4.2 The Derivative. f(x + h) f(x) lim

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1 4.2 Te Derivative Introduction In te previous section, it was sown tat if a function f as a nonvertical tangent line at a point (x, f(x)), ten its slope is given by te it f(x + ) f(x). (*) Tis is potentially very powerful information about te function f. For example, places were a tangent line as slope 0 often correspond to maximum or minimum values of a function. Also, te slope of te tangent line at (x, f(x)) tells ow te function values f(x) are canging at te instant one is passing troug te point (x, f(x)): weter te grap is rising or falling, and ow quickly. Because of te importance of tis slope information, te it ( ), wen it exists, is given a special name: it is called te derivative of f at x, and denoted by f (x) (read f prime of x ). Tis is summarized below. DEFINITIONS differentiable at x; te derivative of f at x; differentiation For a given function f and x D(f), if te it f(x + ) f(x) exists, ten one says tat f is differentiable at x, and writes f (x) := 0 f(x + ) f(x) Te number f (x) is called te derivative of f at x. Te process of finding f (x) is called differentiation.. 193

2 194 copyrigt Dr. Carol JV Fiser Burns ttp:// DEFINITION f, te derivative function Given a function f, we now ave a way to construct a new function, named f. Tis function f is called te derivative of f. Te domain of f is te set of all x D(f) for wic te it f(x + ) f(x) exists. If f is differentiable at every point in its domain, ten D(f ) = D(f). However, f may not be differentiable at every point in its domain. So, te domain of f may be smaller tan te domain of f. In general, all tat can be said is tat D(f ) D(f). Te derivative f takes an input x, and gives as an output te slope of te tangent line to te grap of f at te point (x, f(x)). differentiating f(x) = 2x + 1 Let f be defined by f(x) = 2x + 1. Te grap of f is a line L wit slope 2. At any point on tis line, te tangent line is te line L itself. So, at every point, te slope of te tangent line is 2. Tus, te function f as te same domain as f, and is defined by f (x) = 2. One usually abbreviates te problem as follows: PROBLEM: Differentiate f(x) = 2x + 1. SOLUTION: f (x) = 2. For example, f (0) = 2, f (π) = 2, and f ( ) = 2. differentiating a function tat is defined piecewise PROBLEM: Differentiate { 1 for x 0 f(x) = 1 for x > 0. SOLUTION: For all positive and negative x, tangent lines exist and ave slope zero. However, f is not differentiable at 0. Intuitively, tis is clear; tere is no obvious way to draw a tangent line at te point (0, 1). Tis conclusion is confirmed by investigating te rigt-and it at x = 0: f(0 + ) f(0) 1 ( 1) 2 = =, wic does not exist. Tus, te two-sided it does not exist. Summarizing, { 0 for x 0 f (x) = not defined for x = 0. For example, f (.1) = 0, f (.001) = 0, and f (0) does not exist. In tis example, D(f) = R, and D(f ) = {x x 0}. Using interval notation, one can alternately write D(f ) = (, 0) (0, ). Unfortunately, bot of tese are pretty long expressions for te domain of f. Tere is a simpler expression, tat makes use of set subtraction, discussed next.

3 copyrigt Dr. Carol JV Fiser Burns ttp:// 195 DEFINITION set subtraction Let A and B be sets. Define a new set, denoted by A B, and read as A minus B, by A B := {x x A and x / B}. Tus, te set A B consists of all te elements of A tat are not elements of B. (In oter words, take A and subtract off any elements of B.) set subtraction For example, if ten A = R and B = [1, 2), A B = (, 1) [2, ) and B A =. If A = {1, 2, 3} and B = {3, 4, 5} ten A B = {1, 2} and B A = {4, 5}. Tis notation gives an easier way to describe te domain of f in te previous example: D(f ) = R {0}. EXERCISE 1 1. Wy is it incorrect to say D(f ) = R 0? For eac of te following sets A and B, find bot A B and B A. Be sure to answer using complete matematical sentences. 2. A = R, B = (, 2] 3. A = ( 3, 3], B = [ 1, 4) 4. A = R, B is te set of irrational numbers Find sets A and B so tat S = A B. (Tere is not a unique correct answer.) 5. S = ( 1, 0) (0, 1] 6. S = {1, 2, 3}

4 196 copyrigt Dr. Carol JV Fiser Burns ttp:// differentiating f(x) = x ; a function tat is continuous at a point, but not differentiable tere Consider te function f(x) = x. For x > 0, f(x) = x = x is certainly differentiable wit derivative f (x) = 1. For x < 0, f(x) = x = x is also differentiable wit derivative f (x) = 1. Wen x = 0, tere is no tangent line. In tis case, bot one-sided its exist, but do not agree: f(0 + ) f(0) 0 = = = 1 and = 0 0 = 0 = 1. Since te one-sided its do not agree, te two-sided it does not exist, and f is not differentiable at x = 0. Summarizing, EXERCISE 2 Let f(x) = 3x 1. 1 for x > 0 f (x) = not defined for x = 0 1 for x < 0. Tus, for example, f (1.7) = 1, f ( 4/3) = 1, and f (0) does not exist. 1. Grap f. Wat is D(f)? 2. Wat is te function f? In particular, wat is D(f )? Now, let f(x) = x Give a piecewise description for f. 4. Grap f. Wat is D(f)? 5. For x > 3, wat is f (x)? 6. For x < 3, wat is f (x)? 7. Sow tat f is not defined at x = 3, by investigating te its f(3 + ) f(3) and f(3 + ) f(3) Write down a piecewise description of f. 9. Grap f.

5 copyrigt Dr. Carol JV Fiser Burns ttp:// 197 a patced togeter function tat IS differentiable at te patcing point PROBLEM: Is te function { x 2 x 1 f(x) = 2x 1 x < 1 differentiable at x = 1? SOLUTION: Note tat f(1) = 1 2 = 1. Investigate bot one-sided its: f(1 + ) f(1) (1 + ) 2 1 = = (2 + ) = = 2 and f(1 + ) f(1) (2(1 + ) 1) 1 2 = = = 2. Since bot one-sided its agree, f(1 + ) f(1) exists and equals 2. Tus, f is differentiable at 1, and f (1) = 2. Tat is, te tangent line to te grap of f at te point (1, 1) as slope 2. EXERCISE 3 Let f be defined by 1. Grap f. Wat is D(f)? { x 2 for x 1 f(x) = 2x 1 for x > Does f (x) exist for x < 1? If so, wat is it? 3. Does te it x 1 f (x) exist? If so, wat is it? 4. Does f (x) exist for x > 1? If so, wat is it? 5. Investigate two appropriate one-sided its to decide if f ( 1) exists. If it does, wat is it? 6. Is tere a tangent line to te grap of f at te point wit x-value 1? If so, wat is its slope? 7. Grap f. Wat is D(f )?

6 198 copyrigt Dr. Carol JV Fiser Burns ttp:// a function tat is differentiable at an endpoint of its domain PROBLEM: Consider te function f : [1, 2] R, f(x) = 1 x. Is f differentiable at x = 1? SOLUTION: Note tat f(1) = 1 1 = 1. For f to be differentiable at x = 1, te it f(1 + ) 1 must exist. Does it? Remember tat to investigate tis it, one only considers values of tat are close to 0 and in te domain of te function f(1+) 1. Wen is in te domain of f(1+) 1? Only wen 1+ D(f). And for tis function, 1 + D(f) only wen > 0. Here, te two-sided it is identical to te rigt-and it. Wen a function is only defined on one side of a point, te two-sided it is actually just a one-sided it. Tus, one as f(1 + ) f(1) = 0 + = (line 1) (line 2) 1 1 = (1 + ) (line 3) 1 = = (line 4) Tus, f is differentiable at x = 1, and f (1) = 1. EXERCISE 4 1. Give a reason (or reasons) for eac line of te preceding display. Te lines are numbered for easy reference. Consider te function f : [0, 4] R given by f(x) = (x 1) Grap f. 3. Is f differentiable at x = 0? Tat is, does te it f(0 + ) f(0) exist? Justify your answer. Be sure to write complete matematical sentences.

7 copyrigt Dr. Carol JV Fiser Burns ttp:// 199 a function tat is not differentiable at an endpoint of its domain Let f(x) = x. For all x > 0 one as f(x + ) f(x) x + x = x + x = = 0 (x + ) x ( x + + x) x + + x x + + x (line 2) (line 1) (line 3) 1 = (line 4) 0 x + + x = 1 2. (line 5) x Tus, at eac point (x, x) for x > 0, te tangent line exists and as slope 1 2 x. EXERCISE 5 Give a reason (or reasons) for eac line in te display above. Te lines are numbered for easy reference. Now tink about wat appens wen x = 0. In tis case, te it is actually a rigt-and it, and reduces to For > 0, we can write = ( ) 2, so tat = 0 + = ( ) = But as 0 +, 1 does not approac a specific real number. It gets arbitrarily large. Tere is a vertical tangent line at te point (0, 0), and a vertical line as no slope. So f is not differentiable at 0. Caution! no slope versus zero slope Every orizontal line as zero slope. Coosing any two points on te line, and traveling from one point to te oter via te rule rise, ten run yields rise run = 0 some nonzero number = 0. Every vertical line as no slope; tat is, te slope is undefined. For if any two points are cosen on te line, computation of te slope yields rise some nonzero number = run 0 and division by zero is undefined. Tus, no slope and zero slope ave entirely different meanings. Tis can be confusing, because in Englis, te words no and zero are often used as synonyms.,

8 200 copyrigt Dr. Carol JV Fiser Burns ttp:// reading info from a grap Consider te function f wose grap is sown below: Read te following information from te grap, if possible. If a quantity does not exist, so state. f( 1), f ( 1), f ( 1.5), f (1.5), f (2), f(4), f (6), f (7) SOLUTION: f( 1) = 1 f ( 1) does not exist f ( 1.5) = 0 f (1.5) = 2 (Use te known points (1, 1) and (2, 1) to compute te slope.) f (2) does not exist f(4) = 2 f (6) = 0 f (7) > 0; one migt estimate tat f (7) 1 Now, answer te following questions about f: Wat is D(f)? Wat is R(f)? Were is f continuous? Were is f differentiable? Wat is {x f(x) > 0}? Wat is {x f(x) [ 1, 1]}? Wat is {x f(x) = 1}?

9 copyrigt Dr. Carol JV Fiser Burns ttp:// 201 SOLUTION: For all tese answers, te assumption is made tat te patterns indicated at te four borders of te grap continue. D(f) = R R(f) = R f is continuous at all x in te set (, 1) ( 1, 4) (4, ). A simpler notation for tis set is R { 1, 4}. f is differentiable at all x in te set R { 1, 1, 2, 4}. Some approximation is necessary ere. {x f(x) > 0} = (, 1] (1.5, 3.5) [4, 10) Some approximation is necessary ere. {x f(x) [ 1, 1]} = (, 3.8) {6} (9.5, 10.5) Some approximation is necessary ere. {x f(x) = 1} = (, 1] {2, 6, 9.5} EXERCISE 6 Consider te function f wose grap is sown below. Read te following information from te grap, if possible. Approximate, wen necessary. If a quantity does not exist, so state. Be sure to write complete matematical sentences. 1. f(0), f(1), f (1), f (2), f (1.34), f(3), f(4), f (π), f (1000) 2. Wat is D(f)? 3. Wat is D(f )? 4. Wat is R(f)? 5. Were is f continuous? Classify any discontinuities. 6. Wat is {x f(x) 0}? 7. Wat is {x f (x) < 0}?

10 202 copyrigt Dr. Carol JV Fiser Burns ttp:// reconstructing a function from its derivative Suppose tat a function f as derivative f wose grap is sown below. Wat, if anyting, can be said about te grap of f? SOLUTION: For x < 0, te tangent lines to te grap of f must all ave slope 1. For 0 < x < 1, te tangent lines to te grap of f must all ave slope 0. For x > 1, te tangent lines to te grap of f must all ave slope 1. Tere is not a unique function f tat satisfies tese requirements. For example, any of te following graps would produce te specified derivative: EXERCISE 7 Suppose tat a function f as derivative f wose grap is sown below: 1. Wat, if anyting, can be said about te grap of f? 2. Grap tree different functions f tat could ave te specified derivative.

11 copyrigt Dr. Carol JV Fiser Burns ttp:// 203 QUICK QUIZ sample questions 1. Give a precise definition of f (x). 2. Wat is te difference between f and f (x)? 3. If A = [0, 4) and B = {0, 2, 4}, find A B and B A. 4. For te function given below, find and grap f. 5 TRUE or FALSE: If f is differentiable at x, ten f is defined at x. KEYWORDS for tis section Differentiable at x, f (x) is te derivative of f at x, differentiation, f is te derivative function, finding derivatives using te definition, set subtraction. END-OF-SECTION EXERCISES For eac function f listed below, do te following: Grap f. Wat is D(f)? Find f. Wen necessary, use te definition of derivative. Grap f. Wat is D(f )? 1. f(x) = x 2 { 2 for 3 < x 0 2. f(x) = 1 x for 0 < x < 4 { x 2 for x 1 3. f(x) = 2x for x > 1 Use te definition of te derivative to find f (c) for eac function f and number c D(f). 4. f(x) = 3x 2 1, c = 2 5. f(x) = 1 x 1, c = 2 6. f(x) = x + 1, c = 4 Find te equation of te tangent line to te grap of te function f at te specified point. Feel free to use any earlier results. Te point-slope form may be useful: remember tat y y 1 = m(x x 1 ) is te equation of te line tat as slope m and passes troug te point (x 1, y 1 ). 7. f(x) = x 2, c = 3 8. f(x) = x, c = 0 9. f(x) = (x + 2) 2 + 1, c = 2