(Section.0: Difference Quotients).0. SECTION.0: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES Define average rate of cange (and average velocity) algebraically and grapically. Be able to identify, construct, and evaluate (and simplify) various forms of difference quotients. PART A: DISCUSSION Tis section will revisit Section 0.4 on slopes of lines, Section. on function evaluations, and Section.2 on graps. A difference quotient is used to find te slope of a secant line to a grap or to find an average rate of cange (peraps an average velocity). Difference quotients will form te basis for derivatives, tangent lines, and instantaneous rates of cange in Section.. PART B: SECANT LINES and AVERAGE RATE OF CHANGE Te secant line to te grap of a function f on te interval a, b, were a < b, is te line tat passes troug te points ( a, f ( a) ) and ( b, f ( b) ). Te average rate of cange of f on a, b is equal to te slope of tis secant line, wic is given by: rise run f ( b ) f ( a). b a We call tis a difference quotient, because it as te form: (See Footnote on assumptions about f.) difference of outputs difference of inputs.
(Section.0: Difference Quotients).0.2 PART C: AVERAGE VELOCITY Te following development of average velocity will elp explain te association between slope and average rate of cange. Example (Average Velocity) A car is driven due nort 00 miles during a two-our trip. Wat is te average velocity of te car? Let t te time (in ours) elapsed since te beginning of te trip. Let y st (), were s is te position function for te car (in miles). s gives te signed distance of te car from te starting position. Te position (s) values would be negative if te car were sout of te starting position. Let s( 0) 0, meaning tat y 0 corresponds to te starting position. Terefore, s( 2) 00 (miles). Te average velocity on te time-interval a, b is te average rate of cange of position wit respect to time. Tat is, cange in position cange in time s t were (uppercase delta) denotes cange in sb sa, a difference quotient b a Here, te average velocity on 0, 2 is: s( 2) s( 0) 00 0 2 0 2 50 miles our or mi r or mp TIP : Te unit of velocity is te unit of slope given by: unit of s unit of t.
(Section.0: Difference Quotients).0.3 Te average velocity is 50 mp on 0, 2 in te tree scenarios below. It is te slope of te orange secant line. We will define instantaneous velocity (or simply velocity) in Section.. Here, te velocity is constant (50 mp). Here, te velocity is increasing; te car is accelerating. Here, te car oversoots te destination and ten backtracks. WARNING : Te car s velocity is negative in value wen it is backtracking; tis appens wen te grap falls. In calculus, te Mean Value Teorem for Derivatives will imply tat te car must be going exactly 50 mp at some time value t in ( 0, 2). Te teorem applies in all tree scenarios above, because s is continuous on 0, 2 and is differentiable on ( 0, 2), meaning tat its grap makes no sarp turns and does not exibit infinite steepness on ( 0, 2). Differentiability will be discussed in Section..
(Section.0: Difference Quotients).0.4 PART D: FORMS OF DIFFERENCE QUOTIENTS Te purposes of tese forms will be discussed in Section. on derivatives. Forms of Difference Quotients Form : Fixed interval f ( b) f ( a) b a a, b are constants Form 2: Variable endpoint (x) f ( x) f ( a) x a a is constant; x is variable Form 3: Variable run () f ( a+ ) f ( a) a is constant; is variable Form 4: Variable endpoint (x) and Variable run () f ( x+ ) f ( x) x, are variable Te denominators can be negative. (See Footnote 2 in Section..)
PART E: EVALUATING DIFFERENCE QUOTIENTS Example 2 (Form of a Difference Quotient; Profit) Solution (Section.0: Difference Quotients).0.5 A company sells widgets. Assume tat all widgets produced are sold. Let P be te profit function for te company; P( x) is te profit (in dollars) if x widgets are produced and sold. Our model: P( x) x 2 + 200x 5000. Find te average rate of cange of profit between 60 and 90 widgets. WARNING 2: We will treat te domain of P as 0, ), even toug one could argue tat te domain sould only consist of integers. Be aware of tis issue wit applications suc as tese. P( 90) P 60 90 60 ( 90) 2 + 200 90 5000 60 30 2 + 200 60 5000 WARNING 3: Grouping symbols are essential wen expanding P 60 ere, since we are subtracting an expression wit more tan one term. 800 + 8,000 5000 3600 + 2,000 5000 30 800 + 8,000 5000 + 3600 2,000 + 5000 30 and TIP 2: Instead of completely evaluating P 90 P( 60), it may elp to take advantage of cancellations suc as te one above. 500 30 50 dollars widget
(Section.0: Difference Quotients).0.6 Te practical interpretation is tat, if te company increases its production level from 60 widgets to 90 widgets, ten its profit will increase by 50 dollars, on average. widget Te total increase in profit is $500 over te 30 additional widgets. Te grap of y P( x) is below. Te slope of te orange secant line is also 50 dollars widget. Te fact tat te slope is positive reflects te fact tat te profit function is increasing on 60, 90. We will revisit tis idea in Section. on derivatives.
(Section.0: Difference Quotients).0.7 Example 3 (Form 2 of a Difference Quotient; Revisiting Example 2 on Profit) Solution Again, P( x) x 2 + 200x 5000. Evaluate P ( x ) P 60 x 60 P( x) P( 60) x 60 x 2 + 200x 5000 x2 + 200x 5000 60 x 60 2 + 200 60. (Form 2) 5000 3600 + 2,000 5000 x 60 x2 + 200x 5000 + 3600 2,000 + 5000 x 60 x2 + 200x 8400 x 60 ( x2 200x + 8400) x 60 Te Factor Teorem in Capter 2 will imply tat, if P is polynomial, ten ( x 60) is a factor of te numerator, wic is equivalent to P( x) P( 60). Tis is because 60 is a zero of te numerator: P( 60) P( 60) 0. ( x 40) ( x 60) x 60 ( x 40), x 60 40 x, x 60, ( x 60)
(Section.0: Difference Quotients).0.8 Te unit for te difference quotient is still dollars, toug we tend to omit widget it wen a variable is present in our final expression. If we substitute x 90, we obtain 50 dollars widget, our answer to Example 2.
(Section.0: Difference Quotients).0.9 Example 4 (Form 3 of a Difference Quotient; Revisiting Example 2 on Profit) Again, P( x) x 2 + 200x 5000. Evaluate P ( 60 + ) P( 60). (Form 3) Solution P( 60 + ) P( 60) ( 60 + ) 2 + 200( 60 + ) 5000 60 ( 3600 + 20 + 2 )+ 2,000 + 200 5000 2 + 200 60 5000 3600 + 2,000 5000 3600 20 2 + 2,000 + 200 5000 + 3600 2,000 + 5000 80 2 80 80, 0, ( 0) If we substitute 30, wic corresponds to x 90, we obtain dollars 50 widget, our answer to Example 2.
(Section.0: Difference Quotients).0.0 Example 5 (Form 4 of a Difference Quotient; Revisiting Example 2 on Profit) Again, P( x) x 2 + 200x 5000. Evaluate P ( x+ ) P( x). (Form 4) Solution P( x+ ) P( x) ( x + ) 2 + 200( x + ) 5000 x2 + 200x 5000 ( x2 + 2x + 2 )+ 200x + 200 5000 x2 + 200x 5000 x2 2x 2 + 200x + 200 5000 x2 + 200x 5000 x2 2x 2 + 200x + 200 5000 + x 2 200x + 5000 2x 2 + 200 2x + 200 2x + 200, 0, ( 0) If we substitute x 60 and 30 (wic ten corresponds to x 90 ), dollars we obtain 50 widget, our answer to Example 2.
(Section.0: Difference Quotients).0. FOOTNOTES. Assumptions made about a function. Wen defining te average rate of cange of a function f on an interval a, b, were a < b, sources typically do not state te assumptions made about f. Te formula f ( b ) f ( a) seems only to require te existence of f ( a) and b a f ( b), but we typically assume more tan just tat. Altoug te slope of te secant line on a, b can still be defined, we need more for te existence of derivatives (i.e., te differentiability of f ) and te existence of non-vertical tangent lines, as we will see in Section.. We ordinarily assume tat f is continuous on a, b. Ten, tere are no oles, jumps, or vertical asymptotes on a, b wen f is graped. (See Section.5.) We may also assume tat f is differentiable on a, b. Ten, te grap of f makes no sarp turns and does not exibit infinite steepness (corresponding to vertical tangent lines). However, tis assumption may lead to circular reasoning, because te ideas of secant lines and average rate of cange are used to develop te ideas of derivatives, tangent lines, and instantaneous rate of cange, as we will see in Section.. Differentiability is defined in terms of te existence of derivatives. We may also need to assume tat f, te derivative of f, is continuous on a, b. Ten, te average rate of cange of f on a, b is equal to te average value of f on a, b. In calculus, we will assume tat a function (say, g) is continuous on a, b and b g( x)dx ten define te average value of g on a, b to be a ; te numerator is a b a definite integral, wic will be defined as a limit of sums in calculus (see Section 9.8). b f( x)dx a Ten, te average value of f on a, b is given by:, wic is equal to b a f ( b) f ( a) by te Fundamental Teorem of Calculus. Te teorem assumes tat te b a integrand [function], f, is continuous on a, b.
(Section.: Limits and Derivatives in Calculus).. SECTION.: LIMITS AND DERIVATIVES IN CALCULUS LEARNING OBJECTIVES Be able to develop limit definitions of derivatives and use tem to find derivatives. Understand te grapical interpretation of a derivative as te slope of a tangent line. Understand te practical interpretation of a derivative as an instantaneous rate of cange, possibly velocity. Be able to find equations of tangent lines to graps. Relate derivatives to local linearization of, and marginal cange in, a function. Use derivatives to determine were a function is increasing, decreasing, or constant. PART A: DISCUSSION In Section.5, we discussed limits. In Section.0, we saw ow difference quotients represented slopes of secant lines and average rates of cange. We will now define derivatives as limits of difference quotients. Derivatives represent slopes of tangent lines (wic model or approximate graps locally) and instantaneous rates of cange. Difference quotients on small intervals migt be used to approximate derivatives. Conversely, derivatives migt be used to approximate marginal cange in a function, an idea used in economics. In calculus, we will use derivatives to elp us grap functions by finding were tey are increasing, decreasing, or constant. (See Section.2.) Limits, differentiation (te process of taking derivatives), and integration (wic reverses differentiation; see Section 9.8) are te tree key topics you will find in te first alf of a calculus book, and tey will be key temes trougout.
PART B: TANGENT LINES and DERIVATIVES (Section.: Limits and Derivatives in Calculus)..2 In Section.0, we looked at four forms of difference quotients. Form gave us te average rate of cange of a function f on a fixed interval a, b, were a < b. Tis is equal to te slope of te secant line on a, b. Form : Fixed interval f ( b) f ( a) b a a, b are constants Te oter forms dealt wit variable intervals. Eac form, togeter wit te idea of limits from Section.5, leads to a definition of derivative. Form 2: Variable endpoint (x) f ( x) f ( a) x a a is constant; x is variable If we let x approac a ( denoted by: x a), te corresponding secant lines approac te red tangent line in te following figures. Tis limiting process makes te tangent line a creature of calculus, not just precalculus. (See Footnote on etymology.)
(Section.: Limits and Derivatives in Calculus)..3 Below, we let x approac a Below, we let x approac a from te rigt ( x a + ). from te left ( x a ). (See Footnote 2.) We define te slope of te tangent line to be te (two-sided) limit of te difference quotient as x approaces a, if tat limit exists. We denote tis slope by f( a), read as f prime of (or at) a. f( a), te derivative of f at a, is te slope of te tangent line to te grap of f at te point ( a, f ( a) ), if tat slope exists (as a real number). \ If f( a) exists, we ten say tat f is differentiable at a. Limit Definition of te Derivative at a Point (Based on Form 2) f ( x) f a f( a) lim x a x a Te term point ere tecnically refers to te real number a, wic corresponds to a point on te real number line.
(Section.: Limits and Derivatives in Calculus)..4 Form 3: Variable run () f ( a+ ) f ( a) a is constant; is variable If we let te run approac 0 ( denoted by: 0), te corresponding secant lines approac te red tangent line below. Below, we let approac 0 Below, we let approac 0 from te rigt ( 0 + ). from te left ( 0 ). (See Footnote 2.) Limit Definition of te Derivative at a Point (Based on Form 3) f ( a+ ) f a f( a) lim 0
(Section.: Limits and Derivatives in Calculus)..5 Form 4: Variable endpoint (x) and Variable run () f ( x+ ) f ( x) x, are variable Again, if we let 0, te corresponding secant lines approac te red tangent line below. Te figures are similar to te ones for Form 3, except tat x replaces a. Below, we let approac 0 Below, we let approac 0 from te rigt ( 0 + ). from te left ( 0 ). (See Footnote 2.) provides a rule for f x of f. TIP : Tink of f, te derivative function (or simply derivative) f as a slope function. Limit Definition of te Derivative Function (Based on Form 4) f ( x+ ) f x f( x) lim 0
PART C: INSTANTANEOUS RATE OF CHANGE and INSTANTANEOUS VELOCITY (Section.: Limits and Derivatives in Calculus)..6 Te instantaneous rate of cange of f at a is equal to f( a), if it exists. Te following development of instantaneous velocity will elp explain te association between te slope of a tangent line and instantaneous rate of cange. Example (Velocity) A car is driven due nort for two ours, beginning at noon. How can we find te instantaneous velocity of te car at pm? (If tis is positive, tis can be tougt of as te speedometer reading at pm.) Re Example : Definitions Let t te time (in ours) elapsed since noon. Let y st (), were s is te position function for te car (in miles). In Section.0, we saw tat: Te average velocity on te time-interval a, b is te average rate of cange of position wit respect to time. Tat is, cange in position cange in time s t sb sa b a We will now consider average velocities on variable time intervals of te form a, a +, if > 0, or te form a +, a, if < 0, were is a variable run. (We can let t.)
(Section.: Limits and Derivatives in Calculus)..7 Using Form 3 of a difference quotient, Te average velocity on te time-interval a, a +, if > 0, or a +, a, if < 0, is given by: sa+ ( ) sa Tis equals te slope of te secant line to te grap of s on te interval. (See Footnote 2 on te < 0 case.) Let s assume tere exists a non-vertical tangent line to te grap of s at te point ( a, sa ). Ten, as 0, te slopes of te secant lines will approac te slope of tis tangent line, wic is s( a). Likewise, as 0, te average velocities will approac te instantaneous velocity at a. Below, we let 0 +. Below, we let 0. Te instantaneous velocity (or simply velocity) at a is given by: s sa+ ( ) sa ( a), or v( a) lim 0
(Section.: Limits and Derivatives in Calculus)..8 Using Form 4 of a difference quotient, st+ ( ) st Te velocity function v is defined by: v() t lim 0 () Re Example : Numerical approaces; Approximating derivatives Let s say te position function s is defined by: s() t t 3 on 0, 2. We want to find v(), te instantaneous velocity of te car at t or a. We will first consider average velocities on intervals of te form, +. Here, we let 0 +. Interval Value of (in ours), 2,. 0.,.0 0.0,.00 0.00 0 + Average velocity, s ( + ) s s( 2) s() () 7 mp s(.) s() 3.3 mp 0. s(.0) s() 3.030 mp 0.0 s(.00) s() 3.00300 mp 0.00 3 mp Tese average velocities approac 3 mp, wic is v(). Te reader can verify tat v ( )3 mp in te Exercises. Part D will elp. WARNING : Tables can sometimes be misleading. Te table ere does not represent a rigorous evaluation of v (). Answers are not always integer-valued. If is close to 0, we may be able to use te corresponding average velocity as an approximation for v ().
(Section.: Limits and Derivatives in Calculus)..9 We could also consider tis approac: Interval Value of Average velocity, s ( + ) s() (rounded off to six significant digits), 2 our 7.00000 mp, 60 minute 3.05028 mp, 3600 second 3.00083 mp 0 + 3 mp Here, we let 0. Interval Value of (in ours) 0, 0.9, 0. 0.99, 0.0 0.999, 0.00 0 Average velocity, s ( + ) s s( 0) s() mp () () s( 0.9) s 2.7 mp 0. s( 0.99) s() 2.970 mp 0.0 s( 0.999) s() 2.99700 mp 0.00 3 mp Because of te way we normally look at slopes, we may prefer to rewrite te difference quotient s ( 0 ) s() as s () s( 0), and so fort. (See Footnote 2.)
PART D: FINDING DERIVATIVES (Section.: Limits and Derivatives in Calculus)..0 Example 2 (Profit; Revisiting Examples 2-5 in Section.0) A company sells widgets. Assume tat all widgets produced are sold. Let P be te profit function for te company; P( x) is te profit (in dollars) if x widgets are produced and sold. Our model: P( x) x 2 + 200x 5000. Find te instantaneous rate of cange of profit at 60 widgets. We will ignore integer restrictions on x. Solution We want to find P( 60). We will present tree solutions based on tree different forms of difference quotients we developed in Section.0. Form 2 P( 60) lim x 60 lim x 60 P( x) P( 60) x 60 ( 40 x), x 60 ( by Ex.3 in Section.0) As x approaces 60, we see tat 40 x approaces 40 ( 60), or 80. We can substitute x 60 directly; te restriction x 60 40 ( 60) 80 dollars widget is irrelevant ere. Tis is te slope of te red tangent line below.
(Section.: Limits and Derivatives in Calculus).. Form 3 P( 60 + ) P 60 P( 60) lim 0 lim 80 0, 0 Form 4 80 0 80 dollars widget First, find te formula for P ( by Ex.4 in Section.0) P P( x+ ) P( x) ( x) lim 0 lim 2x + 200 0, 0 2x ( 0)+ 200 2x + 200 Now, evaluate P( 60). P( 60) 260 + 200 80 dollars widget ( x), te rule for te derivative of P. ( by Ex.5 in Section.0) Form 3 Form 4
(Section.: Limits and Derivatives in Calculus)..2 PART E: EQUATIONS OF TANGENT LINES Example 3 (Profit; Revisiting Example 2) Again, P( x) x 2 + 200x 5000. Find Slope-Intercept and Point-Slope Forms of te equation of te tangent line to te grap of y P x point ( 60, P( 60) ). (Review Section 0.4 on tese forms.) Solution Find P( 60), te y-coordinate of te desired point. P( 60) ( 60) 2 + 200( 60) 5000 3400 dollars at te Find P( 60), te slope of te desired tangent line. In Part D, we sowed (tree times) tat: P( 60) 80 dollars widget Find a Point-Slope Form of te equation of te tangent line. y y mx x y 3400 80 x 60 Find te Slope-Intercept Form of te equation of te tangent line. y 80x 400 Note: Some sources would use P instead of y as te dependent variable.
(Section.: Limits and Derivatives in Calculus)..3 PART F: LINEARIZATION and MARGINAL CHANGE ( ) represents Te tangent line to te grap of a function f at te point a, f a te best linear approximation to te function close to a. Te tangent line model linearizes te function locally around a. Te principle of local linearity states tat, if te tangent line exists, te grap of f resembles te line if we zoom in on te point ( a, f ( a) ). Example 4 (Marginal Profit; Revisiting Examples 2 and 3) If P is a profit function, te derivative P is referred to as a marginal profit (MP) function. It approximates te cange in profit if we increase production by unit (from 60 to 6 widgets, for instance). Tat is, P( 60) P( 6) P( 60). Tis is based on te idea tat te tangent line to te grap of P at ( 60, P( 60) ) represents te best linear approximation to P close to x 60 (or a 60). P( 6) P( 60) is te actual rise along te grap of P as x canges from 60 to 6. We will approximate tis by te rise along te tangent line as x canges from 60 to 6. Tis rise is given by P( 60), te slope of te tangent line, because, if run, slope rise rise. (We saw tis in Section 0.4.) run (Te blue grap of P below as been distorted for convenience. Te true grap is very close to te red tangent line.)
(Section.: Limits and Derivatives in Calculus)..4 PART G: GRAPHING FUNCTIONS USING DERIVATIVES Example 5 (Profit; Revisiting Example 2) Solution Again, P( x) x 2 + 200x 5000. Grap P. We will discuss ow to grap quadratic functions suc as P in Section 2.2. For now, we can use te derivative function P to elp us grap P. In Part D (Example 2, Form 4), we found tat P( x) 2x + 200. Tink of tis as a slope function tat gives us slopes of tangent lines. Find any point(s) on te grap were te tangent line is orizontal; its slope is 0. Solve: P( x) 0 2x + 200 0 x 00 Tere is a orizontal tangent line at te point ( 00, P( 00) ), or ( 00, 5000). WARNING 2: Evaluate P( 00), not P( 00), to obtain te y-coordinate of te point. Wat is P( 00)? In calculus, we call 00 a critical number of P, because it is a number in Dom( P) were P is eiter 0 in value or is undefined. We are interested in critical numbers, because tose are te possible x-coordinates of turning points on te grap of P. WARNING 3: Some precalculus sources use te term critical number differently; it is were P, itself, is eiter 0 in value or is undefined. P values could cange sign around critical numbers and discontinuities.
(Section.: Limits and Derivatives in Calculus)..5 Find x-coordinates of points were te tangent lines slope upward; teir slopes are positive. Solve: P( x)> 0 2x + 200 > 0 2x > 200 x < 00 Terefore, P is increasing on te x-interval ( 0,00). In fact, we can say tat P is increasing on 0,00 by a continuity argument. (See Section.2, Part H on wen a function increases on an interval.) Find x-coordinates of points were te tangent lines slope downward; teir slopes are negative. Solve: P( x)< 0 2x + 200 < 0 2x < 200 x > 00 Terefore, P is decreasing on te x-interval ( 00, ). In fact, we can say tat P is decreasing on 00, ) by a continuity argument. Here is te grap of P; te orizontal tangent line is in red:
(Section.: Limits and Derivatives in Calculus)..6 FOOTNOTES. Etymology. Te word secant comes from te Latin secare (to cut). Te word tangent comes from te Latin tangere (to touc). A secant line to te grap of f must intersect it in at least two points. A tangent line only need intersect te grap in one point; te tangent line migt just touc te grap at tat point (toug tere could be oter intersection points). 2. Difference quotients wit negative denominators. Our forms of difference quotients allow negative denominators, as well. Tey still represent slopes of secant lines. Form Form 2 Form ( b < a) slope rise run f ( a ) f b a b Form 2 ( x < a) slope rise run f ( a ) f x a x f ( b ) f a f ( b ) f a b a b a f ( x ) f a f ( x ) f a x a x a Form 3 Form 4 Form 3 ( < 0) slope rise run f ( a ) f a+ a a + Form 4 ( < 0) slope rise run f ( x ) f x+ x x + f ( a+ ) f ( a) f ( a+ ) f a f ( x+ ) f ( x) f ( x+ ) f x