MA123, Chapter 3: The idea of its (pp. 47-67, Gootman) Chapter Goals: Evaluate its. Evaluate one-sided its. Understand the concepts of continuity and differentiability and their relationship. Assignments: Assignment 04 Assignment 05 Earlier, the idea of its came up naturally in the course of defining the derivative of a function at a point. We now study its more systematically. Computing a it means computing what happens to the value of a function as the variable in the epression gets closer and closer to (but does not equal) aparticularvalue. The basic definition of it: Let f be a function of. Theepression f() =L means that as gets closer and closer to c, throughvaluesbothsmallerandlargerthanc, butnotequaltoc, then the values of f() getcloserandclosertothevaluel. Note: It may sometimes happen that the it does not eist. 2 +8 Eample 1 (a): Use the tables to help evaluate 2 +2. gets close to 2 from the left 1.8 1.9 1.99 1.999 2 +8 +2 gets close to 2 from the right 2.001 2.01 2.1 2.2 2 +8 +2 Eample 1 (b): Suppose that, instead of calculating all the values in the above tables, you simply substitute the value =2 into 2 +8. What do you find? +2 Note: The method of substituting in the iting value of the variable works because the operations of arithmetic, namely, addition, subtraction, multiplication, and division, all behave reasonably with respect to the idea of getting closer to as long as nothing illegal happens. The one illegality you will mainly have to watch out for is division by zero. More precisely, if f and g are two functions one has: ( ) f()+g() = f() + g() ( ) ( ) ( ) f() g() = f() g() ( ) f() g() = f() g() f() g() = f() g() as long as g() 0 21 1/10 Chapter3.pdf
Eample 2: Compute 1 ( ( 2 +4 +3) (2 4) ). 2 2 +1 Eample 3: Compute. 1 +1 Eample 4: Suppose 3 f() = 2 and 3 g() =4. 3 ( ( +1) f() 2 + +2 g() Determine ). Some complications with the definition of its: The previous eamples seem to imply that computing a it is the same thing as evaluating a function. This is only true if the function in the it is nice enough ( nice enough will be defined more precisely in a few pages). The net few eamples will illustrate that the computation of f() doesnotalwaysreducetothemere substitution of the value of c in place of in the epression defining f(). The unusual functions described in what follows are introduced to emphasize the fact that the notion of it really involves what happens to the values of f() as gets closer to the fied value c, andnotwhatthevalueoff() at = c is. In addition, the most interesting its generally arise precisely when substitution gives an illegal epression involving division by 0, or even an epression of the form 0 0.Thelattercaseoccursforeamplewhencomputingthederivative of a function. How can a it fail to eist? There are two basic ways that a it can fail to eist. (a) The function attempts to approach multiple values as c. Geometrically, this behavior can be seen as a jump in the graph ofafunction. Algebraically, this behavior typically arises with piecewise defined funtions. (b) The function grows without bound as c. Geometrically, this behavior can be seen as a vertical asymptote in the graph of a function. Algebraically, this behavior typically arises when the denominator of a function approaches zero. 22 2/10 Chapter3.pdf (2/10)
Eample 5: The graph of the function { 2 3, h() = 2 +7, is shown to the right. if > 2; if 2 y 3 1 Analyze h(). 2 2 The previous eample showed that the it of h() asthevariableapproached 2 didnoteist. Ontheother hand, the function appears to have well defined iting behavior on either side of = 2. This brings us to the following notions: One-sided its: Aone-sideditepresseswhathappenstothevaluesofanepression as the variable in the epression gets closer and closer to some particular value c from either the left on the number line (that is, through values less than c) or from the right on the number line(that is, through values greater than c). The notation is: f() } {{} it from the left of c f() } + {{} it from the right of c Fact: f() eists if and only if both f() and f() + eist and have the same value. Eample 6: The graph of the function 4 g() = 2, if 1; +1 3, if =1. is shown to the right. g() 0.8 1.9512195 0.9 1.9889503 0.999 1.999999 y 3 2 1 Compute g(). 1 1.001 1.999999 1.1 1.9909502 1.2 1.9672131 23 3/10 Chapter3.pdf (3/10)
The problem of division by zero and a finite nonzero numerator: When this happens, it is standard to say that the epression is getting arbitrarily large (inthepositiveornegativedirection) oris going to (positive or negative) infinity, denoted by ±. As infinity is not really a number, the epression is not really getting close to any particular real number. Thus, technically speaking, the it does not eist. In the web homework system, infinite its should be entered as DNE. Eample 7: Analyze 1 5 ( 1) 2. 2 Eample 8: Analyze 1 1. 2 Eample 9: Analyze the it 0 Be sure to graph the functions in each of the last three eamples, and notice the graphs have vertical asymptotes at =1,=1, and =0,respectively. 24 4/10 Chapter3.pdf (4/10)
The case 0 : The most interesting and important situation with its is when a substitution yields 0 0. This is precisely the situation we are confronted with when attempting to compute derivatives from the 0 definition. The result 0 yields absolutely no information about the it. It does not eventellusthattheit 0 does not eist. The only thing it tells us is that we have to do more work to determine the it. 4 Eample 10: Find the it 0. ( 2 Eample 11: Find the it 0 + 5 2 ). Eample 12: Find the it 2 2 3. 3 3 y 4 3 (h 3) 2 9 Eample 13: Find the it. h 0 h 25 5/10 Chapter3.pdf (5/10)
Eample 14: Find the its 3 6 2 + 2 3 6 2 2 3 6 2 2. p() Limits at infinity: Afunctionf() issaidtobearational function if it is of the type q(), where p() andq() arebothpolynomialsin. Sometimesweareinterestedindeterminingthebehaviorofarational function for large (positive or negative) values of the variable. This will be the case, for eample, in Chapter 9. There is a general principle that makes computing these its easy. The idea is that, for very large (positive or negative) values of, thetermwiththehighestpowerof has the most influence on the behavior of the polynomial. In other words, when is very large, the term with the highest power dominates the other terms. Theorem: Let p() andq() bepolynomials.then ± p() q() = ± highest order term of p() highest order term of q(). Eample 15: y Let p() = 4 2. Find the its +1 + 4 2 +1 4 2 +1. (2 +1) 2 Eample 16: Find the it 5 2 +2 +1. (3 +2) 2 (5 +1) 4 Eample 17: Find the it 6 +1 ( +1)(2 +3) 2 (4 +5) 3 26 6/10 Chapter3.pdf (6/10)
Continuity and differentiability: We first give a brief, non-rigorous and intuitive eplanation of two fundamental notions in Calculus whose definitions involve its. We then discuss how these two notions relate to each other. Definition of continuity: Afunctionf is continuous at a point = c if f() =f(c). Afunctionf is continuous on an interval if it is continuous at every point of that interval. Note: Geometrically, this means that the graph of f has no holes, jumps, or gaps at any point in the domain of f. Thusyoucandrawthegraphoff from one end of the interval to the other without lifting your pencil off the paper. Analytically, this means the value of the function at = c can be recovered if one knows the values of f() for near = c. In other words, the values of a continuous function cannot change abruptly. Fact: If f and g are continuous functions at c then kf(), f()+g(), f() g() and constant f(), where g(c) 0, are continuous at c. g() Eamples: Many of the standard algebraic functions are continuous. Polynomials are continuous at every point. Rational functions are continuous at every point in their domain. (i.e., rational functions are continuous away from zeros of their denominators) Eample 18: Consider the function f() = Graph the function f when B =4andB = 1. Find a value of B such that the function is continuous at =1. { 2 3, if 1; 2 + B, if >1 27 7/10 Chapter3.pdf (7/10)
Eample 19: Let f() =[[]] be the function that associates to any value of the greatest integer less than or equal to. Computethe[[0.5]], [[1.99]], [[2]], [[2.01]], [[4.87]], [[ 1.5]]. Make a graph of the function y =[[]]. Compute and 2 [[]] 2 +[[]]. y Look back at eample 18 in Chapter 1 for more information about thegreatestintegerfunction. Definition of differentiability: Differentiability is concerned with whether the graph of the function can be well approimated by a straight line. Geometrically, we say that y = f() is differentiable at = c provided that the graph of y = f() is(nearly)indistinguishablefromthegraphofastraightline, provided we restrict attention to values that are sufficiently close to = c. More precisely, we say that f is differentiable at = c if the it f (c) = h 0 f(c + h) f(c) h eists. In other words, a function is differentiable at = c if the derivative eists at = c. Note: Geometrically, f is differentiable at if there is a well defined tangent line to y = f() atthepoint(, f()) (so the graph is smooth there, and does not have a sharp point), and furthermorethetangentlineisnotvertical. Analytically, a function is differentiable if the function does not abruptly change direction. Eamples: Many of the standard algebraic functions are differentiable. Polynomials are differentiable at every point. Rational functions are differentiable at every point in their domain. (i.e., rational functions are differentiable away from zeros of their denominators) The net few eamples illustrate how continuity and differentiability are related. 28 8/10 Chapter3.pdf (8/10)
Eample 20: f at the point =0? Consider the function f() =.Whatcanyousayaboutthetangentlinetothegraphof Eample 21: f at the point =0? Consider the function f() =. Whatcanyousayaboutthetangentlinetothegraphof Theorem: If f is differentiable at = c, thenf is also continuous at = c. Equivalently, iff is not continuous at = c, thenf is not differentiable at = c. Note: If f is not differentiable at = c, thenf may or may not be continuous at = c. (a) f() = is not differentiable at =0,butitiscontinuousat =0. (b) f() = 1 is not differentiable at =0anditisnotcontinuousat =0. Look at the graph of the Dow Jones Industrial Average for a real lifeeampleofafunctionthatiscontinuous everywhere but which has many points of non-differentiability. Eample 22: Let f() = { 2, if 2; m + b, if >2. Find the values of m and b that make f differentiable at =2. 29 9/10 Chapter3.pdf (9/10)
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