Rolle s Theorem and the Mean Value Theorem. Rolle s Theorem

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1 0_00qd //0 0:50 AM Page 7 7 CHAPTER Applications o Dierentiation Section ROLLE S THEOREM French mathematician Michel Rolle irst published the theorem that bears his name in 9 Beore this time, however, Rolle was one o the most vocal critics o calculus, stating that it gave erroneous results and was based on unsound reasoning Later in lie, Rolle came to see the useulness o calculus Rolle s Theorem and the Mean Value Theorem Understand and use Rolle s Theorem Understand and use the Mean Value Theorem Rolle s Theorem The Etreme Value Theorem (Section ) states that a continuous unction on a closed interval a, b must have both a minimum and a maimum on the interval Both o these values, however, can occur at the endpoints Rolle s Theorem, named ater the French mathematician Michel Rolle (5 79), gives conditions that guarantee the eistence o an etreme value in the interior o a closed interval EXPLORATION Etreme Values in a Closed Interval Sketch a rectangular coordinate plane on a piece o paper Label the points, and 5, Using a pencil or pen, draw the graph o a dierentiable unction that starts at, and ends at 5, Is there at least one point on the graph or which the derivative is zero? Would it be possible to draw the graph so that there isn t a point or which the derivative is zero? Eplain our reasoning Relative maimum THEOREM Rolle s Theorem Let be continuous on the closed interval a, b and dierentiable on the open interval a, b I a b then there is at least one number c in a, b such that c 0 d a c (a) is continuous on a, b and dierentiable on a, b Relative maimum b Proo Let a d b Case : I d or all in a, b, is constant on the interval and, b Theorem, 0 or all in a, b Case : Suppose > d or some in a, b B the Etreme Value Theorem, ou know that has a maimum at some c in the interval Moreover, because c > d, this maimum does not occur at either endpoint So, has a maimum in the open interval a, b This implies that c is a relative maimum and, b Theorem, c is a critical number o Finall, because is dierentiable at c, ou can conclude that c 0 Case : I < d or some in a, b, ou can use an argument similar to that in Case, but involving the minimum instead o the maimum d a c b (b) is continuous on a, b Figure 8 From Rolle s Theorem, ou can see that i a unction is continuous on a, b and dierentiable on a, b, and i a b, there must be at least one -value between a and b at which the graph o has a horizontal tangent, as shown in Figure 8(a) I the dierentiabilit requirement is dropped rom Rolle s Theorem, will still have a critical number in a, b, but it ma not ield a horizontal tangent Such a case is shown in Figure 8(b)

2 0_00qd //0 0:50 AM Page 7 SECTION Rolle s Theorem and the Mean Value Theorem 7 EXAMPLE Illustrating Rolle s Theorem () = + (, 0) (, 0) ( ) = 0 Horizontal tangent The -value or which 0 is between the two -intercepts Figure 9 Find the two -intercepts o and show that ) 0 at some point between the two -intercepts Solution Note that is dierentiable on the entire real line Setting equal to 0 produces 0 0 Set equal to 0 Factor So, 0, and rom Rolle s Theorem ou know that there eists at least one c in the interval, such that c 0 To ind such a c, ou can solve the equation 0 Set equal to 0 and determine that 0 when Note that the -value lies in the open interval,, as shown in Figure 9 Rolle s Theorem states that i satisies the conditions o the theorem, there must be at least one point between a and b at which the derivative is 0 There ma o course be more than one such point, as shown in the net eample ( ) = 8 8 () = (0) = 0 ( ) = 0 () = 0 () = 8 0 or more than one -value in the interval, Figure 0 EXAMPLE Illustrating Rolle s Theorem Let Find all values o c in the interval, such that c 0 Solution To begin, note that the unction satisies the conditions o Rolle s Theorem That is, is continuous on the interval, and dierentiable on the interval, Moreover, because 8, ou can conclude that there eists at least one c in, such that c 0 Setting the derivative equal to 0 produces 0 Set equal to 0 0 0,, Factor -values or which 0 So, in the interval,, the derivative is zero at three dierent values o, as shown in Figure 0 Figure TECHNOLOGY PITFALL A graphing utilit can be used to indicate whether the points on the graphs in Eamples and are relative minima or relative maima o the unctions When using a graphing utilit, however, ou should keep in mind that it can give misleading pictures o graphs For eample, use a graphing utilit to graph With most viewing windows, it appears that the unction has a maimum o when (see Figure ) B evaluating the unction at, however, ou can see that 0 To determine the behavior o this unction near, ou need to eamine the graph analticall to get the complete picture

3 0_00qd //0 0:50 AM Page 7 7 CHAPTER Applications o Dierentiation The Mean Value Theorem Rolle s Theorem can be used to prove another theorem the Mean Value Theorem Slope o tangent line = (c) THEOREM The Mean Value Theorem I is continuous on the closed interval a, b and dierentiable on the open interval a, b, then there eists a number c in a, b such that c b a b a Figure Mar Evans Picture Librar (a, (a)) a c b Tangent line Secant line (b, (b)) JOSEPH-LOUIS LAGRANGE (7 8) The Mean Value Theorem was irst proved b the amous mathematician Joseph-Louis Lagrange Born in Ital, Lagrange held a position in the court o Frederick the Great in Berlin or 0 ears Aterward, he moved to France, where he met emperor Napoleon Bonaparte, who is quoted as saing, Lagrange is the lot pramid o the mathematical sciences Proo Reer to Figure The equation o the secant line containing the points a, a and b, b is b a b a a a Let g be the dierence between and Then g B evaluating g at a and b, ou can see that ga 0 gb Because is continuous on a, b it ollows that g is also continuous on a, b Furthermore, because is dierentiable, g is also dierentiable, and ou can appl Rolle s Theorem to the unction g So, there eists a number c in a, b such that gc 0, which implies that 0 gc c So, there eists a number c in a, b such that c b a b a b a b a b a b a a a NOTE The mean in the Mean Value Theorem reers to the mean (or average) rate o change o in the interval a, b Although the Mean Value Theorem can be used directl in problem solving, it is used more oten to prove other theorems In act, some people consider this to be the most important theorem in calculus it is closel related to the Fundamental Theorem o Calculus discussed in Chapter For now, ou can get an idea o the versatilit o this theorem b looking at the results stated in Eercises in this section The Mean Value Theorem has implications or both basic interpretations o the derivative Geometricall, the theorem guarantees the eistence o a tangent line that is parallel to the secant line through the points a, a and b, b, as shown in Figure Eample illustrates this geometric interpretation o the Mean Value Theorem In terms o rates o change, the Mean Value Theorem implies that there must be a point in the open interval a, b at which the instantaneous rate o change is equal to the average rate o change over the interval a, b This is illustrated in Eample

4 0_00qd //0 0:50 AM Page 75 SECTION Rolle s Theorem and the Mean Value Theorem 75 EXAMPLE Finding a Tangent Line Tangent line (, ) (, ) Secant line () = 5 (, ) The tangent line at, is parallel to the secant line through, and, Figure Given 5, ind all values o c in the open interval, such that c Solution The slope o the secant line through, and, is Because satisies the conditions o the Mean Value Theorem, there eists at least one number c in, such that c Solving the equation ields which implies that ± So, in the interval,, ou can conclude that c, as shown in Figure EXAMPLE Finding an Instantaneous Rate o Change 5 miles t = minutes t = 0 Not drawn to scale At some time t, the instantaneous velocit is equal to the average velocit over minutes Figure Two stationar patrol cars equipped with radar are 5 miles apart on a highwa, as shown in Figure As a truck passes the irst patrol car, its speed is clocked at 55 miles per hour Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 miles per hour Prove that the truck must have eceeded the speed limit (o 55 miles per hour) at some time during the minutes Solution Let t 0 be the time (in hours) when the truck passes the irst patrol car The time when the truck passes the second patrol car is t 0 5 hour B letting st represent the distance (in miles) traveled b the truck, ou have s0 0 and s5 5 So, the average velocit o the truck over the ive-mile stretch o highwa is s5 s0 Average velocit miles per hour 5 Assuming that the position unction is dierentiable, ou can appl the Mean Value Theorem to conclude that the truck must have been traveling at a rate o 75 miles per hour sometime during the minutes A useul alternative orm o the Mean Value Theorem is as ollows: I is continuous on a, b and dierentiable on a, b, then there eists a number c in a, b such that b a b ac Alternative orm o Mean Value Theorem NOTE When doing the eercises or this section, keep in mind that polnomial unctions, rational unctions, and trigonometric unctions are dierentiable at all points in their domains

5 0_00qd //0 0:50 AM Page 7 7 CHAPTER Applications o Dierentiation Eercises or Section In Eercises, eplain wh Rolle s Theorem does not appl to the unction even though there eist a and b such that a b (0, 0),, (, 0) In Eercises 5 8, ind the two -intercepts o the unction and show that 0 at some point between the two -intercepts Rolle s Theorem In Eercises 9 and 0, the graph o is shown Appl Rolle s Theorem and ind all values o c such that c 0 at some point between the labeled intercepts 9 0 () = + (, 0) (, 0) cot (0, 0) (, 0) (, 0),, () = sin (, 0 ) See wwwcalcchatcom or worked-out solutions to odd-numbered eercises In Eercises 5 8, use a graphing utilit to graph the unction on the closed interval [a, b] Determine whether Rolle s Theorem can be applied to on the interval and, i so, ind all values o c in the open interval a, b such that c 0 5,,, 0, 7 tan,, 8 sin,, 0 9 Vertical Motion The height o a ball t seconds ater it is thrown upward rom a height o eet and with an initial velocit o 8 eet per second is t t 8t (a) Veri that (b) According to Rolle s Theorem, what must be the velocit at some time in the interval,? Find that time 0 Reorder Costs The ordering and transportation cost C or components used in a manuacturing process is approimated b C 0 where C is measured in thousands, o dollars and is the order size in hundreds (a) Veri that C C (b) According to Rolle s Theorem, the rate o change o the cost must be 0 or some order size in the interval, Find that order size In Eercises and, cop the graph and sketch the secant line to the graph through the points a, a and b, b Then sketch an tangent lines to the graph or each value o c guaranteed b the Mean Value Theorem To print an enlarged cop o the graph, go to the website wwwmathgraphscom In Eercises, determine whether Rolle s Theorem can be applied to on the closed interval [a, b] I Rolle s Theorem can be applied, ind all values o c in the open interval a, b such that c 0, 0, 5,,,,,, 5, 8, 8, 0, 7, 8,,, 9 sin, 0, 0 cos, 0, sin, 0, cos,, tan, 0, sec,, Writing In Eercises, eplain wh the Mean Value Theorem does not appl to the unction on the interval [0, ] 5 a 5 b 5 5 a 5 b

6 0_00qd //0 0:50 AM Page 77 SECTION Rolle s Theorem and the Mean Value Theorem 77 7 Mean Value Theorem Consider the graph o the unction (a) Find the equation o the secant line joining the points, and, 5 (b) Use the Mean Value Theorem to determine a point c in the interval, such that the tangent line at c is parallel to the secant line (c) Find the equation o the tangent line through c (d) Then use a graphing utilit to graph, the secant line, and the tangent line Figure or 7 Figure or 8 8 Mean Value Theorem Consider the graph o the unction (a) Find the equation o the secant line joining the points, and, 0 (b) Use the Mean Value Theorem to determine a point c in the interval, such that the tangent line at c is parallel to the secant line (c) Find the equation o the tangent line through c (d) Then use a graphing utilit to graph, the secant line, and the tangent line In Eercises 9, determine whether the Mean Value Theorem can be applied to on the closed interval [a, b] I the Mean Value Theorem can be applied, ind all values o c in the b a open interval a, b such that c b a 9,, 0,,, 0,,, 7,, 5 sin, 0, sin sin, 0, In Eercises 7 50, use a graphing utilit to (a) graph the unction on the given interval, (b) ind and graph the secant line through points on the graph o at the endpoints o the given interval, and (c) ind and graph an tangent lines to the graph o that are parallel to the secant line 7 8 sin,,, 9,, , 0, 5 (, ) 5 () = +, (, 5) () = + (, ) 0, (, 0), 5 Vertical Motion The height o an object t seconds ater it is dropped rom a height o 500 meters is st 9t 500 (a) Find the average velocit o the object during the irst seconds (b) Use the Mean Value Theorem to veri that at some time during the irst seconds o all the instantaneous velocit equals the average velocit Find that time 5 Sales A compan introduces a new product or which the number o units sold S is St t where t is the time in months (a) Find the average value o St during the irst ear (b) During what month does St equal the average value during the irst ear? Writing About Concepts 5 Let be continuous on a, b and dierentiable on a, b I there eists c in a, b such that c 0, does it ollow that a b? Eplain 5 Let be continuous on the closed interval a, b and dierentiable on the open interval a, b Also, suppose that a b and that c is a real number in the interval such that c 0 Find an interval or the unction g over which Rolle s Theorem can be applied, and ind the corresponding critical number o g ( k is a constant) (a) g k (b) g k (c) g k 55 The unction 0,, 0 0 < is dierentiable on 0, and satisies 0 However, its derivative is never zero on 0, Does this contradict Rolle s Theorem? Eplain 5 Can ou ind a unction such that,, and < or all Wh or wh not? 57 Speed A plane begins its takeo at :00 PM on a 500-mile light The plane arrives at its destination at 7:0 PM Eplain wh there are at least two times during the light when the speed o the plane is 00 miles per hour 58 Temperature When an object is removed rom a urnace and placed in an environment with a constant temperature o 90F, its core temperature is 500F Five hours later the core temperature is 90F Eplain wh there must eist a time in the interval when the temperature is decreasing at a rate o F per hour 59 Velocit Two bicclists begin a race at 8:00 AM The both inish the race hours and 5 minutes later Prove that at some time during the race, the bicclists are traveling at the same velocit 0 Acceleration At 9: AM, a sports car is traveling 5 miles per hour Two minutes later, the car is traveling 85 miles per hour Prove that at some time during this two-minute interval, the car s acceleration is eactl 500 miles per hour squared

7 0_00qd //0 0:50 AM Page CHAPTER Applications o Dierentiation Graphical Reasoning The igure shows two parts o the graph o a continuous dierentiable unction on 0, The derivative is also continuous To print an enlarged cop o the graph, go to the website wwwmathgraphscom (a) Eplain wh must have at least one zero in 0, (b) Eplain wh must also have at least one zero in the interval 0, What are these zeros called? (c) Make a possible sketch o the unction with one zero o on the interval 0, (d) Make a possible sketch o the unction with two zeros o on the interval 0, (e) Were the conditions o continuit o and necessar to do parts (a) through (d)? Eplain Consider the unction (a) Use a graphing utilit to graph and (b) Is a continuous unction? Is a continuous unction? (c) Does Rolle s Theorem appl on the interval,? Does it appl on the interval,? Eplain (d) Evaluate, i possible, lim and lim Think About It In Eercises and, sketch the graph o an arbitrar unction that satisies the given condition but does not satis the conditions o the Mean Value Theorem on the interval [5, 5] is continuous on 5, 5 is not continuous on 5, 5 In Eercises 5 and, use the Intermediate Value Theorem and Rolle s Theorem to prove that the equation has eactl one real solution cos 0 7 Determine the values a, b, and c such that the unction satisies the hpotheses o the Mean Value Theorem on the interval 0,, a b, c, cos 0 0 < < 8 Determine the values a, b, c, and d so that the unction satisies the hpotheses o the Mean Value Theorem on the interval, a,, b c, d, Dierential Equations In Eercises 9 7, ind a unction that has the derivative and whose graph passes through the given point Eplain our reasoning 9 0,, 5 70, 0, 7,, 0 7,, 0 True or False? In Eercises 7 7, determine whether the statement is true or alse I it is alse, eplain wh or give an eample that shows it is alse 7 The Mean Value Theorem can be applied to on the interval, 7 I the graph o a unction has three -intercepts, then it must have at least two points at which its tangent line is horizontal 75 I the graph o a polnomial unction has three -intercepts, then it must have at least two points at which its tangent line is horizontal 7 I 0 or all in the domain o, then is a constant unction 77 Prove that i a > 0 and n is an positive integer, then the polnomial unction p n a b cannot have two real roots 78 Prove that i 0 or all in an interval a, b, then is constant on a, b 79 Let p A B C Prove that or an interval a, b, the value c guaranteed b the Mean Value Theorem is the midpoint o the interval 80 (a) Let and g Then g and g Show that there is at least one value c in the interval, where the tangent line to at c, c is parallel to the tangent line to g at c, gc Identi c (b) Let and g be dierentiable unctions on a, b where a ga and b gb Show that there is at least one value c in the interval a, b where the tangent line to at c, c is parallel to the tangent line to g at c, gc 8 Prove that i is dierentiable on, and < or all real numbers, then has at most one ied point A ied point o a unction is a real number c such that c c 8 Use the result o Eercise 8 to show that cos has at most one ied point 8 Prove that cos a cos b a b or all a and b 8 Prove that sin a sin b a b or all a and b 85 Let 0 < a < b Use the Mean Value Theorem to show that b a < b a a < 0 0 < <