CHAPTER 3: Derivatives

Transcription

1 CHAPTER 3: Derivatives 3.: Derivatives, Tangent Lines, and Rates of Cange 3.2: Derivative Functions and Differentiability 3.3: Tecniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Cain Rule 3.7: Implicit Differentiation 3.8: Related Rates Derivatives represent slopes of tangent lines and rates of cange (suc as velocity). In tis capter, we will define derivatives and derivative functions using limits. We will develop sort cut tecniques for finding derivatives. Tangent lines correspond to local linear approximations of functions. Implicit differentiation is a tecnique used in applied related rates problems.

2 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3.. SECTION 3.: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES Relate difference quotients to slopes of secant lines and average rates of cange. Know, understand, and apply te Limit Definition of te Derivative at a Point. Relate derivatives to slopes of tangent lines and instantaneous rates of cange. Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES For now, assume tat f is a polynomial function of x. (We will relax tis assumption in Part B.) Assume tat a is a constant. Temporarily fix an arbitrary real value of x. (By arbitrary, we mean tat any real value will do). Later, instead of tinking of x as a fixed (or single) value, we will tink of it as a moving or varying variable tat can take on different values. Te secant line to te grap of f on te interval [ a, x], were a < x, is te line tat passes troug te points ( a, f ( a) ) and ( x, f ( x) ). secare is Latin for to cut. Te slope of tis secant line is given by: rise run = f ( x ) f ( a). x a difference of outputs We call tis a difference quotient, because it as te form: difference of inputs.

3 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..2 PART B: TANGENT LINES and DERIVATIVES If we now treat x as a variable and let x a, te corresponding secant lines approac te red tangent line below. tangere is Latin for to touc. A secant line to te grap of f must intersect it in at least two distinct points. A tangent line only need intersect te grap in one point, were te line migt just touc te grap. (Tere could be oter intersection points). Tis limiting process makes te tangent line a creature of calculus, not just precalculus. Below, we let x approac a Below, we let x approac a from te rigt ( x a + ). from te left ( x a ). (See Footnote.) We define te slope of te tangent line to be te (two-sided) limit of te difference quotient as x 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). \ f is differentiable at a f a exists. Polynomial functions are differentiable everywere on. (See Section 3.2.) Te statements of tis section apply to any function tat is differentiable at a.

4 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..3 Limit Definition of te Derivative at a Point a (Version ) f ( x) f ( a) f( a), if it exists x a x a If f is continuous at a, we ave te indeterminate Limit Form 0 0. Continuity involves limits of function values, wile differentiability involves limits of difference quotients. Version : Variable endpoint (x) Slope of secant line: f ( x) f ( a) x a a is constant; x is variable A second version, were x is replaced by a +, is more commonly used. Version 2: Variable run () Slope of secant line: f ( a+ ) f ( a) a is constant; is variable If we let te run 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 ). (Footnote.)

5 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..4 Limit Definition of te Derivative at a Point a (Version 2) f ( a+ ) f ( a) f( a), if it exists 0 Version 3: Two-Sided Approac Limit Definition of te Derivative at a Point a (Version 3) f ( a+ ) f ( a ) f( a), if it exists 0 2 Te reader is encouraged to draw a figure to understand tis approac. Principle of Local Linearity Te tangent line to te grap of f at te point ( a, f ( a) ), if it exists, represents te best local linear approximation to te function close to a. Te grap of f resembles tis line if we zoom in on te point ( a, f ( a) ). Te tangent line model linearizes te function locally around a. We will expand on tis in Section 3.5. (Te figure on te rigt is a zoom in on te box in te figure on te left.)

6 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..5 PART C: FINDING DERIVATIVES USING THE LIMIT DEFINITIONS Example (Finding a Derivative at a Point Using Version of te Limit Definition) Let f ( x)= x 3. Find f () using Version of te Limit Definition of te Derivative at a Point. Solution f ( x) f f ()= lim x x x x 3 x () () 3 ( Here, a =. ) TIP : Te brackets ere are unnecessary, but better safe tan sorry. x x 3 x Limit Form 0 0 We will factor te numerator using te Difference of Two Cubes template and ten simplify. Syntetic Division can also be used. (See Capter 2 in te Precalculus notes). () x ( x 2 + x + ) x ( x ) () x x 2 + x + () 2 + ()+ = = 3

7 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..6 Example 2 (Finding a Derivative at a Point Using Version 2 of te Limit Definition; Revisiting Example ) Let f ( x)= x 3, as in Example. Find f () using Version 2 of te Limit Definition of te Derivative at a Point. Solution f ( + ) f f ()= lim () () 3 ( Here, a =. ) We will use te Binomial Teorem to expand ( + ) 3. (See Capter 9 in te Precalculus notes.) () () () 0 ( ) 2 () = ( 0) 2 = 3 () We obtain te same result as in Example : f ()= 3.

8 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..7 PART D: FINDING EQUATIONS OF TANGENT LINES Example 3 (Finding Equations of Tangent Lines; Revisiting Examples and 2) Solution Find an equation of te tangent line to te grap of y = x 3 at te point were x =. (Review Section 0.4: Lines in te Precalculus notes.) Let f ( x)= x 3, as in Examples and 2. Find f (), te y-coordinate of te point of interest. f ()= () 3 = Te point of interest is ten: (, f () )= (, ). Find f (), te slope (m) of te desired tangent line. In Part C, we sowed (twice) tat: f ()= 3. Find a Point-Slope Form for te equation of te tangent line. y y = mx x y = 3 x Find te Slope-Intercept Form for te equation of te tangent line. y = 3x 3 y = 3x 2 Observe ow te red tangent line below is consistent wit te equation above.

9 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..8 Te Slope-Intercept Form can also be obtained directly. Remember te Basic Principle of Graping: Te grap of an equation consists of all points (suc as, ere) wose coordinates satisfy te equation. PART E: NORMAL LINES y = mx + b ()= ( 3) ()+ b ( Solve for b. ) b = 2 y = 3x 2 Assume tat P is a point on a grap were a tangent line exists. Te normal line to te grap at P is te line tat contains P and tat is perpendicular to te tangent line at P. Example 4 (Finding Equations of Normal Lines; Revisiting Example 3) Find an equation of te normal line to te grap of y = x 3 at P(, ). Solution In Examples and 2, we let f ( x)= x 3, and we found tat te slope of te tangent line at, f was given by: ()= 3. Te slope of te normal line at, of te slope of te tangent line. is ten 3, te opposite reciprocal A Point-Slope Form for te equation of te normal line is given by: y y = mx x y = ( 3 x ) Te Slope-Intercept Form is given by: y = 3 x

10 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..9 WARNING : Te Slope-Intercept Form for te equation of te normal line at P cannot be obtained by taking te Slope-Intercept Form for te equation of te tangent line at P and replacing te slope wit its opposite reciprocal, unless P lies on te y-axis. In tis Example, te normal line is not given by: y = 3 x 2. PART F: NUMERICAL APPROXIMATION OF DERIVATIVES Te Principle of Local Linearity implies tat te slope of te tangent line at te point ( a, f ( a) ) can be well approximated by te slope of te secant line on a small interval containing a. Wen using Version 2 of te Limit Definition of te Derivative, tis implies tat: f( a) f ( a+ ) f ( a) wen 0. Example 5 (Numerically Approximating a Derivative; Revisiting Example 2) Let f ( x)= x 3, as in Example 2. We will find approximations of f (See Example 8 in Part H.) f ( + ) f () ()., or ( 0) ( See Example 2. ) If we only ave a table of values for a function f instead of a rule for f ( x), we may ave to resort to numerically approximating derivatives.

11 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..0 PART G: AVERAGE RATE OF CHANGE Te average rate of cange of f on a, b is equal to te slope of te secant line on a, b, wic is given by: rise run = f ( b ) f ( a). (See Footnotes 2 and 3.) b a Example 6 (Average Velocity) Average velocity is a common example of an average rate of cange. Let s say 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 Here, te average velocity of te car on 0, 2 is: s( 2) s( 0) 00 0 = were (uppercase delta) denotes cange in = sb sa, a difference quotient b a = 50 miles our or mi r or mp TIP 2: Te unit of velocity is te unit of slope given by: unit of s unit of t.

12 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3.. Te average velocity is 50 mp on 0, 2 in te tree scenarios below. It is te slope of te orange secant line. (Axes are scaled differently.) Below, te velocity is constant (50 mp). (We are not requiring te car to slow down to a stop at te end.) Below, te velocity is increasing; te car is accelerating. Below, te car breaks te rules, backtracks, and goes sout. WARNING 2: Te car s velocity is negative in value wen it is backtracking; tis appens wen te grap falls. Note: Te Mean Value Teorem for Derivatives in Section 4.2 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

13 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..2 PART H: INSTANTANEOUS RATE OF CHANGE Te instantaneous rate of cange of f at a is equal to f( a), if it exists. Example 7 (Instantaneous Velocity) Instantaneous velocity (or simply velocity) is a common example of an instantaneous rate of cange. Let s say 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.) Let t = te time (in ours) elapsed since noon. Let y = st (), were s is te position function for te car (in miles). 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.) 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 on te < 0 case.)

14 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..3 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: sa+ sa s( a), or v( a), if it exists 0 In our Example, te instantaneous velocity of te car at pm is given by: s( + ) s() s(), or v()= lim 0 Let s say st ()= t 3. Example 2 ten implies tat s()= v()= 3 mp.

15 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..4 Example 8 (Numerically Approximating an Instantaneous Velocity; Revisiting Examples 5 and 7) Again, let s say te position function s is defined by: st ()= t 3 on 0, 2. We will approximate v(), te instantaneous velocity of te car at pm. We will first compute average velocities on intervals of te form, +. Here, we let 0 +. Interval Value of (in ours), 2,. 0.,.0 0.0, Average velocity, s ( + ) s s( 2) s() () = 7 mp s(.) s() = 3.3 mp 0. s(.0) s() = mp 0.0 s(.00) s() = mp mp Tese average velocities approac 3 mp, wic is v(). WARNING 3: Tables can sometimes be misleading. Te table ere does not represent a rigorous evaluation of v (). Answers are not always integer-valued.

16 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..5 We could also consider tis approac: Interval Value of Average velocity, s ( + ) s() (rounded off to six significant digits), 2 our mp, 60 minute mp, 3600 second mp mp Here, we let 0. Interval Value of (in ours) 0, 0.9, , , Average velocity, s ( + ) s s( 0) s() = mp () () s( 0.9) s = 2.7 mp 0. s( 0.99) s() = mp 0.0 s( 0.999) s() = mp mp Because of te way we normally look at slopes, we may prefer to rewrite te first difference quotient s ( 0 ) s() as s () s( 0), and so fort. (See Footnote.)

17 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..6 Example 9 (Rate of Cange of a Profit Function) Solution A company sells widgets. Assume tat all widgets produced are sold. P( x), te profit (in dollars) if x widgets are produced and sold, is modeled by: P( x)= x x Find te instantaneous rate of cange of profit at 60 widgets. (In economics, tis is referred to as marginal profit.) WARNING 4: 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. We want to find P( 60). P 60 P( 60 + ) P( 60) 0 ( 60 + ) ( 60 + ) WARNING 5: Grouping symbols are essential wen expanding P 60 ere, because we are subtracting an expression wit more tan one term. 2 ( )+ 2, , TIP 3: Instead of simplifying witin te brackets immediately, we will take advantage of cancellations , ,

18 () 0 0 = 80 0 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..7 ( 80 ) () ( 80 ) = 80 dollars widget Tis is te slope of te red tangent line below. If we produce and sell one more widget (from 60 to 6), we expect to make about $80 more in profit. Wat would be your business strategy if marginal profit is positive?

19 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) 3..8 FOOTNOTES. Difference quotients wit negative denominators. Our forms of difference quotients allow negative denominators ( runs ), as well. Tey still represent slopes of secant lines. Left figure ( x < a): slope = rise run = f ( a ) f ( x) = f ( x ) f a a x ( x a) Rigt figure ( < 0): slope = rise run = f ( a ) f ( a+ ) f ( a+ ) f ( a) = a a + = f ( x ) f a x a = f ( a+ ) f a 2. Average rate of cange and 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 b a te existence of f ( a) and 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. 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 2.8.) 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. Differentiability is defined in terms of te existence of derivatives. We may also need to assume tat f is continuous on a, b. (See Footnote 3.)

20 (Section 3.: Derivatives, Tangent Lines, and Rates of Cange) Average rate of cange of f as te average value of f. Assume tat a, b f is continuous on. Ten, te average rate of cange of f on a, b is equal to te average value of f on a, b. In Capter 5, we will assume tat a function (say, g) is continuous on a, b b g( x)dx a and ten define te average value of g on a, b to be ; te numerator is a b a definite integral, wic will be defined as a limit of sums. Ten, te average value of b f( x)dx a a, b is given by:, wic is equal to f ( b ) f ( a) by te Fundamental b a b a Teorem of Calculus. Te teorem assumes tat te integrand [function], on a, b. f on f, is continuous