SECTION 9 Differentials 5 Section 9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph f In the same viewing window, graph the tangent line to the graph of f at the point, Zoom in twice on the point of tangenc Does our graphing utilit distinguish between the two graphs? Use the trace feature to compare the two graphs As the -values get closer to, what can ou sa about the -values? Differentials Understand the concept of a tangent line approimation Compare the value of the differential, d, with the actual change in, Estimate a propagated error using a differential Find the differential of a function using differentiation formulas Tangent Line Approimations Newton s Method (Section 8) is an eample of the use of a tangent line to a graph to approimate the graph In this section, ou will stud other situations in which the graph of a function can be approimated b a straight line To begin, consider a function f that is differentiable at c The equation for the tangent line at the point c, f c is given b f c fc c f c fc c and is called the tangent line approimation (or linear approimation) of f at c Because c is a constant, is a linear function of Moreover, b restricting the values of to be sufficientl close to c, the values of can be used as approimations (to an desired accurac) of the values of the function f In other words, as c, the limit of is f c EXAMPLE Using a Tangent Line Approimation Find the tangent line approimation of Tangent line f sin at the point 0, Then use a table to compare the -values of the linear function with those of f on an open interval containing 0 π f() = + sin π π The tangent line approimation of f at the point 0, Figure 65 Solution The derivative of f is f cos First derivative So, the equation of the tangent line to the graph of f at the point 0, is f 0 f0 0 0 Tangent line approimation The table compares the values of given b this linear approimation with the values of f near 0 Notice that the closer is to 0, the better the approimation is This conclusion is reinforced b the graph shown in Figure 65 Editable Graph 05 0 00 0 00 0 05 f sin 05 0900 0990000 0099998 0998 79 05 09 099 0 5 Tr It Eploration A Open Eploration NOTE Be sure ou see that this linear approimation of f sin depends on the point of tangenc At a different point on the graph of f, ou would obtain a different tangent line approimation
6 CHAPTER Applications of Differentiation (c +, f(c + )) (c, f(c)) c c + f f (c) f(c) f(c + ) When is small, fc fc is approimated b fc Figure 66 Differentials When the tangent line to the graph of f at the point c, f c f c fc c Tangent line at c, f c is used as an approimation of the graph of f, the quantit c is called the change in, and is denoted b, as shown in Figure 66 When is small, the change in (denoted b ) can be approimated as shown f c f c Actual change in fc Approimate change in For such an approimation, the quantit is traditionall denoted b d, and is called the differential of The epression fd is denoted b d, and is called the differential of Definition of Differentials Let f represent a function that is differentiable on an open interval containing The differential of (denoted b d) is an nonzero real number The differential of (denoted b d) is d f d Video In man tpes of applications, the differential of can be used as an approimation of the change in That is, d or fd = = d (, ) The change in,, is approimated b the differential of, d Figure 67 EXAMPLE Comparing and d Let Find d when and d 00 Compare this value with for and 00 Solution Because f, ou have f, and the differential d is given b d f d f00 00 00 Now, using 00, the change in is f f f0 f Differential of 0 000 Figure 67 shows the geometric comparison of d and Tr comparing other values of d and You will see that the values become closer to each other as d or approaches 0 Tr It Eploration A In Eample, the tangent line to the graph of f at is or g Tangent line to the graph of f at For -values near, this line is close to the graph of f, as shown in Figure 67 For instance, f 0 0 00 and g0 0 0
SECTION 9 Differentials 7 Error Propagation Phsicists and engineers tend to make liberal use of the approimation of b d One wa this occurs in practice is in the estimation of errors propagated b phsical measuring devices For eample, if ou let represent the measured value of a variable and let represent the eact value, then is the error in measurement Finall, if the measured value is used to compute another value f, the difference between f and f is the propagated error Measurement Propagated error error f f Eact value Measured value EXAMPLE Estimation of Error The radius of a ball bearing is measured to be 07 inch, as shown in Figure 68 If the measurement is correct to within 00 inch, estimate the propagated error in the volume V of the ball bearing 07 Ball bearing with measured radius that is correct to within 00 inch Figure 68 Solution The formula for the volume of a sphere is V r, where r is the radius of the sphere So, ou can write r 07 Measured radius and 00 r 00 Possible error To approimate the propagated error in the volume, differentiate V to obtain dvdr r and write V dv Approimate V b dv r dr 07 ±00 Substitute for r and dr ±00658 cubic inch So, the volume has a propagated error of about 006 cubic inch Tr It Eploration A Eploration B Video Video Would ou sa that the propagated error in Eample is large or small? The answer is best given in relative terms b comparing dv with V The ratio dv V r dr r dr r 07 ±00 ±009 Ratio of dv to V Simplif Substitute for dr and r is called the relative error The corresponding percent error is approimatel 9%
8 CHAPTER Applications of Differentiation Calculating Differentials Each of the differentiation rules that ou studied in Chapter can be written in differential form For eample, suppose u and v are differentiable functions of B the definition of differentials, ou have du u d and dv v d So, ou can write the differential form of the Product Rule as shown below duv d uv d d uv vu d uv d vu d u dv v du Differential of uv Product Rule Differential Formulas Let u and v be differentiable functions of Constant multiple: dcu c du Sum or difference: du ± v du ± dv Product: Quotient: d uv u dv v du d u v du u dv v v EXAMPLE Finding Differentials Function Derivative Differential a b c d sin cos d d d cos d d sin cos d d d d d d cos d d sin cos d d d Tr It Eploration A GOTTFRIED WILHELM LEIBNIZ (66 76) Both Leibniz and Newton are credited with creating calculus It was Leibniz, however, who tried to broaden calculus b developing rules and formal notation He often spent das choosing an appropriate notation for a new concept MathBio The notation in Eample is called the Leibniz notation for derivatives and differentials, named after the German mathematician Gottfried Wilhelm Leibniz The beaut of this notation is that it provides an eas wa to remember several important calculus formulas b making it seem as though the formulas were derived from algebraic manipulations of differentials For instance, in Leibniz notation, the Chain Rule d d du d du d would appear to be true because the du s divide out Even though this reasoning is incorrect, the notation does help one remember the Chain Rule
SECTION 9 Differentials 9 EXAMPLE 5 Finding the Differential of a Composite Function f sin f cos d f d cos d Original function Appl Chain Rule Differential form Tr It EXAMPLE 6 Eploration A Finding the Differential of a Composite Function f d f d f d Original function Appl Chain Rule Differential form Tr It Eploration A Eploration B Technolog Differentials can be used to approimate function values To do this for the function given b f, ou use the formula f f d f f d which is derived from the approimation f f d The ke to using this formula is to choose a value for that makes the calculations easier, as shown in Eample 7 EXAMPLE 7 Approimating Function Values Use differentials to approimate 65 Solution Using f, ou can write f f f d d Now, choosing 6 and d 05, ou obtain the following approimation f 65 6 6 05 8 065 6 g() = + 8 f() = (6, ) 8 6 0 Tr It Eploration A The tangent line approimation to f at 6 is the line g 8 For -values near 6, the graphs of f and g are close together, as shown in Figure 69 For instance, f 65 65 060 and g65 65 065 8 Figure 69 In fact, if ou use a graphing utilit to zoom in near the point of tangenc 6,, ou will see that the two graphs appear to coincide Notice also that as ou move farther awa from the point of tangenc, the linear approimation is less accurate
0 CHAPTER Applications of Differentiation Eercises for Section 9 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises 6, find the equation of the tangent line T to the graph of f at the given point Use this linear approimation to complete the table f T In Eercises 7 0, use the information to evaluate and compare and d In Eercises 0, find the differential d of the given function 9 In Eercises, use differentials and the graph of f to approimate (a) f 9 and (b) f 0 To print an enlarged cop of the graph, select the MathGraph button 5 f, f 6, 9 99 0 (, ),, f 5, f,,, 5 6 f sin, f csc,, sin, csc 7 8 9 0 0 d 0 d 0 d 00 d 00 5 6 7 cot 8 sin 9 0 sec cos 6 f 5 5 f (, ) 5 5 In Eercises 5 and 6, use differentials and the graph of approimate (a) g9 and (b) g given that g 8 5 6 (, ) 5 g f 5 (, ) 5 7 Area The measurement of the side of a square is found to be inches, with a possible error of 6 inch Use differentials to approimate the possible propagated error in computing the area of the square 8 Area The measurements of the base and altitude of a triangle are found to be 6 and 50 centimeters, respectivel The possible error in each measurement is 05 centimeter Use differentials to approimate the possible propagated error in computing the area of the triangle 9 Area The measurement of the radius of the end of a log is found to be inches, with a possible error of inch Use differentials to approimate the possible propagated error in computing the area of the end of the log 0 Volume and Surface Area The measurement of the edge of a cube is found to be inches, with a possible error of 00 inch Use differentials to approimate the maimum possible propagated error in computing (a) the volume of the cube and (b) the surface area of the cube Area The measurement of a side of a square is found to be 5 centimeters, with a possible error of 005 centimeter (a) Approimate the percent error in computing the area of the square (b) Estimate the maimum allowable percent error in measuring the side if the error in computing the area cannot eceed 5% Circumference The measurement of the circumference of a circle is found to be 56 inches, with a possible error of inches (a) Approimate the percent error in computing the area of the circle 5 (, ) 5 f (, ) g g to
SECTION 9 Differentials (b) Estimate the maimum allowable percent error in measuring the circumference if the error in computing the area cannot eceed % Volume and Surface Area The radius of a sphere is measured to be 6 inches, with a possible error of 00 inch Use differentials to approimate the maimum possible error in calculating (a) the volume of the sphere, (b) the surface area of the sphere, and (c) the relative errors in parts (a) and (b) Profit The profit P for a compan is given b P 500 77 000 Approimate the change and percent change in profit as production changes from 5 to 0 units Volume In Eercises 5 and 6, the thickness of each shell is 0 centimeter Use differentials to approimate the volume of each shell 5 0 cm 6 5 cm 0 cm 7 Pendulum The period of a pendulum is given b T L g where L is the length of the pendulum in feet, g is the acceleration due to gravit, and T is the time in seconds The pendulum has been subjected to an increase in temperature such that the length has increased b % (a) Find the approimate percent change in the period (b) Using the result in part (a), find the approimate error in this pendulum clock in da 8 Ohm s Law A current of I amperes passes through a resistor of R ohms Ohm s Law states that the voltage E applied to the resistor is E IR If the voltage is constant, show that the magnitude of the relative error in R caused b a change in I is equal in magnitude to the relative error in I 9 Triangle Measurements The measurement of one side of a right triangle is found to be 95 inches, and the angle opposite that side is with a possible error of 5 (a) Approimate the percent error in computing the length of the hpotenuse (b) Estimate the maimum allowable percent error in measuring the angle if the error in computing the length of the hpotenuse cannot eceed % 0 Area Approimate the percent error in computing the area of the triangle in Eercise 9 65 0 cm 00 cm Projectile Motion The range R of a projectile is where v 0 is the initial velocit in feet per second and is the angle of elevation If v 0 00 feet per second and is changed from 0 to, use differentials to approimate the change in the range Surveing A surveor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as 75 How accuratel must the angle be measured if the percent error in estimating the height of the tree is to be less than 6%? In Eercises 6, use differentials to approimate the value of the epression Compare our answer with that of a calculator 99 6 5 6 6 99 Writing In Eercises 7 and 8, give a short eplanation of wh the approimation is valid 7 0 00 8 tan 005 0 005 In Eercises 9 5, verif the tangent line approimation of the function at the given point Then use a graphing utilit to graph the function and its approimation in the same viewing window 9 50 5 5 R v 0 sin Function f f f tan f Approimation Writing About Concepts Point 0,, 0, 0 0, 5 Describe the change in accurac of d as an approimation for when is decreased 5 When using differentials, what is meant b the terms propagated error, relative error, and percent error? True or False? In Eercises 55 58, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 55 If c, then d d 56 If a b, then dd 57 If is differentiable, then lim d 0 0 58 If f, f is increasing and differentiable, and > 0, then d
CHAPTER Applications of Differentiation Review Eercises for Chapter The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph Give the definition of a critical number, and graph a function f showing the different tpes of critical numbers Consider the odd function f that is continuous and differentiable and has the functional values shown in the table f 5 (a) Determine f (b) Determine f (c) Plot the points and make a possible sketch of the graph of f on the interval 6, 6 What is the smallest number of critical points in the interval? Eplain (d) Does there eist at least one real number c in the interval 6, 6 where fc? Eplain (e) Is it possible that lim f does not eist? Eplain 0 (f) Is it necessar that f eists at? Eplain In Eercises and, find the absolute etrema of the function on the closed interval Use a graphing utilit to graph the function over the given interval to confirm our results g 5 cos, 0, f 0,, In Eercises 5 and 6, determine whether Rolle s Theorem can be applied to f on the closed interval [a, b] If Rolle s Theorem can be applied, find all values of c in the open interval a, b such that fc 0 f 5 f,, 6, 0, f 7 Consider the function (a) Graph the function and verif that f f 7 (b) Note that f is not equal to zero for an in, 7 Eplain wh this does not contradict Rolle s Theorem 8 Can the Mean Value Theorem be applied to the function f on the interval,? Eplain In Eercises 9, find the point(s) guaranteed b the Mean Value Theorem for the closed interval [a, b] 9 f 0 f,, 8, f cos, f,, 0 6 0 0, 0, For the function f A B C, determine the value of c guaranteed b the Mean Value Theorem on the interval, Demonstrate the result of Eercise for f on the interval 0, In Eercises 5 8, find the critical numbers (if an) and the open intervals on which the function is increasing or decreasing 5 6 f g 7 8 h, f sin cos, > 0 0, In Eercises 9 and 0, use the First Derivative Test to find an relative etrema of the function Use a graphing utilit to verif our results 9 0 ht t 8t g sin, Harmonic Motion The height of an object attached to a spring is given b the harmonic equation cos t sin t where is measured in inches and t is measured in seconds (a) Calculate the height and velocit of the object when t 8 second (b) Show that the maimum displacement of the object is inch (c) Find the period P of Also, find the frequenc f (number of oscillations per second) if f P Writing The general equation giving the height of an oscillating object attached to a spring is A sin k m t B cos k m t where k is the spring constant and m is the mass of the object (a) Show that the maimum displacement of the object is A B (b) Show that the object oscillates with a frequenc of f k m In Eercises and, determine the points of inflection and discuss the concavit of the graph of the function f cos, 0, f In Eercises 5 and 6, use the Second Derivative Test to find all relative etrema 5 g 6 ht t t 0, 5
REVIEW EXERCISES Think About It In Eercises 7 and 8, sketch the graph of a function f having the given characteristics 7 f 0 f 6 0 f f5 0 f > 0 if < f > 0 if < < 5 f < 0 if > 5 f < 0 if < or > f > 0 if < < 8 f 0, f 6 0 f < 0 if < or > f does not eist f 0 f > 0 if < < f < 0 if 9 Writing A newspaper headline states that The rate of growth of the national deficit is decreasing What does this mean? What does it impl about the graph of the deficit as a function of time? 0 Inventor Cost The cost of inventor depends on the ordering and storage costs according to the inventor model C Q s r Determine the order size that will minimize the cost, assuming that sales occur at a constant rate, Q is the number of units sold per ear, r is the cost of storing one unit for ear, s is the cost of placing an order, and is the number of units per order Modeling Data Outlas for national defense D (in billions of dollars) for selected ears from 970 through 999 are shown in the table, where t is time in ears, with t 0 corresponding to 970 (Source: US Office of Management and Budget) t D t D 0 5 0 5 0 90 0 55 790 8 5 6 7 8 9 099 07 098 0 0 (a) Use the regression capabilities of a graphing utilit to fit a model of the form D at bt ct dt e to the data (b) Use a graphing utilit to plot the data and graph the model (c) For the ears shown in the table, when does the model indicate that the outla for national defense is at a maimum? When is it at a minimum? (d) For the ears shown in the table, when does the model indicate that the outla for national defense is increasing at the greatest rate? Modeling Data The manager of a store recorded the annual sales S (in thousands of dollars) of a product over a period of 7 ears, as shown in the table, where t is the time in ears, with t 7 corresponding to 997 t S (a) Use the regression capabilities of a graphing utilit to find a model of the form S at bt ct d for the data (b) Use a graphing utilit to plot the data and graph the model (c) Use calculus to find the time t when sales were increasing at the greatest rate (d) Do ou think the model would be accurate for predicting future sales? Eplain In Eercises 0, find the limit lim lim 5 5 5 lim 6 lim 5 5 cos 7 lim 8 lim 6 9 lim 0 lim cos sin In Eercises, find an vertical and horizontal asmptotes of the graph of the function Use a graphing utilit to verif our results h g 5 f f In Eercises 5 8, use a graphing utilit to graph the function Use the graph to approimate an relative etrema or asmptotes 5 f 6 f 7 f 8 g cos cos 7 8 9 0 5 69 5 55 90 0 6 In Eercises 9 66, analze and sketch the graph of the function 9 f 50 f 5 f 6 5 f 5 f 5 f 55 f 56 f 57 f 58 f
CHAPTER Applications of Differentiation 59 60 6 f f f 77 Distance A hallwa of width 6 feet meets a hallwa of width 9 feet at right angles Find the length of the longest pipe that can be carried level around this corner [Hint: If L is the length of the pipe, show that L 6 csc 9 csc 6 66 f f 9 f 6 6 65 f cos, f sin sin, 67 Find the maimum and minimum points on the graph of (a) without using calculus (b) using calculus 68 Consider the function f n for positive integer values of n (a) For what values of n does the function have a relative minimum at the origin? (b) For what values of n does the function have a point of inflection at the origin? 69 Distance At noon, ship A is 00 kilometers due east of ship B Ship A is sailing west at kilometers per hour, and ship B is sailing south at 0 kilometers per hour At what time will the ships be nearest to each other, and what will this distance be? 70 Maimum Area Find the dimensions of the rectangle of maimum area, with sides parallel to the coordinate aes, that can be inscribed in the ellipse given b 6 0 6 0 7 Minimum Length A right triangle in the first quadrant has the coordinate aes as sides, and the hpotenuse passes through the point, 8 Find the vertices of the triangle such that the length of the hpotenuse is minimum 7 Minimum Length The wall of a building is to be braced b a beam that must pass over a parallel fence 5 feet high and feet from the building Find the length of the shortest beam that can be used 7 Maimum Area Three sides of a trapezoid have the same length s Of all such possible trapezoids, show that the one of maimum area has a fourth side of length s 7 Maimum Area Show that the greatest area of an rectangle inscribed in a triangle is one-half that of the triangle 75 Distance Find the length of the longest pipe that can be carried level around a right-angle corner at the intersection of two corridors of widths feet and 6 feet (Do not use trigonometr) 76 Distance Rework Eercise 75, given corridors of widths a meters and b meters where is the angle between the pipe and the wall of the narrower hallwa] 78 Length Rework Eercise 77, given that one hallwa is of width a meters and the other is of width b meters Show that the result is the same as in Eercise 76 Minimum Cost In Eercises 79 and 80, find the speed v, in miles per hour, that will minimize costs on a 0-mile deliver trip The cost per hour for fuel is C dollars, and the driver is paid W dollars per hour (Assume there are no costs other than wages and fuel) 79 Fuel cost: C v 80 Fuel cost: C v 600 500 Driver: W $5 Driver: W $750 In Eercises 8 and 8, use Newton s Method to approimate an real zeros of the function accurate to three decimal places Use the zero or root feature of a graphing utilit to verif our results 8 f 8 f In Eercises 8 and 8, use Newton s Method to approimate, to three decimal places, the -value(s) of the point(s) of intersection of the equations Use a graphing utilit to verif our results 8 8 sin In Eercises 85 and 86, find the differential d 85 cos 86 6 87 Surface Area and Volume The diameter of a sphere is measured to be 8 centimeters, with a maimum possible error of 005 centimeter Use differentials to approimate the possible propagated error and percent error in calculating the surface area and the volume of the sphere 88 Demand Function A compan finds that the demand for its commodit is p 75 If changes from 7 to 8, find and compare the values of p and dp
PS Problem Solving 5 PS Problem Solving The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph Graph the fourth-degree polnomial p a for various values of the constant a (a) Determine the values of a for which p has eactl one relative minimum (b) Determine the values of a for which p has eactl one relative maimum (c) Determine the values of a for which p has eactl two relative minima (d) Show that the graph of p cannot have eactl two relative etrema (a) Graph the fourth-degree polnomial p a 6 for a,,, 0,,, and For what values of the constant a does p have a relative minimum or relative maimum? (b) Show that p has a relative maimum for all values of the constant a (c) Determine analticall the values of a for which p has a relative minimum (d) Let,, p be a relative etremum of p Show that, lies on the graph of Verif this result graphicall b graphing together with the seven curves from part (a) Let f c Determine all values of the constant c such that f has a relative minimum, but no relative maimum (a) Let f a b c, a 0, be a quadratic polnomial How man points of inflection does the graph of f have? (b) Let f a b c d, a 0, be a cubic polnomial How man points of inflection does the graph of f have? (c) Suppose the function f satisfies the equation d where k and L are positive constants Show that the graph of f has a point of inflection at the point where L (This equation is called the logistic differential equation) 5 Prove Darbou s Theorem: Let f be differentiable on the closed interval a, b such that fa and fb If d lies between and, then there eists c in a, b such that fc d 6 Let f and g be functions that are continuous on a, b and differentiable on a, b Prove that if f a ga and g > f for all in a, b, then gb > fb 7 Prove the following Etended Mean Value Theorem If f and are continuous on the closed interval a, b, and if eists in the open interval a, b, then there eists a number c in a, b such that f d k L f b f a fab a fcb a f 8 (a) Let V Find dv and V Show that for small values of, the difference V dv is ver small in the sense that there eists such that V dv, where 0 as 0 (b) Generalize this result b showing that if f is a differentiable function, then d, where 0 as 0 9 The amount of illumination of a surface is proportional to the intensit of the light source, inversel proportional to the square of the distance from the light source, and proportional to sin, where is the angle at which the light strikes the surface A rectangular room measures 0 feet b feet, with a 0-foot ceiling Determine the height at which the light should be placed to allow the corners of the floor to receive as much light as possible 0 Consider a room in the shape of a cube, meters on each side A bug at point P wants to walk to point Q at the opposite corner, as shown in the figure Use calculus to determine the shortest path Can ou solve the problem without calculus? P The line joining P and Q crosses the two parallel lines, as shown in the figure The point R is d units from P How far from Q should the point S be chosen so that the sum of the areas of the two shaded triangles is a minimum? So that the sum is a maimum? P m S d m R Q m Q ft d ft θ 0 f 5 ft
6 CHAPTER Applications of Differentiation The figures show a rectangle, a circle, and a semicircle inscribed in a triangle bounded b the coordinate aes and the first-quadrant portion of the line with intercepts, 0 and 0, Find the dimensions of each inscribed figure such that its area is maimum State whether calculus was helpful in finding the required dimensions Eplain our reasoning (a) Prove that lim (b) Prove that lim 0 (c) Let L be a real number Prove that if lim f L, then Find the point on the graph of (see figure) where the tangent line has the greatest slope, and the point where the tangent line has the least slope lim f 0 L r r r = + 5 (a) Let be a positive number Use the table feature of a graphing utilit to verif that < (b) Use the Mean Value Theorem to prove that < for all positive real numbers 6 (a) Let be a positive number Use the table feature of a graphing utilit to verif that sin < (b) Use the Mean Value Theorem to prove that sin < for all positive real numbers 7 The police department must determine the speed limit on a bridge such that the flow rate of cars is maimum per unit time The greater the speed limit, the farther apart the cars must be in order to keep a safe stopping distance Eperimental data on the stopping distance d (in meters) for various speeds v (in kilometers per hour) are shown in the table v 0 0 60 80 00 d 5 7 7 66 r (a) Convert the speeds v in the table to the speeds s in meters per second Use the regression capabilities of a graphing utilit to find a model of the form ds as bs c for the data (b) Consider two consecutive vehicles of average length 55 meters, traveling at a safe speed on the bridge Let T be the difference between the times (in seconds) when the front bumpers of the vehicles pass a given point on the bridge Verif that this difference in times is given b (c) Use a graphing utilit to graph the function T and estimate the speed s that minimizes the time between vehicles (d) Use calculus to determine the speed that minimizes T What is the minimum value of T? Convert the required speed to kilometers per hour (e) Find the optimal distance between vehicles for the posted speed limit determined in part (d) 8 A legal-sized sheet of paper (85 inches b inches) is folded so that corner P touches the opposite inch edge at R Note: PQ C P T ds s (a) Show that C 85 (b) What is the domain of C? (c) Determine the -value that minimizes C (d) Determine the minimum length C 9 The polnomial P c 0 c a c a is the quadratic approimation of the function f at a, f a if Pa f a, Pa fa, and Pa fa (a) Find the quadratic approimation of f C 55 s R in Q 85 in at 0, 0 (b) Use a graphing utilit to graph P and f in the same viewing window