0 Section 35 Basic Differentiation Rules 7 Key concepts derivative of a constant derivative of the identity function derivative of a linear function derivative of a power function positive integer power derivative of a constant multiple of a function derivatives of sums and differences derivative at a break point 7 Comments Compare to limit rules Again these rules have two categories derivatives we can actually compute constant function identity function linear function positive integer power function and derivatives we can compute if we can compute the parts constant multiple sum and difference Recognize the difference between these 9 Because of the way our rules work we must always multiply out functions so that we have it written as a sum of simpler functions that we know how to differentiate All of these rules have relatively simple proofs They all rely on the de nition of the derivative 9 The derivative of piecewise functions use the rules on the function de ned away from a break point But you have to be even more careful when you are at the break point You have to check continuity Then you have to compare the derivative on the left and on the right to see if they agree 7 Reappearances Polynomials Even though there is a chapter on this later we now have everything we need to do polynomials Rational functions Except for the quotient rule we have what we need to nd 19 and qm which are used in the quotient rule The sum and difference rule will reappear very frequently in the second semester We also will nally meet the product rule and a rule for compositions the chain rule 7 Common errors Too many students fall for the temptation to differentiate each term in a product We must multiply it out and then differentiate using the sum rule If the rule isn t known then we must return to the de nition of the derivative The best example right now is fz x 2m 1 or functions like this We don t have a rule for these so you have to use the de nition For piecewise functions it is easy to forget to check continuity It is then just as easy to forget to see if the derivatives on the left and right agree 0 Section 36 Three Theorems about Tangent Lines 7 Key concepts derivative at a local extremum critical points Rolle s theorem Mean Value Theorem 7 Comments There has been some confusion about how to think about theorems like this These are called existence theorems This means that they provide one way to guarantee that some point with a property exists For example Rolle s theorem says that we can guarantee there is a point c in a b where fc 0 if we can rst demonstrate that f is continuous on 11 and differentiable on a b There may be times when fc 0 for some 0 E a b where these hypotheses are false But the theorem isn t concerned about that You must guarantee that every part of the hypothesis is true before you are allowed to state that the conclusion is true It is not the reverse See common errors for Section 05 To use the MVT you rst calculate the average rate of change on an interval and then verify that the function is continuous and differentiable Then you can assume that fc equals this average rate for some 0 in the interval 7 Reappearances The Derivative at a Local Extremum is used anytime we look for a maximum or minimum by nding critical points Rolle s Theorem is used when we said that there is a turning point between any two roots of a polynomial The Mean Value Theorem is used to show that f is increasing when f m is positive on some interval Similar for concavity 7 Common errors It might be easy to confuse the names of the Intermediate Value Theorem and Mean Value Theorem The MVT involves the average rate of change while the lVT involves points between two heights of a graph 0 Section 37 The First Derivative and Function Behavior 7 Key concepts increasingdecreasing functions with same derivative integration con stants nding intervals rst derivative test for extrema 7 Comments 7 Reappearances The basic strategy of nding intervals where a derivative is positivenegative will be used for every function class we consider 7 Common errors Some assume that the sign must change every time we cross a critical point This will not happen if the critical point is a root with even multiplicity It is possible to have intervals 7 To show a critical point is a maximum or minimum you must use either the rst derivative test or second derivative test Section 38 0 Section 38 The Second Derivative and Function Behavior 7 Key concepts second derivative and concavity in ection points second derivative test for extrema detailed curve sketching 7 Comments The second derivative is the derivative of 1quot That means if we have a graph of f z we can determine f z as the slope of the graph of f z at each x The of cial de nition of concavity involves intervals where f m is increasingdecreasing Because the value of f z is related to this the concavity is also determined from Mm The same strategy for nding local extrema is used to nd in ection points except that we use f z instead of f z So in ection points of 1 correspond to what kind of points of f Detailed curve sketching is most directly summarized by number line summaries of f f and 1 However in a real problem you need to translate the number lines into statements about intervals which you then convert into a graphical picture 7 Reappearances Curve sketching is a key part of our understanding of every type of function from here on Second derivative test is a quick way sometimes to identify a critical point as a maximum or minimum 7 Common errors Chapter 4 lncomplete factorization will make you miss roots and so all intervals are potentially wrong 9 Second derivative test You should always think of f c in terms of concavity That is f c gt 0 means concave up so that we would have a minimum f c lt 0 means concave down so that we have a maximum The line numbers for f f and 1 usually have different values creating interval end points They are always the roots or discontinuites of the function That means we have to factor three different polynomials to do curve sketching analysis of polynomials I often see students who just use the roots of f m for both the f and 1 number lines You need the roots of f z as well 9 Many students get intervals that seem to have no relation to the graph That is if an interval analysis states that the function is increasing on an interval the graph needs to be increasing in that interval If the interval analysis shows f z lt 0 on an interval then the graph better appear concave down 0 Section 41 The Algebra of Power Functions 7 Key concepts integer exponents recursive de nition negative exponents fractional exponents roots rational exponents rules of exponents manipulations common de nominator and conjugate extraneous solutions power functions domain 7 Comments In addition to knowing what is a power function be sure you understand functions that look like power functions in one way or another but aren t eg exponentials and compositions Finding domain is a revisiting of an old topic Worry about zeros in denominators negative exponents and negative numbers in even roots If we have an even root in a denominator then we can only allow strictly positive numbers inside 7 Reappearances Power functions with positive integer powers are the building blocks of polynomials although polynomials are not power functions do you know why Dealing with exponents will reappear next semester with algebraic functions and exponential and logarithmic functions 7 Common errors Using the conjugate of a sum or difference involving roots We aren t just squaring both terms or cubing them as the case may be we are multiplying by 1 in dis guise which leaves the expression with the same value in a special way so that we get a product like a 7 ba b a2 7 b2 0 Section 42 Limits of Power Functions 7 Key concepts continuity and limits of power functions one sided limits for even roots and for negative powers global behavior depends on sign of exponent limits with com positions indeterminate forms involving 00 7 oo 7 Comments When doing a limit involving a composition in other words a power of a more complicated expression than just m you must rst identify the domain of the new function so that you can see if the limit is in the interior of the domain or at the endpoints where only one sided limits make sense When trying to remember limits near zero or limits at in nity it is often a good idea to think about what happens when you divide by a very small number or when you raise a large number to a power or divide by a large number These thoughts should help you keep straight when limits are zero or in nite 7 Reappearances The same concepts that are used for limits of power functions are used when we evaluate limits of polynomials and rational functions except that we need to worry about the sums and quotients 7 Common errors Ignoring the domain can make you overlook one sided limits at end points Confusing when a limit is zero with limits of in nity Forgetting to factor the largest power of z when dealing with a sum of limits as x 7 00 but this is the only time we want to factor like this 0 Section 43 Derivatives of Power Functions 7 Key concepts using the de nition often involves common denominator conjugates or both power rule di erentiability everywhere in domain need to simplify rst antiderivatives 7 Comments You must recognize times when the power rule does not apply when in a fraction unless you can rewrite as a simple power function or when involving a composition If the exponent is less than one but positive the function will be continuous but not differentiable Why What is the limit See Example 420 7 Reappearances Key to understanding polynomials rational functions and later algebraic functions 7 Common errors Frequently mistakes involving using the de nition center on algebra errors incorrect interpretation of negative exponents and incorrect methods involving the conjugate Derivatives of terms like fz 72 you need to rewrite in the form of a power function rst fz m4 to apply the power rule correctly 16 0 Section 44 Graphs of Power Functions with Integer Powers 7 Key concepts graphs domain and range even and odd curve sketching analysis trans formations and thinking in terms of reciprocals modeling a graph with a function nd a good function 7 Comments Many times thinking about limits as x 7 oo remind us about the other features of the graph If the exponent is positive then the limit is in nite but if the exponent is negative the limit is zero a horizontal asymptote Similarly we remember what happens when x 7 O The left side will follow by appropriate symmetry To do curve sketching analysis you need to consider the functions f f and f as described in Section 38 A positive number raised to any power will be positive 7 Reappearances Rational powers next section are slightly more complicated than integer powers 7 Common errors Forgetting to do all three number lines and then interpreting these explicitly in terms of intervals where f has different behaviors 0 Section 45 Graphs of Power Functions with Rational Powers 7 Key concepts reduced rational power graphs symmetry domain and range curve sketching analysis negative powers inverse functions modeling graphs with power func tions 7 Comments You should be able to do a curve sketching analysis from scratch You should also be able to sketch a power function directly using properties we know about the function 7 Reappearances 7 Common errors The root which determines if the function has negative numbers in the domain is based on the denominator of the exponent Sometimes these get mixed up fz mil2 has domain 0 00 but gz z23 has domain 700 00 9 Modeling graphs Each point on the graph corresponds to an equation Two points allow you to solve for both the constant coefficient usually written A as well as the power k Only one point typically suggests you get to choose the power k to match the basic features such as an appropriate domain symmetry differentiability then you use the point to solve for A You can t just guess or write down A and k You need to write down the equations and then solve for A and k Chapter 5 0 Section 51 The Algebra of Polynomial Functions Key concepts polynomial degree coef cients leading coef cient roots and factors splitting theorem reducible and irreducible quadratics factor theorem Fundamental Theorem of Algebra number of roots factoring strategies grouping guessing integer roots and lnteger Root Theorem synthetic division Comments The real power house theorem in this section is the Splitting Theorem lt guarantees that we can factor a polynomial The Factor Theorem equates each simple linear factor in the Splitting Theorem with the roots of the polynomial The Fundamental Theorem simply counts the number of possible roots Synthetic division is very useful Evaluates the polynomial at z c at the same time that it divides the polynomial by the factor x 7 c This makes it easy to factor a polynomial whenever you can successfully guess the roots Reappearances Factoring is used to sketch graphs of polynomials quickly to nd roots of a poly nomial f to nd critical points roots of f and to nd potential in ection points roots of f to identify discontinuities in rational functions and to cancel common factors and evaluate limits Common errors ln synthetic division In the rst step you add 0 to the leading coef cient not just write 1 in the rst spot To divide a polynomial by z 7 3 you use 3 in the synthetic division Some still do not realize the relationship between synthetic division and factor ing A quadratic is irreducible if it has no real roots You must check the quadratic formula or at least the discriminant The list of possible integer roots is not the list of all possible roots Showing that none of the possible integer roots are roots is not enough to say there are no roots They might be fractions or irrational numbers 0 Section 52 Limits and Derivatives of Polynomial Functions Key concepts continuity everywhere limits at in nity degree and leading coef cient derivatives differentiable everywhere positionvelocity acceleration problems local behavior roots and local extrema turning points how many Comments Really this section has nothing we didn t already know It just summarizes this information Counting turning points gives a clue to the degreeipay attention Reappearances Similar strategies occur for rational functions especially in limits at in nity Common errors Not expanding product for derivatives we can t deal with products of factors yet Antiderivatives 7 it s easy to get confused about the coef cient in front 18 0 Section 53 Graphing Polynomial Functions 7 Key concepts repeated roots sketching polynomials with linear factors accurate curve sketching analysis using sign of f f and 1 modeling graphs with polynomial func tions 7 Comments Modeling graphs with polynomials usually leads to a system of equations These strategies were discussed albeit brie ly in Section 02 A key skill for this is factoring If we can factor we can immediately identify where a function is positivenegative lrreducible quadratic factors might introduce extra wiggles but not new roots Factoring the derivatives we can do a quick sketch of f or f to identify the sign of f or 1 without testing a single point Detailed analysis tells us where speci c points are such as extreme values and in ection points 7 Reappearances Similar strategies are used for rational functions 7 Common errors Failure to analyze the rst two derivatives to sketch a detailed graph Failure to relate analysis to a curve that is sketched For example if you have a number line representation for jquot which you should you need to state the intervals where f is increasing and decreasing and your graph needs to agree with this You can t just nd the zeros You need to factor in order to nd out the degree of the roots and to guarantee you didn t miss any roots 0 Section 54 Optimization with Polynomial Functions 7 Key concepts word problems interpreting a problem in a mathematical way optimiza tion on an interval difference between open and closed intervals comparing endpoints and critical points position problems optimization with constraints 7 Comments Perhaps the most dreaded type of problem unfortunately application problems such as these are some of the most important reasons for understanding calculus We can solve problems that have fundamentally important solutions Constraints allow us deal with problems involving more than one unknown The constraint allows us to solve for one variable in terms of the other 9 The book uses limits when dealing with intervals having open endpoints because that is of cially what we are doing We are dealing with a function restricted to the domain so the restricted function isn t really de ned at that endpoint Also this is what we ll need to do with rational functions especially in the presence of a discontinuity 7 Reappearances Polynomials are one simple function for optimization But we ll see these again next semester for other types of functions 7 Common errors Ignoring end points of intervals 7 closed ends can be extreme values open ends can not be extreme values and may make it so no extreme value lncomplete analysis 7 missing a critical point usually by not factoring or not identifying if a maximum or minimum Optimizing wrong function Chapter 6 0 Section 61 The Algebra of Rational Functions 7 Key concepts rational function factoring cancelling holes removable discontinuities domains and roots long division proper and improper comparing degrees 7 Comments Long division works always Synthetic division only works when we divide by some thing of the exact form x 7 0 Synthetic division won t work for a divisor 2x 7 1 or m2 7 1 Most concepts from this section should have been review 7 Reappearances 7 Common errors Canceling factors after canceling a common factor you must ensure the domain is preserved There is either a hole or a vertical asymptote to worry about You can t cancel an asymptote away Terms in a polynomial that aren t represented need to have token place holders zero coef cients in both schemes of division long and synthetic 0 Section 62 Limits and Asymptotes of Rational Functions 7 Key concepts limits and cancellation including holes reduced rational functions no common factors vertical asymptotes horizontal asymptotes slant asymptotes curve asymptotes 7 Comments Almost always best to factor completely top and bottom Then cancel common factors noting the domainl Many concepts relate to comparing the degree of numerator and denominator lim its at in nity horizontal slant and curve asymptotes 7 Reappearances 7 Common errors lndeterminate and unde ned are not the same We only factor out largest power of z for limits at in nity Otherwise we try to plug it in Unde ned limits division by zero require exploration of left and right limits 20 0 Section 63 Derivatives of Rational Functions 7 Key concepts derivative by de nition quotient rule local extrema using critical points global extrema using critical points endpoints and watching for asymptotes 7 Comments The formula for quotient rule is not easy But it is important to keep the order right Critical points occur where f z 0 or f m does not exist But rational func tions are differentiable meaning that f z does not exist only at points not in the domain 9 After computing a derivative we still have a rational function But nding critical points involves only the roots of the numerator which is a polynomial where f z 0 and the roots of the denominator also a polynomial where f z does not exist Remember a function can changes sign at zeros and discontinuities So for a rational function it can change sign at zeros of the numerator zeros or of the denominator discontinuities Otherwise our common strategy with number lines still works 7 Reappearances 7 Common errors Do not just take the derivative of the top and the bottom This almost never gives the right answer and it is always wrong to do Finding intervals where a function is positive or negative you must not miss any factors for the numerator or denominator Don t just check integers7factor both polynomials completely Checking for extreme values 7 vertical asymptotes may make it so there is no maximum no minimum or neither one Watch for them 0 Section 63 Graphs of Rational Functions 7 Key concepts quick sketches from factorization accurate graphs curve sketching anal ysis modeling graphs with rational functions 7 Comments Factoring and canceling tells us about zeros vertical asymptotes and discontinuities Quick sketches of the polynomials pz and qz tells most of the behavior of f pq zeros of p match zeros of 1 unless in q also7cancel rst zeros of q lead to discontinuities and you can directly see the left and right limits by comparing the sign ofp and q to the left and right of the zero of q Comparing degrees of top and bottom tells us about right and left limits at in nity Long division tells us about any slant or curve asymptotes only if top has higher degree 7 Reappearances 7 Common errors If you cancel a factor you need to be sure the domain is correct hole or asymptote 21
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