3.2 Differentiability

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1 Section 3 Differentiability 09 3 Differentiability What you will learn about How f (a) Might Fail to Eist Differentiability Implies Local Linearity Numerical Derivatives on a Calculator Differentiability Implies Continuity Intermeiate Value Theorem for Derivatives an why Graphs of ifferentiable functions can be approimate by their tangent lines at points where the erivative eists How f œ (a) Might Fail to Eist A function will not have a erivative at a point P(a, f(a)) where the slopes of the secant lines,, - a fail to approach a it as approaches a Figures 3 34 illustrate four ifferent instances where this occurs For eample, a function whose graph is otherwise smooth will fail to have a erivative at a point where the graph has a corner, where the one-sie erivatives iffer; Eample: f() = ƒ ƒ [ 3, 3] by [, ] [ 3, 3] by [, ] Figure 3 There is a corner at = 0 Figure 3 There is a cusp at = 0 a cusp, where the slopes of the secant lines approach from one sie an - q from the other (an etreme case of a corner); Eample: qf() = >3 3 a vertical tangent, where the slopes of the secant lines approach either q or - q from both sies (in this eample, q); Eample: f() = 3 How rough can the graph of a continuous function be? The graph of the absolute value function fails to be ifferentiable at a single point If you graph y = sin - (sin ()) on your calculator, you will see a continuous function with an infinite number of points of nonifferentiability But can a continuous function fail to be ifferentiable at every point? The answer, surprisingly enough, is yes, as Karl Weierstrass showe in 87 One of his formulas (there are many like it) was q f () = a a ncos n=0 3 b (9 n p), a formula that epresses f as an infinite (but converging) sum of cosines with increasingly higher frequencies By aing wiggles to wiggles infinitely many times, so to speak, the formula prouces a function whose graph is too bumpy in the it to have a tangent anywhere! [ 3, 3] by [, ] Figure 33 There is a vertical tangent line at = 0 4 a iscontinuity (which will cause one or both of the one-sie erivatives to be noneistent) Eample: The Unit Step Function In this eample, the left-han erivative fails to eist: U(0 + h) - U(0) - U() = e -, 6 0, Ú 0 (-) - () = - [ 3, 3] by [, ] Figure 34 There is a iscontinuity at = 0 = h:0 - - h = q

2 0 Chapter 3 Derivatives Later in this section we will prove a theorem that states that a function must be continuous at a to be ifferentiable at a This theorem woul provie a quick an easy verification that U is not ifferentiable at = 0 EXAMPLE Fining Where a Function Is Not Differentiable Fin all points in the omain of f() = ƒ - ƒ + 3 where f is not ifferentiable Think graphically! The graph of this function is the same as that of y = ƒ ƒ, translate units to the right an 3 units up This puts the corner at the point (, 3 ), so this function is not ifferentiable at = At every other point, the graph is (locally) a straight line an f has erivative + or - ( again, just like y = ƒ ƒ ) Now Try Eercise Most of the functions we encounter in calculus are ifferentiable wherever they are efine, which means that they will not have corners, cusps, vertical tangent lines, or points of iscontinuity within their omains Their graphs will be unbroken an smooth, with a well-efine slope at each point Polynomials are ifferentiable, as are rational functions, trigonometric functions, eponential functions, an logarithmic functions Composites of ifferentiable functions are ifferentiable, an so are sums, proucts, integer powers, an quotients of ifferentiable functions, where efine We will see why all of this is true as Chapters 3 an 4 unfol [ 4, 4] by [ 3, 3] (a) Differentiability Implies Local Linearity A goo way to think of ifferentiable functions is that they are locally linear; that is, a function that is ifferentiable at a closely resembles its own tangent line very close to a In the jargon of graphing calculators, ifferentiable curves will straighten out when we zoom in on them at a point of ifferentiability (See Figure 35) EXPLORATION Zooming in to See Differentiability [7, 3] by [7, ] (b) [93, 07] by [85, 95] (c) Figure 35 Three ifferent views of the ifferentiable function f() = cos (3) We have zoome in here at the point (, 9) Is either of these functions ifferentiable at = 0? (a) f() = ƒ ƒ + (b) g() = We alreay know that f is not ifferentiable at = 0; its graph has a corner there Graph f an zoom in at the point ( 0, ) several times Does the corner show signs of straightening out? Now o the same thing with g Does the graph of g show signs of straightening out? We will learn a quick way to ifferentiate g in Section 36, but for now suffice it to say that it is ifferentiable at = 0, an in fact has a horizontal tangent there 3 How many zooms oes it take before the graph of g looks eactly like a horizontal line? 4 Now graph f an g together in a stanar square viewing winow They appear to be ientical until you start zooming in The ifferentiable function eventually straightens out, while the nonifferentiable function remains impressively unchange

3 Section 3 Differentiability y m = f(a + h) f(a h) a h a a + h m = f(a + h) f(a) h tangent line Figure 36 The symmetric ifference quotient (slope m ) usually gives a better approimation of the erivative for a given value of h than oes the regular ifference quotient (slope m ), which is why the symmetric ifference quotient is use in the numerical erivative Numerical Derivatives on a Calculator For small values of h, the ifference quotient f(a + h) - f(a) h is often a goo numerical approimation of f (a) However, as suggeste by Figure 36, the same value of h will usually yiel a better approimation of f (a) if we use the symmetric ifference quotient f(a + h) - f(a - h) to compute the slope between two nearby points on opposite sies of a In fact, this approimation (which calculators can easily compute) is close enough to be use as a substitute for the erivative at a point in most applications DEFINITION The Numerical Derivative The numerical erivative of f at a, which we will enote NDER (f(), a), is the number f(a + 000) - f(a - 000) 000 The numerical erivative of f, which we will enote NDER (f(), ), is the function f( + 000) - f( - 000) 000 Some calculators have a name for the numerical erivative, like nderiv (f(),, a ), which is similar to the one we use in the efinition Others use the Leibniz notation for the actual erivative at a: (f()) ` =a We o not want to suggest that the numerical erivative an the erivative are the same, so we will continue to use our generic term NDER when referring to the numerical erivative Thus, in this tetbook, while Be aware, however, that on your calculator might well refer to the numerical erivative EXAMPLE Computing a Numerical Derivative If f() = 3, use the numerical erivative to approimate f () f () = NDER (f(), a) = (f()) ` f(a + h) - f(a) = =a (f()) ` =a f(a + 000) - f(a - 000), 000 (3 ) ` L NDER( 3, ) = (00)3 - (999) 3 = = 000 Now Try Eercise 7

4 Chapter 3 Derivatives In Eample of Section 3, we foun the erivative of 3 to be 3, whose value at = is 3() = The numerical erivative is accurate to 5 ecimal places Not ba for the push of a button Eample gives ramatic evience that NDER is very accurate when h = 000 Such accuracy is usually the case, although it is also possible for NDER to prouce some surprisingly inaccurate results, as in Eample 3 EXAMPLE 3 Fooling the Symmetric Difference Quotient Compute NDER ( ƒ ƒ, 0), the numerical erivative of ƒ ƒ at = 0 We saw at the start of this section that y = ƒ ƒ is not ifferentiable at = 0, since its right-han an left-han erivatives at = 0 are not the same Nonetheless, Even in the it, NDER( ƒ ƒ, 0) = h: = 000 = 0 ƒ 0 + h ƒ - ƒ 0 - h ƒ ƒ ƒ - ƒ ƒ (000) 0 = h:0 = 0 This proves that the erivative cannot be efine as the it of the symmetric ifference quotient The symmetric ifference quotient, which works on opposite sies of 0, has no chance of etecting the corner! Now Try Eercise 3 In light of Eample 3, it is worth repeating here that the symmetric ifference quotient actually oes approach f (a) when f (a) eists, an in fact approimates it quite well (as in Eample ) EXPLORATION Looking at the Symmetric Difference Quotient Analytically Let f() = an let h = 00 Fin f(0 + h) - f(0) h How close is it to f (0)? Fin f(0 + h) - f(0 - h) How close is it to f (0)? 3 Repeat this comparison for f() = 3 EXAMPLE 4 Graphing a Derivative Using NDER Let f() = ln Use NDER to graph y = f () Can you guess what function f () is by analyzing its graph? continue

5 Section 3 Differentiability 3 The graph is shown in Figure 37a The shape of the graph suggests, an the table of values in Figure 37b supports, the conjecture that this is the graph of y = > We will prove in Section 39 (using analytic methos) that this is inee the case Now Try Eercise X X = [, 4] by [, 3] Y (a) (b) Figure 37 (a) The graph of NDER (ln (), ) an (b) a table of values What graph coul this be? (Eample 4) Differentiability Implies Continuity We began this section with a look at the typical ways that a function coul fail to have a erivative at a point As one eample, we inicate graphically that a iscontinuity in the graph of f woul cause one or both of the one-sie erivatives to be noneistent It is actually not ifficult to give an analytic proof that continuity is an essential conition for the erivative to eist, so we inclue that as a theorem here THEOREM Differentiability Implies Continuity If f has a erivative at = a, then f is continuous at = a Proof Our task is to show that :a :a f() = f(a), or, equivalently, that [] = 0 Using the Limit Prouct Rule (an noting that - a is not zero), we can write [] = c( - a) :a :a - a = ( - a) # :a :a - a = 0 # f (a) = 0 The converse of Theorem is false, as we have alreay seen A continuous function might have a corner, a cusp, or a vertical tangent line, an hence not be ifferentiable at a given point Intermeiate Value Theorem for Derivatives Not every function can be a erivative A erivative must have the Intermeiate Value Property, as state in the following theorem (the proof of which can be foun in avance tets) THEOREM Intermeiate Value Theorem for Derivatives If a an b are any two points in an interval on which f is ifferentiable, then f takes on every value between f (a) an f (b) EXAMPLE 5 Applying Theorem Does any function have the Unit Step Function (see Figure 34) as its erivative? No Choose some a 6 0 an some b 7 0 Then U(a) = - an U(b) =, but U oes not take on any value between - an Now Try Eercise 37 The question of when a function is a erivative of some function is one of the central questions in all of calculus The answer, foun by Newton an Leibniz, woul revolutionize the worl of mathematics We will see what that answer is when we reach Chapter 6

6 4 Chapter 3 Derivatives Quick Review 3 (For help, go to Sections an ) Eercise numbers with a gray backgroun inicate problems that the authors have esigne to be solve without a calculator In Eercises 5, tell whether the it coul be use to efine f (a) (assuming that f is ifferentiable at a) f(a + h) - f(a) f(a + h) - f(h) 3 :a - a 4 f(a + h) + f(a - h) 5 :a f(a) - f() a - 6 Fin the omain of the function y = 4>3 7 Fin the omain of the function y = 3>4 8 Fin the range of the function y = ƒ - ƒ Fin the slope of the line y - 5 = 3( + p) 0 If f() = 5, fin f( ) - f(3-000) 000 Section 3 Eercises In Eercises 4, compare the right-han an left-han erivatives to show that the function is not ifferentiable at the point P Fin all points where f is not ifferentiable y y y f () y y f() y P(0, 0) y 3 y 4 y f() 0 P(, ) y y In Eercises 5 0, the graph of a function over a close interval D is given At what omain points oes the function appear to be (a) ifferentiable? (b) continuous but not ifferentiable? (c) neither continuous nor ifferentiable? 5 y f () y 6 y y f () D: 3 D: y y f () 8 D: y y 0 y P(, ) y f() P(, ) y y y f () D: y 0 y f() D: 0 y y f() D: In Eercises 6, the function fails to be ifferentiable at = 0 Tell whether the problem is a corner, a cusp, a vertical tangent, or a iscontinuity y = e tan-, Z 0 y = 4>5, = 0 3 y = y = y = 3 - ƒ ƒ - 6 y = 3 ƒ ƒ In Eercises 7 6, fin the numerical erivative of the given function at the inicate point Use h = 000 Is the function ifferentiable at the inicate point? 7 f() = 4 -, = 0 8 f() = 4 -, = 3 9 f() = 4 -, = 0 f() = 3-4, = 0 f() = 3-4, = - f() = 3-4, = 3 f() = >3, = 0 4 f() = ƒ - 3 ƒ, = 3 5 f() = >5, = 0 6 f() = 4>5, = 0 Group Activity In Eercises 7 30, use NDER to graph the erivative of the function If possible, ientify the erivative function by looking at the graph 7 y = -cos 8 y = y = ƒ ƒ 30 y = -ln ƒ cos ƒ In Eercises 3 36, fin all values of for which the function is ifferentiable f() = 3 h() = P() = sin ( ƒ ƒ ) - 34 Q() = 3 cos ( ƒ ƒ ) ( + ), 0 35 g() = c +, (4 - ), Ú 3

7 Section 3 Differentiability 5 36 C() = ƒ ƒ 37 Show that the function 0, f() = e, 0 is not the erivative of any function on the interval - 38 Writing to Learn Recall that the numerical erivative (NDER) can give meaningless values at points where a function is not ifferentiable In this eercise, we consier the numerical erivatives of the functions > an > at = 0 (a) Eplain why neither function is ifferentiable at = 0 (b) Fin NDER at = 0 for each function (c) By analyzing the efinition of the symmetric ifference quotient, eplain why NDER returns wrong responses that are so ifferent from each other for these two functions 39 Let f be the function efine as f() = e 3 -, 6 a + b, Ú where a an b are constants (a) If the function is continuous for all, what is the relationship between a an b? (b) Fin the unique values for a an b that will make f both continuous an ifferentiable Stanarize Test Questions You may use a graphing calculator to solve the following problems 40 True or False If f has a erivative at = a, then f is continuous at = a Justify your answer 4 True or False If f is continuous at = a, then f has a erivative at = a Justify your answer 4 Multiple Choice Which of the following is true about the graph of f() = 4>5 at = 0? (A) It has a corner (B) It has a cusp (C) It has a vertical tangent (D) It has a iscontinuity (E) f(0) oes not eist 43 Multiple Choice Let f() = 3 - At which of the following points is f (a) Z NDER (f(),, a)? (A) a = (B) a = - (C) a = (D) a = - (E) a = 0 In Eercises 44 an 45, let +, 0 f() = e +, Multiple Choice Which of the following is equal to the lefthan erivative of f at = 0? (A) (B) (C) 0 (D) - q (E) q 45 Multiple Choice Which of the following is equal to the righthan erivative of f at = 0? (A) (B) (C) 0 (D) - q (E) q Eplorations 46 (a) Enter the epression 6 0 into Y of your calculator using 6 from the TEST menu Graph Y in DOT MODE in the winow [-47, 47] by [-3, 3] (b) Describe the graph in part (a) (c) Enter the epression Ú 0 into Y of your calculator using Ú from the TEST menu Graph Y in DOT MODE in the winow [-47, 47] by [-3, 3] () Describe the graph in part (c) 47 Graphing Piecewise Functions on a Calculator Let f() = e, 0, 7 0 (a) Enter the epression (X )(X 0)+(X)(X70) into Y of your calculator an raw its graph in the winow [-47, 47] by [-3, 5] (b) Eplain why the values of Y an f() are the same (c) Enter the numerical erivative of Y into Y of your calculator an raw its graph in the same winow Turn off the graph of Y () Use TRACE to calculate NDER( Y,, -0), NDER( Y,, 0 ), an NDER( Y,, 0 ) Compare with Section 3, Eample 6 Etening the Ieas 48 Oscillation There is another way that a function might fail to be ifferentiable, an that is by oscillation Let f() = c sin, Z 0 0, = 0 (a) Show that f is continuous at = 0 (b) Show that f(0 + h) - f(0) = sin h h (c) Eplain why f(0 + h) - f(0) oes not eist () Does f have either a left-han or right-han erivative at = 0? (e) Now consier the function g() = c sin, Z 0 0, = 0 Use the efinition of the erivative to show that g is ifferentiable at = 0 an that g (0) = 0

x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)

x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x) Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane