Tangent Line Approximations. y f c f c x c. y f c f c x c. Find the tangent line approximation of. f x 1 sin x

Size: px
Start display at page:

Download "Tangent Line Approximations. y f c f c x c. y f c f c x c. Find the tangent line approximation of. f x 1 sin x"

Transcription

1 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 f sin 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

2 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 f Now, using 00, the change in is f f f0 f Differential of 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 and g0 0 0

3 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%

4 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

5 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 g() = + 8 f() = (6, ) 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 and g 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

6 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, (, ),, f 5, f,,, 5 6 f sin, f csc,, sin, csc d 0 d 0 d 00 d 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 (, ) 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

7 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 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 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 Writing In Eercises 7 and 8, give a short eplanation of wh the approimation is valid tan 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 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 If f, f is increasing and differentiable, and > 0, then d

8 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, 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

9 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 (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 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 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

10 CHAPTER Applications of Differentiation 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 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 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 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 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

11 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

12 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 d 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

Tangent Line Approximations

Tangent Line Approximations 60_009.qd //0 :8 PM Page SECTION.9 Dierentials Section.9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph. In the same viewing window, graph the tangent line to the graph o at the point,.

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval................... 0 Section. Rolle s Theorem and the Mean Value Theorem...... 0 Section. Increasing and Decreasing Functions and

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative

More information

Review Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4.

Review Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4. Review Eercises for Chapter 6. r v 0 sin. Let f, 00, d 0.6. v 0 00 ftsec changes from 0 to dr 00 cos d 6 0 d 0 r dr 80 00 6 96 feet 80 cos 0 96 feet 8080 f f fd d f 99. 00 0.6 9.97 00 Using a calculator:

More information

Increasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video

Increasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl

More information

Infinite Limits. Let f be the function given by. f x 3 x 2.

Infinite Limits. Let f be the function given by. f x 3 x 2. 0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 78 Section. Rolle s Theorem and the Mean Value Theorem. 8 Section. Increasing and Decreasing Functions and the First

More information

1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs

1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs 0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals

More information

One of the most common applications of Calculus involves determining maximum or minimum values.

One of the most common applications of Calculus involves determining maximum or minimum values. 8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..

More information

Finding Limits Graphically and Numerically. An Introduction to Limits

Finding Limits Graphically and Numerically. An Introduction to Limits 8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use

More information

Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?

Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5? 3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear

More information

Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)

Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.) Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,

More information

Fitting Integrands to Basic Rules. x x 2 9 dx. Solution a. Use the Arctangent Rule and let u x and a dx arctan x 3 C. 2 du u.

Fitting Integrands to Basic Rules. x x 2 9 dx. Solution a. Use the Arctangent Rule and let u x and a dx arctan x 3 C. 2 du u. 58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration rules Fitting Integrands

More information

5.5 Worksheet - Linearization

5.5 Worksheet - Linearization AP Calculus 4.5 Worksheet 5.5 Worksheet - Linearization All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. Consider the function = sin. a) Find the equation

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Properties of Limits

Properties of Limits 33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate

More information

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

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

More information

AP Calc AB First Semester Review

AP Calc AB First Semester Review AP Calc AB First Semester Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. 1) lim (7-7) 7 A) -4 B) -56 C) 4 D) 56 1) Determine

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Choose the answer that represents the graph of its derivative.

More information

Finding Limits Graphically and Numerically. An Introduction to Limits

Finding Limits Graphically and Numerically. An Introduction to Limits 60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Chapter Practice Dicsclaimer: The actual eam is different. On the actual eam ou must show the correct reasoning to receive credit for the question. SHORT ANSWER. Write the word or phrase that best completes

More information

MATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MATH 0/GRACEY PRACTICE FINAL Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the graph that represents the given function without using

More information

Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE

Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test

More information

P.4 Lines in the Plane

P.4 Lines in the Plane 28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables

More information

(b) Find the difference quotient. Interpret your result. 3. Find the average rate of change of ƒ(x) = x 2-3x from

(b) Find the difference quotient. Interpret your result. 3. Find the average rate of change of ƒ(x) = x 2-3x from 6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 67 CHAPTER Summar 67 69. ƒ() = 3 70. ƒ() = -5 (b) Find the difference quotient. Interpret our result. 7. ƒ() = - 7. ƒ() = 0 73. ƒ() = + 74. ƒ() = -3 + 4 75. ƒ() =

More information

PreCalculus Final Exam Review Revised Spring 2014

PreCalculus Final Exam Review Revised Spring 2014 PreCalculus Final Eam Review Revised Spring 0. f() is a function that generates the ordered pairs (0,0), (,) and (,-). a. If f () is an odd function, what are the coordinates of two other points found

More information

Math 2413 General Review for Calculus Last Updated 02/23/2016

Math 2413 General Review for Calculus Last Updated 02/23/2016 Math 243 General Review for Calculus Last Updated 02/23/206 Find the average velocity of the function over the given interval.. y = 6x 3-5x 2-8, [-8, ] Find the slope of the curve for the given value of

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

Derivatives of Multivariable Functions

Derivatives of Multivariable Functions Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit

More information

Review Exercises for Chapter 2

Review Exercises for Chapter 2 Review Eercises for Chapter 367 Review Eercises for Chapter. f 1 1 f f f lim lim 1 1 1 1 lim 1 1 1 1 lim 1 1 lim lim 1 1 1 1 1 1 1 1 1 4. 8. f f f f lim lim lim lim lim f 4, 1 4, if < if (a) Nonremovable

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus.4 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. What is a difference quotient?. How do you find the slope of a curve (aka slope

More information

Find the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis

Find the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais

More information

I. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.

I. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3. 0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.

More information

9.1 Practice A. Name Date sin θ = and cot θ = to sketch and label the triangle. Then evaluate. the other four trigonometric functions of θ.

9.1 Practice A. Name Date sin θ = and cot θ = to sketch and label the triangle. Then evaluate. the other four trigonometric functions of θ. .1 Practice A In Eercises 1 and, evaluate the si trigonometric functions of the angle. 1.. 8 1. Let be an acute angle of a right triangle. Use the two trigonometric functions 10 sin = and cot = to sketch

More information

AP Calculus AB/BC ilearnmath.net

AP Calculus AB/BC ilearnmath.net CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or

More information

Basic Differentiation Rules and Rates of Change. The Constant Rule

Basic Differentiation Rules and Rates of Change. The Constant Rule 460_00.q //04 4:04 PM Page 07 SECTION. Basic Differentiation Rules an Rates of Change 07 Section. The slope of a horizontal line is 0. Basic Differentiation Rules an Rates of Change Fin the erivative of

More information

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a: SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert

More information

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least Problems to practice for FINAL. Below is the graph of a function () At which of the marked values ( and ) is: (a) () greatest = (b) () least = (c) () the greatest = (d) () the least = (e) () = = (f) ()

More information

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites: 8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in

More information

Exam. Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Exam. Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given epression (), sketch the general shape of the graph of = f(). [Hint: it ma

More information

Functions. Introduction CHAPTER OUTLINE

Functions. Introduction CHAPTER OUTLINE Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of

More information

4.3 Mean-Value Theorem and Monotonicity

4.3 Mean-Value Theorem and Monotonicity .3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What is the definition of a derivative?. What is the alternative definition of a

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

Quadratic Functions and Models

Quadratic Functions and Models Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Use quadratic functions

More information

Study Guide and Intervention

Study Guide and Intervention 6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a

More information

Unit 10 - Graphing Quadratic Functions

Unit 10 - Graphing Quadratic Functions Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( ) Water seeps out of a conical filter at the constant rate of 5 cc / sec. When the height of water level in the cone is 5 cm, find the rate at which the height

More information

4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4

4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward

More information

Chapter 1 Prerequisites for Calculus

Chapter 1 Prerequisites for Calculus Section. Chapter Prerequisites for Calculus Section. Lines (pp. ) Quick Review.. + ( ) + () +. ( +). m. m ( ) ( ). (a) ( )? 6 (b) () ( )? 6. (a) 7? ( ) + 7 + Yes (b) ( ) + 9 No Yes No Section. Eercises.

More information

In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work.

In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work. Calculus I Eam File Fall 7 Test # In #-5, find the indicated limits. For each one, if it does not eist, tell why not. Show all necessary work. lim sin.) lim.) 3.) lim 3 3-5 4 cos 4.) lim 5.) lim sin 6.)

More information

1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS

1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS .6 Continuit of Trigonometric, Eponential, and Inverse Functions.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS In this section we will discuss the continuit properties of trigonometric

More information

3.2 Introduction to Functions

3.2 Introduction to Functions 8 CHAPTER Graphs and Functions Write each statement as an equation in two variables. Then graph each equation. 97. The -value is more than three times the -value. 98. The -value is - decreased b twice

More information

+ 2 on the interval [-1,3]

+ 2 on the interval [-1,3] Section.1 Etrema on an Interval 1. Understand the definition of etrema of a function on an interval.. Understand the definition of relative etrema of a function on an open interval.. Find etrema on a closed

More information

CHAPTER 2: Partial Derivatives. 2.2 Increments and Differential

CHAPTER 2: Partial Derivatives. 2.2 Increments and Differential CHAPTER : Partial Derivatives.1 Definition of a Partial Derivative. Increments and Differential.3 Chain Rules.4 Local Etrema.5 Absolute Etrema 1 Chapter : Partial Derivatives.1 Definition of a Partial

More information

Find the following limits. For each one, if it does not exist, tell why not. Show all necessary work.

Find the following limits. For each one, if it does not exist, tell why not. Show all necessary work. Calculus I Eam File Spring 008 Test #1 Find the following its. For each one, if it does not eist, tell why not. Show all necessary work. 1.) 4.) + 4 0 1.) 0 tan 5.) 1 1 1 1 cos 0 sin 3.) 4 16 3 1 6.) For

More information

Functions. Introduction

Functions. Introduction Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of

More information

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1 Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

More information

This is only a list of questions use a separate sheet to work out the problems. 1. (1.2 and 1.4) Use the given graph to answer each question.

This is only a list of questions use a separate sheet to work out the problems. 1. (1.2 and 1.4) Use the given graph to answer each question. Mth Calculus Practice Eam Questions NOTE: These questions should not be taken as a complete list o possible problems. The are merel intended to be eamples o the diicult level o the regular eam questions.

More information

2018 Midterm Review Trigonometry: Midterm Review A Missive from the Math Department Trigonometry Work Problems Study For Understanding Read Actively

2018 Midterm Review Trigonometry: Midterm Review A Missive from the Math Department Trigonometry Work Problems Study For Understanding Read Actively Summer . Fill in the blank to correctl complete the sentence..4 written in degrees and minutes is..4 written in degrees and minutes is.. Find the complement and the supplement of the given angle. The complement

More information

CHAPTER P Preparation for Calculus

CHAPTER P Preparation for Calculus CHAPTER P Preparation for Calculus Section P. Graphs and Models...................... Section P. Linear Models and Rates of Change............ Section P. Functions and Their Graphs................. Section

More information

AP Calculus AB Information and Summer Assignment

AP Calculus AB Information and Summer Assignment AP Calculus AB Information and Summer Assignment General Information: Competency in Algebra and Trigonometry is absolutely essential. The calculator will not always be available for you to use. Knowing

More information

Differentiation and applications

Differentiation and applications FS O PA G E PR O U N C O R R EC TE D Differentiation and applications. Kick off with CAS. Limits, continuit and differentiabilit. Derivatives of power functions.4 C oordinate geometr applications of differentiation.5

More information

Additional Topics in Differential Equations

Additional Topics in Differential Equations 6 Additional Topics in Differential Equations 6. Eact First-Order Equations 6. Second-Order Homogeneous Linear Equations 6.3 Second-Order Nonhomogeneous Linear Equations 6.4 Series Solutions of Differential

More information

D u f f x h f y k. Applying this theorem a second time, we have. f xx h f yx k h f xy h f yy k k. f xx h 2 2 f xy hk f yy k 2

D u f f x h f y k. Applying this theorem a second time, we have. f xx h f yx k h f xy h f yy k k. f xx h 2 2 f xy hk f yy k 2 93 CHAPTER 4 PARTIAL DERIVATIVES We close this section b giving a proof of the first part of the Second Derivatives Test. Part (b) has a similar proof. PROOF OF THEOREM 3, PART (A) We compute the second-order

More information

Methods for Advanced Mathematics (C3) Coursework Numerical Methods

Methods for Advanced Mathematics (C3) Coursework Numerical Methods Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on

More information

Additional Topics in Differential Equations

Additional Topics in Differential Equations 0537_cop6.qd 0/8/08 :6 PM Page 3 6 Additional Topics in Differential Equations In Chapter 6, ou studied differential equations. In this chapter, ou will learn additional techniques for solving differential

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the graph to evaluate the limit. ) lim 0 f() ) - - - - - - - - A) - B) 0 C)does not eist

More information

1. sin 2. Honors Pre-Calculus Final Exam Review 2 nd semester June TRIGONOMETRY Solve for 0 2. without using a calculator: 2. csc 2 3.

1. sin 2. Honors Pre-Calculus Final Exam Review 2 nd semester June TRIGONOMETRY Solve for 0 2. without using a calculator: 2. csc 2 3. Honors Pre-Calculus Name: Final Eam Review nd semester June 05 TRIGONOMETRY Solve for 0 without using a calculator:. sin. csc. tan. cos ) ) ) ) Solve for in degrees giving all solutions. 5. sin 6. cos

More information

Analytic Trigonometry

Analytic Trigonometry CHAPTER 5 Analtic Trigonometr 5. Fundamental Identities 5. Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines It

More information

Homework Assignments Math /02 Spring 2015

Homework Assignments Math /02 Spring 2015 Homework Assignments Math 1-01/0 Spring 015 Assignment 1 Due date : Frida, Januar Section 5.1, Page 159: #1-, 10, 11, 1; Section 5., Page 16: Find the slope and -intercept, and then plot the line in problems:

More information

Applications of Derivatives

Applications of Derivatives 58_Ch04_pp86-60.qd /3/06 :35 PM Page 86 Chapter 4 Applications of Derivatives A n automobile s gas mileage is a function of man variables, including road surface, tire tpe, velocit, fuel octane rating,

More information

1. sin 2. csc 2 3. tan 1 2. Cos 8) Sin 10. sec. Honors Pre-Calculus Final Exam Review 2 nd semester. TRIGONOMETRY Solve for 0 2

1. sin 2. csc 2 3. tan 1 2. Cos 8) Sin 10. sec. Honors Pre-Calculus Final Exam Review 2 nd semester. TRIGONOMETRY Solve for 0 2 Honors Pre-Calculus Final Eam Review nd semester Name: TRIGONOMETRY Solve for 0 without using a calculator: 1 1. sin. csc 3. tan 1 4. cos 1) ) 3) 4) Solve for in degrees giving all solutions. 5. sin 1

More information

I K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists

I K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists LEVEL I. A circular metal plate epands under heating so that its radius increase by %. Find the approimate increase in the area of the plate, if the radius of the plate before heating is 0cm.. The length

More information

PRECALCULUS FINAL EXAM REVIEW

PRECALCULUS FINAL EXAM REVIEW PRECALCULUS FINAL EXAM REVIEW Evaluate the function at the indicated value of. Round our result to three decimal places.. f () 4(5 ); 0.8. f () e ; 0.78 Use the graph of f to describe the transformation

More information

Try It Exploration A Exploration B Open Exploration. Fitting Integrands to Basic Rules. A Comparison of Three Similar Integrals

Try It Exploration A Exploration B Open Exploration. Fitting Integrands to Basic Rules. A Comparison of Three Similar Integrals 58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Basic Integration Rules Review procedures for fitting an integrand to one of the basic integration rules Fitting

More information

Multiple Choice. Circle the best answer. No work needed. No partial credit available. is continuous.

Multiple Choice. Circle the best answer. No work needed. No partial credit available. is continuous. Multiple Choice. Circle the best answer. No work needed. No partial credit available. + +. Evaluate lim + (a (b (c (d 0 (e None of the above.. Evaluate lim (a (b (c (d 0 (e + + None of the above.. Find

More information

CHAPTER 3 Polynomial Functions

CHAPTER 3 Polynomial Functions CHAPTER Polnomial Functions Section. Quadratic Functions and Models............. 7 Section. Polnomial Functions of Higher Degree......... 7 Section. Polnomial and Snthetic Division............ Section.

More information

AP Calculus AB Summer Assignment Mrs. Berkson

AP Calculus AB Summer Assignment Mrs. Berkson AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre- Calculus skills required for AP Calculus AB. Net ear we will be starting off the

More information

CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions

CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Differentiation.... 9 Section 5. The Natural Logarithmic Function: Integration...... 98

More information

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity. Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive

More information

CHAPTER 2 Polynomial and Rational Functions

CHAPTER 2 Polynomial and Rational Functions CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............

More information

Instructor: Imelda Valencia Course: A3 Honors Pre Calculus

Instructor: Imelda Valencia Course: A3 Honors Pre Calculus Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression

More information

Identifying end behavior of the graph of a polynomial function

Identifying end behavior of the graph of a polynomial function 56 Polnomial Functions 3.1.1 Eercises For a link to all of the additional resources available for this section, click OSttS Chapter 3 materials. In Eercises 1-10, find the degree, the leading term, the

More information

Review of Exponent Rules

Review of Exponent Rules Page Review of Eponent Rules Math : Unit Radical and Rational Functions Rule : Multipling Powers With the Same Base Multipl Coefficients, Add Eponents. h h h. ( )( ). (6 )(6 ). (m n )(m n ). ( 8ab)( a

More information

Technical Calculus I Homework. Instructions

Technical Calculus I Homework. Instructions Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the

More information

With the Chain Rule. y x 2 1. and. with respect to second axle. dy du du dx. Rate of change of first axle. with respect to third axle

With the Chain Rule. y x 2 1. and. with respect to second axle. dy du du dx. Rate of change of first axle. with respect to third axle 0 CHAPTER Differentiation Section The Chain Rule Fin the erivative of a composite function using the Chain Rule Fin the erivative of a function using the General Power Rule Simplif the erivative of a function

More information

9 (0, 3) and solve equations to earn full credit.

9 (0, 3) and solve equations to earn full credit. Math 0 Intermediate Algebra II Final Eam Review Page of Instructions: (6, ) Use our own paper for the review questions. For the final eam, show all work on the eam. (-6, ) This is an algebra class do not

More information

Fitting Integrands to Basic Rules

Fitting Integrands to Basic Rules 6_8.qd // : PM Page 8 8 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration

More information

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

More information

Final Exam Review / AP Calculus AB

Final Exam Review / AP Calculus AB Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f 5 4 1 y - - -1 - - -4-5 ) lim f(), - lim f(), and + lim f 8 6 4 y -4 - - -1-1 4 5-4

More information

Derivatives of Multivariable Functions

Derivatives of Multivariable Functions Chapter 10 Derivatives of Multivariable Functions 10.1 Limits Motivating Questions What do we mean b the limit of a function f of two variables at a point (a, b)? What techniques can we use to show that

More information

Precalculus Honors - AP Calculus A Information and Summer Assignment

Precalculus Honors - AP Calculus A Information and Summer Assignment Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou

More information

2Polynomial and. Rational Functions

2Polynomial and. Rational Functions Polnomial and Rational Functions A ballista was used in ancient times as a portable rock-throwing machine. Its primar function was to destro the siege weaponr of opposing forces. Skilled artiller men aimed

More information

Section 3.3 Graphs of Polynomial Functions

Section 3.3 Graphs of Polynomial Functions 3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section

More information

Review for Test 2 Calculus I

Review for Test 2 Calculus I Review for Test Calculus I Find the absolute etreme values of the function on the interval. ) f() = -, - ) g() = - + 8-6, ) F() = -,.5 ) F() =, - 6 5) g() = 7-8, - Find the absolute etreme values of the

More information

CHAPTER 6 Applications of Integration

CHAPTER 6 Applications of Integration PART II CHAPTER Applications of Integration Section. Area of a Region Between Two Curves.......... Section. Volume: The Disk Method................. 7 Section. Volume: The Shell Method................

More information

v t t t t a t v t d dt t t t t t 23.61

v t t t t a t v t d dt t t t t t 23.61 SECTION 4. MAXIMUM AND MINIMUM VALUES 285 The values of f at the endpoints are f 0 0 and f 2 2 6.28 Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value

More information