Differentiation: Basic Concepts Capter 1. Te Derivative: Slope and Rates 2. Tecniques of Differentiation 3. Te Product and Quotient Rules 4. Marginal Analsis: Approimation b Increments 5. Te Cain Rule 6. Te Second Derivative 7. Implicit Differentiation and Related Rates Capter Summar and Review Problems
1 Te Derivative: Slope and Rates 98 Capter 2 Differentiation: Basic Concepts Calculus is te matematics of cange, and te primar tool for stuing rates of cange is a procedure called differentiation. In tis section, we describe tis procedure and sow ow it can be used in rate problems and to find te slope of a tangent line to a curve. A FALLING BODY PROBLEM As an illustration of te ideas we sall eplore, consider te motion of an object falling from a great eigt. In psics, it is sown tat after t seconds, te object will ave fallen s(t) 16t 2 feet. Suppose we wis to compute te velocit of te object after, sa, 2 seconds. Unless te falling object as a speedometer, it is ard to simpl read its velocit, but we can measure te distance it falls between time t 2 and time t 2 and compute te average velocit over te time period (2, 2 ) b te ratio v ave distance traveled elapsed time 16(2 )2 16(2) 2 64 162 64 16 s(2 ) s(2) 16(4 4 2 ) 16(4) If te elapsed time of seconds is small, we would epect te average velocit to be ver close to te instantaneous velocit at t 2. Tus, it is reasonable to compute te instantaneous velocit v ins b te limit Eplore! Te graping calculator can simulate secant lines approacing a tangent line. For a simple eample, store f() 2 2 into Y1 of te equation editor, selecting a bold graping stle. In Y2 write L1*X 2. Using te stat edit menu, input te values 3.6, 2.4, 1.4,.6, 0. Grap in sequential mode, using a window of size [ 4.7, 4.7]1 b [ 2.2, 12.2]1. Describe wat ou observe. Wat is te limiting tangent line? v ins fi 0 v ave fi 0 (64 6) 64 Tat is, after 2 seconds, te falling object is traveling at te rate of 64 feet per second. Te procedure we ave just described is illustrated geometricall in Figure 2.1. Figure 2.1a sows te grap of te distance function s 16t 2, along wit te points P(2, 64) and Q(2, 16(2 ) 2 ). Te line joining P and Q is called a secant line of te grap and as slope m sec 16(2 )2 64 (2 ) 2 64 16
Capter 2 Section 1 Te Derivative: Slope and Rates 99 (a) Te secant line troug P(2, 64) and Q(2, 16(2 ) 2 ). s Q(2 +, 16(2 + ) 2 ) s = 16t 2 P(2, 64) 2 2 + t (b) As 0 te secant line PQ tends toward te tangent line at P. s Secant lines Q Tangent line P 2 2 + t FIGURE 2.1 Te grap of s 16t 2. As indicated in Figure 2.1b, wen we take smaller and smaller, te corresponding secant lines PQ tend toward te position of wat we intuitivel tink of as te tangent line at P. Tis suggests tat we compute te slope of te tangent line m tan b finding te limiting value of te slopes of te approimating secant lines PQ; tat is, m tan fi 0 m sec fi 0 (64 16) 64 Tus, te slope of te tangent line to te grap of s(t) 16t 2 at te point were t 2 is eactl te same as te instantaneous rate of cange of s wit respect to t wen t 2. RATES OF CHANGE AND SLOPE Te procedure illustrated for te falling bo function s(t) 16t 2 applies to a variet of oter functions f(). In particular, te average rate of cange in f() over te interval (, ) is given b te ratio f( ) f()
100 Capter 2 Differentiation: Basic Concepts wic can be interpreted geometricall as te slope of te secant line troug te points P(, f()) and Q(, f( )) (Figure 2.2a). To find te instantaneous rate of cange of f() at, we compute te limit f( ) f() lim fi 0 wic also gives te slope of te tangent line to te grap of f() at te point P(, f()), as indicated in Figure 2.2b. (a) Te grap of f() wit a secant line troug points P(, f()) and Q(, f( )). P (, f()) Secant line Q ( +, f( + )) Tangent line (b) As 0 te secant lines tend toward te tangent line at P. Eplore! Store f() 2 into Y1 of te equation editor and grap using te window [ 4.7, 4.7]1 b [ 2.2, 12.2]1. Access te tangent line option of te DRAW ke (second PRGM) and use te rigt arrow to trace te cursor to te point (2, 4) on our grap. Press ENTER and observe wat appens. Were does tis line intercept te ais? (, f()) P Tangent FIGURE 2.2 Secant lines approimating a tangent line. Secants Here is an eample from economics illustrating te relationsip between rate of cange and slope. EXAMPLE 1.1 Te grap sown in Figure 2.3 gives te relationsip between te percentage of unemploment U and te corresponding percentage of inflation I. Use te grap to estimate
Capter 2 Section 1 Te Derivative: Slope and Rates 101 te rate at wic I canges wit respect to U wen te level of unemploment is 3% and again wen it is 10%. Solution From te figure, we estimate te slope of te tangent line at te point (3, 15), corresponding to U 3, to be approimatel 14. Tat is, wen unemploment is 3%, inflation I is decreasing at te rate of 14 percentage points for eac percentage point increase in unemploment U. 50 40 30 Inflation, % 20 10 7 7 Slope = = 14 0.5 0 0.5 1 Slope = 2.5 = 0.4 Tangent lines 2.5 1 10 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Unemploment, % FIGURE 2.3 Inflation as a function of unemploment. Source: Adapted from Robert Eisner, Te Misunderstood Econom: Wat Counts and How to Count It, Boston, MA: Harvard Business Scool Press, 1994, page 173. At te point (10, 5), te slope of te tangent line is approimatel 0.4, wic means tat wen tere is 10% unemploment, inflation is decreasing at te rate of onl 0.4 percentage point for eac percentage point increase in unemploment.
102 Capter 2 Differentiation: Basic Concepts THE DERIVATIVE Te epression f( ) f() tat appears in bot slope and rate of cange computation is called a difference quotient of te function f. Specificall, in bot applications, we compute te limit of a difference quotient as approaces 0. To unif te stu of tese and oter similar applications, we introduce te following terminolog and notation. Te Derivative of a Function Te derivative of te function f() wit respect to is te function f () (read as f prime of ) given b f( ) f() f () fi 0 and te process of computing te derivative is called differentiation. We sa tat f() is differentiable at c if f (c) eists (tat is, if te limit of te difference quotient eists wen c). Te advantage of te derivative notation is tat te observations made earlier in tis section about slope and rates of cange can be summarized in te following compact form. Slope as a Derivative Te slope of te tangent line to te curve f() at te point (c, f(c)) is given b m tan f (c). Instantaneous Rate of Cange as a Derivative Te quantit f() canges at te rate f (c) wit respect to wen c. In te first of te following two eamples, we find te equation of a tangent line. Ten in te second, we consider a business application involving rates. EXAMPLE 1.2 First compute te derivative of f() 3, and ten use it to find te slope of te tangent line to te curve 3 at te point were 1. Wat is te equation of te tangent line at tis point? Solution According to te definition of te derivative
Capter 2 Section 1 Te Derivative: Slope and Rates 103 = 3 f( ) f() f () fi 0 3 2 ( ) 3 3 fi 0 ( 3 3 2 3 2 3 ) 3 (3 2 3 2 ) fi 0 fi 0 ( 1, 1) Tus, te slope of te tangent line to te curve 3 at te point were 1 is f ( 1) 3( 1) 2 3 (Figure 2.4). To find an equation for te tangent line, we also need te coordinate of te point of tangenc; namel, ( 1) 3 1. Terefore, te tangent line passes troug te point ( 1, 1) wit slope 3. B appling te point-slope formula, we get FIGURE 2.4 Te grap of 3. or ( 1) 3[ ( 1)] 3 2 EXAMPLE 1.3 A manufacturer estimates tat wen units of a certain commodit are produced and sold, te revenue derived will be R() 0.5 2 3 2 tousand dollars. At wat rate is te revenue canging wit respect to te level of production wen 3 units are being produced? Is te revenue increasing or decreasing at tis time? Solution First, since represents te number of units produced, we must ave 0. Te difference quotient of R() is R (tousands of dollars) R( ) R() [0.5(2 2 2 ) 3( ) 2] [0.5 2 3 2] 0.52 3 0.5 3 Tus, te derivative of R() is 3 (units produced) and since R( ) R() R () ( 0.5 3) 3 fi 0 fi 0 R (3) (3) 3 6 FIGURE 2.5 Te grap of R() 0.5 2 3 2, for 0. it follows tat revenue is canging at te rate of $6,000 per unit wit respect to te level of production wen 3 units are being produced.
104 Capter 2 Differentiation: Basic Concepts Since R (3) 6 is positive, te tangent line at te point on te grap of te revenue function were 3 must be sloped upward. Tis observation suggests tat revenue is increasing wen 3, as confirmed b te grap of R() sown in Figure 2.5. DERIVATIVE NOTATION Eplore! Man graping calculators ave a special utilit for computing derivatives numericall, called te numerical derivative (nderiv). It can be accessed via te MATH ke. Tis derivative can also be accessed troug te CALC (second TRACE) ke, especiall if a grapical presentation is desired. For instance, store f() into Y1 of te equation editor and displa its grap using a decimal window. Use te option of te CALC ke d and observe te numerical derivative value at 1. Te derivative f () of f() is sometimes written as (read as dee, dee ), d and in tis notation, te value of te derivative at c (tat is, f (c)) is written as For eample, if 2, ten and te value of tis derivative at 3 is Te notation for derivative suggests slope,, and can also be tougt of as d te rate of cange of wit respect to. Sometimes it is convenient to condense a statement suc as b writing simpl 2 d d 3 2 3 wen 2, ten 2 d d d (2 ) 2 2( 3) 6 wic reads, te derivative of 2 wit respect to is 2. d c Te following eample illustrates ow te different notational forms for te derivative can be used. EXAMPLE 1.4 First compute te derivative of f(), ten use it to (a) Find te equation of te tangent line to te curve 4. at te point were
Capter 2 Section 1 Te Derivative: Slope and Rates 105 (b) Find te rate at wic is canging wit respect to wen 1. Solution Te derivative of wit respect to is given b (a) Wen 4, te corresponding coordinate on te grap of f() is 1 4 2, so te point of tangenc is P(4, 2). Since f (), te slope of te 2 tangent line to te grap of f() at te point P(4, 2) is given b and b substituting into te point-slope formula, we find tat te equation of te tangent line at P is or d f( ) f() d fi 0 fi 0 fi 0 ( )( ) ( ) fi 0 ( ) fi 0 fi 0 1 1 2 2 1 ( 4) 4 (b) Te rate of cange of wen 1 is f (4) 1 2 4 1 4 d 1 1 4 1 1 2 1 1 2 ( ) DIFFERENTIABILITY AND CONTINUITY If a function f() is differentiable at te point P( 0, f( 0 )), ten te grap of f() as a nonvertical tangent line at P and at all points near P. Intuitivel, tis suggests tat a function must be continuous at an point were it is differentiable, since a grap cannot ave a ole or gap at an point were a well-defined tangent can be drawn.
106 Capter 2 Differentiation: Basic Concepts (a) (b) (c) = 0 = 1 0 0 = 2/3 Eplore! Store f() abs(x) into Y1 of te equation editor. Te absolute value function can be obtained troug te MATH ke b accessing te NUM menu. Use a decimal window and compute te numerical derivative at d 0. Wat do ou observe and ow does tis answer reconcile wit Figure 2.6(b)? Te curve is too sarp at te point (0, 0) to possess a well-defined tangent line tere. Hence, te derivative of f() abs() does not eist at 0. Note tat te numerical derivative must be used wit caution at cusps and unusual points. Tr computing te 1 numerical derivative of at 0 and eplaining ow suc a result could occur numericall. FIGURE 2.6 Tree functions tat are not differentiable at (0, 0). (a) Te grap as a gap at 0. (b) Tere is a sarp corner at (0, 0). (c) Tere is a cusp at (0, 0). Te converse, owever, is not true; tat is, a continuous function need not be everwere differentiable. For instance, consider te tree graps sown in Figure 2.6. Te first as a gap at 0, and certainl as no tangent tere. Te graps in Figures 2.6b and 2.6c are continuous at 0. In bot cases, owever, tere are sarp points at (0, 0) (a corner in Figure 2.6b and a cusp in Figure 2.6c), wic prevent te construction of a well-defined tangent line tere. In general, te functions ou encounter in tis tet will be differentiable at almost all points. In particular, polnomials are everwere differentiable and rational functions are differentiable werever te are defined. P. R. O. B. L. E. M. S 2.1 P. R. O. B. L. E. M. S 2.1 In Problems 1 troug 8, compute te derivative of te given function and find te slope of te line tat is tangent to its grap for te specified value of te independent variable. 1. f() 5 3; 2 2. f() 2 1; 1 3. f() 2 2 3 5; 0 4. f() 3 1; 2 2 1 5. g(t) 6. f() ; 2 t ; t 1 2 2 1 7. f() ; 9 8. (u) ; u 4 u In Problems 9 troug 12, compute te derivative of te given function and find te equation of te line tat is tangent to its grap for te specified value of 0. 9. f() 2 1; 0 2 10. f() 3 ; 0 2 3 11. f() 2 ; 0 1 2 12. f() 2 ; 0 4
Capter 2 Section 1 Te Derivative: Slope and Rates 107 In Problems 13 troug 16, find te rate of cange were 0. d 13. 3; 0 2 14. 6 2; 0 3 15. (1 ); 0 1 16. 1 ; 0 3 17. Suppose f() 3. (a) Compute te slope of te secant line joining te points on te grap of f wose coordinates are 1 and 1.1. (b) Use calculus to compute te slope of te line tat is tangent to te grap wen 1 and compare tis slope wit our answer in part (a). 18. Suppose f() 2. (a) Compute te slope of te secant line joining te points on te grap of f wose coordinates are 2 and 1.9. (b) Use calculus to compute te slope of te line tat is tangent to te grap wen 2 and compare tis slope wit our answer in part (a). P 2 15 PROBLEM 21 MAXIMIZATION OF PROFIT ANIMAL BEHAVIOR In Problems 19 and 20, sketc te grap of te function f(). Determine te values of for wic te derivative is zero. Wat appens to te grap at te corresponding points? 19. f() 3 3 2 20. f() 3 2 21. In Eample 4.5 of Capter 1, we obtained te profit function P() 400(15 ) ( 2) for te production of ig-grade blank videocassettes. Te grap of P() is te downward opening parabola sown in te accompaning figure. (a) Find P (). (b) Find were P () 0. Tis is were te grap of te profit function as a orizontal tangent. Wat can be said about te profit at te corresponding value of? 22. Sketc te grap of te function 2 3 and use calculus to find its lowest point. 23. Sketc te grap of te function 1 2 and use calculus to find its igest point. 24. A manufacturer can produce tape recorders at a cost of $20 apiece. It is estimated tat if te tape recorders are sold for dollars apiece, consumers will bu 120 of tem eac mont. Use calculus to determine te price at wic te manufacturer s profit will be te greatest. 25. Eperiments indicate tat wen a flea jumps, its eigt (in meters) after t seconds is given b te function H(t) (4.4)t (4.9)t 2
108 Capter 2 Differentiation: Basic Concepts RENEWABLE RESOURCES Volume of lumber V (units) 60 50 40 30 20 10 0 10 20 30 40 50 60 Time (ears) PROBLEM 26 Grap sowing ow te volume of lumber V in a tree varies wit time t. Source: Adapted from Robert H. Frank, Microeconomics and Beavior, 2nd ed., New York, NY: McGraw-Hill, Inc., 1994, page 623. t Using calculus, determine te time at wic te flea will be at te top of its jump. Wat is te maimum eigt reaced b te flea? 26. Te accompaning grap sows ow te volume of lumber V in a tree varies wit time t (te age of te tree). Use te grap to estimate te rate at wic V is canging wit respect to time wen t 30 ears. Wat seems to be appening to te rate of cange of V as t increases witout bound (tat is, in te long run )? 27. (a) Find te derivative of te linear function f() 3 2. (b) Find te equation of te tangent line to te grap of tis function at te point were 1. (c) Eplain ow te answers to parts (a) and (b) could ave been obtained from geometric considerations wit no calculation watsoever. 28. (a) Find te derivatives of te functions 2 and 2 3 and account geometricall for teir similarit. (b) Witout furter computation, find te derivative of te function 2 5. 29. (a) Find te derivative of te function 2 3. (b) Find te derivatives of te functions 2 and 3 separatel. (c) How is te derivative in part (a) related to tose in part (b)? (d) In general, if f() g() (), wat would ou guess is te relationsip between te derivative of f and tose of g and? 30. (a) Compute te derivatives of te functions 2 and 3. (b) Eamine our answers in part (a). Can ou detect a pattern? Wat do ou tink is te derivative of 4? How about te derivative of 27? 31. Eplain w te grap of a function f() is rising over an interval a b if f () 0 trougout te interval. Wat can ou sa about te grap if f () 0 trougout te interval a b? In Problems 32 and 33, sketc te grap of a function f tat as all of te given properties. You ma need to refer to te result of Problem 31. 32. (a) f () 0 wen 1 and wen 5 (b) f () 0 wen 1 5 (c) f (1) 0 and f (5) 0 33. (a) f () 0 wen 2 and wen 2 3 (b) f () 0 wen 3 (c) f ( 2) 0 and f (3) 0
Capter 2 Section 2 Tecniques of Differentiation 109 UNEMPLOYMENT 34. In economics, te grap in Figure 2.3 is called te Pillips curve, after A. W. Pillips, a New Zealander associated wit te London Scool of Economics. Until Pillips publised is ideas in te 1950s, man economists believed tat unemploment and inflation were linearl related. Read an article on te Pillips curve (te source cited wit Eample 1.1 would be a good place to start) and write a paragrap on te nature of unemploment in te U.S. econom. 35. Find te slope of te line tat is tangent to te grap of te function f() 2 2 3 at te point were 3.85 b filling in te following cart. Record all calculations using five decimal places. 0.02 0.01 0.001 0 0.001 0.01 0.02 f() f( ) f( ) f() 2 Tecniques of Differentiation 36. Find te values at wic te peaks and valles of te grap of 2 3 0.8 2 4 occur. Use four decimal places. 2 1 37. Sow tat f() is not differentiable at 1. 1 If we ad to use te limit definition ever time we wanted to compute a derivative, it would be bot tedious and difficult to use calculus in applications. Fortunatel, tis is not necessar, and in tis section and te net, we develop tecniques tat greatl simplif te process of differentiation. We begin wit a rule for te derivative of a constant. = c Slope 0 Te Constant Rule For an constant c, d (c) 0 d Tat is, te derivative of a constant is zero. FIGURE 2.7 Te grap of f() c. You can see tis b considering te grap of a constant function f() c, wic is a orizontal line (see Figure 2.7). Since te slope of suc a line is 0 at all its points, it follows tat f () 0. Here is a proof using te limit definition: