OptimizationManufacturing tires Your company can

Problem 126PE Applications and Examples Can the integrations in (a) and (b) both be correct? Explain.

Problem 6AAE If for all x, what is the most the values of ƒ can increase on [0, 6]? Give reasons for your answer.

Problem 4QGY What are the hypotheses and conclusion of Rolle’s Theorem? Are the hypotheses really necessary? Explain.

Problem 4AAE Functions and Derivatives Local extrema Use the sign pattern for the derivative to identify the points where ƒ has local maximum and minimum values.

Problem 4PE Extreme Values Find values of a and b such that the function has a local extreme value of 1 at x = 3. Is this extreme value a local maximum, or a local minimum? Give reasons for your answer.

Problem 7AAE Bounding a function Suppose that ƒ is continuous on [a, b] and that c is an interior point of the interval. Show that if on [a, c) and on (c, b], then ƒ(x) is never less than ƒ(c) on [a, b]. Bounding a function Suppose that ƒ is continuous on [a, b] and that c is an interior point of the interval. Show that if on on (c, b], then ƒ(x) is never less than ƒ(c) on [a, b].

Problem 7PE Extreme Values Find the absolute maximum and absolute minimum values of ƒ over the interval.

Problem 6QGY State the Mean Value Theorem’s three corollaries.

Problem 8AAE An inequality a. Show that for every value of x. b. Suppose that ƒ is a function whose derivative is Use the result in part (a) to show that for any a and b.

Problem 21PE The Mean Value Theorem Show that even though Doesn’t this contradict Corollary 2 of the Mean Value Theorem? Give reasons for your answer

Problem 21QGY Can a function have more than one antiderivative? If so, how are the antiderivatives related? Explain.

Problem 23PE Analyzing Graphs Use the graph to answer the questions. Identify any global extreme values of ƒ and the values of x at which they occur.

Problem 11QGY What is the Second Derivative Test for Local Extreme Values? Give examples of how it is applied.

Problem 23QGY How can you sometimes solve a differential equation of the form

Problem 24AAE Theory and Examples The family of straight lines y = ax + b (a, b arbitrary constants) can be characterized by the relation Find a similar relation satisfied by the family of all circles where h and r are arbitrary constants. (Hint: Eliminate h and r from the set of three equations including the given one and two obtained by successive differentiation.)

Problem 24PE Analyzing Graphs Estimate the intervals on which the function is a. increasing. b. decreasing. c. Use the given graph of to indicate where any local extreme values of the function occur, and whether each extreme is a relative maximum or minimum.

Problem 11PE Extreme Values The function does not take on either a maximum or a minimum on the interval even though the function is continuous on this interval. Does this contradict the Extreme Value Theorem for continuous functions? Why?

Problem 26PE Analyzing Graphs Each of the graphs in Exercises 25 and 26 is the graph of the position function of an object moving on a coordinate line (t represents time). At approximately what times (if any) is each object’s (a) velocity equal to zero? (b) acceleration equal to zero? During approximately what time intervals does the object move (c) forward? (d) backward?

Problem 29AAE Theory and Examples Can there be a curve satisfying the following conditions? is everywhere equal to zero and, when x = 0, y = 0 and dy/dx = 1.Give a reason for your answer

Problem 30AAE Theory and Examples Find the equation for the curve in the xy-plane that passes through the point (1, -1) if its slope at x is always 3x2 + 2.

Graph the curves in Exercises 27– 42. \(y=-x^{3}+6 x^{2}-9 x+3\) Equation Transcription: Text Transcription: y=-x^3+6 x^2-9x + 3

Problem 54PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 52PE Graphs and Graphing Graph each function. Then use the function’s first derivative to explain what you see.

Problem 53PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 59PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 60PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 61PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 58PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 62PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 63PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 64PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 65PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 66PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 67PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 68PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Use l’Hôpital’s Rule to find the limits in Exercises 61–72. \(\lim_{x\rightarrow0}(\csc\ x-\cot\ x)\) Equation Transcription: Text Transcription: lim_x rightarrow 0 (csc x-cot x)

Find the limits in Exercises 73–84. \(\lim _{x \rightarrow 0} \frac{10^{x}-1}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 10^x - 1 / x

Problem 74PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 103PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 75PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 104PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 109PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 105PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 110PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 112PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 111PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 114PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 113PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 116PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 115PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 117PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 118PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 119PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 120PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 121PE Initial Value Problems Solve the initial value problems.

Problem 122PE Initial Value Problems Solve the initial value problems.

Problem 125PE Applications and Examples Can the integrations in (a) and (b) both be correct? Explain.

Problem 124PE Initial Value Problems Solve the initial value problems.

Problem 7QGY How can you sometimes identify a function ƒ(x) by knowing and knowing the value of ƒ at a point Give an example.

Problem 24QGY What is an initial value problem? How do you solve one? Give an example.

Problem 8QGY What is the First Derivative Test for Local Extreme Values? Give examples of how it is applied.

Problem 8PE Extreme Values Find the absolute maximum and absolute minimum values of ƒ over the interval.

Problem 25PE Analyzing Graphs Each of the graphs in Exercises 25 and 26 is the graph of the position function of an object moving on a coordinate line (t represents time). At approximately what times (if any) is each object’s (a) velocity equal to zero? (b) acceleration equal to zero? During approximately what time intervals does the object move (c) forward? (d) backward?

Problem 25AAE Theory and Examples Free fall in the fourteenth century In the middle of the fourteenth century, Albert of Saxony (1316–1390) proposed a model of free fall that assumed that the velocity of a falling body was proportional to the distance fallen. It seemed reasonable to think that a body that had fallen 20 ft might be moving twice as fast as a body that had fallen 10 ft. And besides, none of the instruments in use at the time were accurate enough to prove otherwise. Today we can see just how far off Albert of Saxony’s model was by solving the initial value problem implicit in his model. Solve the problem and compare your solution graphically with the equation You will see that it describes a motion that starts too slowly at first and then becomes too fast too soon to be realistic.

Problem 25QGY If you know the acceleration of a body moving along a coordinate line as a function of time, what more do you need to know to find the body’s position function? Give an example.

Problem 9AAE The derivative of but ƒ is not a constant function. Doesn’t this contradict the corollary of the Mean Value Theorem that says that functions with zero derivatives are constant? Give reasons for your answer.

Problem 9PE Extreme Values The greatest integer function defined for all values of x, assumes a local maximum value of 0 at each point of [0, 1). Could any of these local maximum values also be local minimum values of ƒ? Give reasons for your answer.

Problem 9QGY How do you test a twice-differentiable function to determine where its graph is concave up or concave down? Give examples.

Problem 26AAE Theory and Examples Group blood testing During World War II it was necessary to administer blood tests to large numbers of recruits. There are two standard ways to administer a blood test to N people. In method 1, each person is tested separately. In method 2, the blood samples of x people are pooled and tested as one large sample. If the test is negative, this one test is enough for all x people. If the test is positive, then each of the x people is tested separately, requiring a total of x + 1 tests. Using the second method and some probability theory it can be shown that, on the average, the total number of tests y will be With q = 0.99 and N = 1000, find the integer value of x that minimizes y. Also find the integer value of x that maximizes y. (This second result is not important to the real-life situation.) The group testing method was used in World War II with a savings of 80% over the individual testing method, but not with the given value of q.

Problem 27AAE Theory and Examples Assume that the brakes of an automobile produce a constant deceleration of k ft/sec2 (a) Determine what k must be to bring an automobile traveling 60 mi/hr (88 ft/sec ) to rest in a distance of 100 ft from the point where the brakes are applied. (b) With the same k, how far would a car traveling 30 mi/hr travel before being brought to a stop?

Problem 6PE Extreme Values Does have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.

Problem 23AAE Theory and Examples To find we apply Newton’s method to Here we assume that a is a positive real number and q is a positive integer. Show that is a “weighted average” of and find the coefficients such that What conclusion would you reach if were equal? What would be the value of x1 in that case?

Problem 10AAE Extrema and inflection points Let h = ƒg be the product of two differentiable functions of x. a. If ƒ and g are positive, with local maxima at x = a, and if and change sign at a, does h have a local maximum at a? b. If the graphs of ƒ and g have inflection points at x = a does the graph of h have an inflection point at a?

Problem 10PE Extreme Values a. Give an example of a differentiable function ƒ whose first derivative is zero at some point c even though ƒ has neither a local maximum nor a local minimum at c. b. How is this consistent with Theorem 2 in Section 4.1? Give reasons for your answer.

Problem 10QGY What is an inflection point? Give an example. What physical significance do inflection points sometimes have?

Problem 27PE Theory and Examples Assume that the brakes of an automobile produce a constant deceleration of k ft/sec2 (a) Determine what k must be to bring an automobile traveling 60 mi/hr (88 ft/sec ) to rest in a distance of 100 ft from the point where the brakes are applied. (b) With the same k, how far would a car traveling 30 mi/hr travel before being brought to a stop?

Problem 28AAE Theory and Examples Let ƒ(x), g(x) be two continuously differentiable functions satisfying the relationships Let If h(0) = 5, find h(10).

Problem 28PE Graphs and Graphing Graph the curves

Problem 70PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 71PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 72PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 123PE Initial Value Problems Solve the initial value problems.

Problem 21AAE Theory and Examples Suppose that it costs a company y = a + bx dollars to produce x units per week. It can sell x units per week at a price of P = c - ex dollars per unit. Each of a, b, c, and e represents a positive constant. (a) What production level maximizes the profit? (b) What is the corresponding price? (c) What is the weekly profit at this level of production? (d) At what price should each item be sold to maximize profits if the government imposes a tax of t dollars per item sold? Comment on the difference between this price and the price before the tax.

Problem 11AAE In either case, if the answer is yes, give a proof. If the answer is no, give a counterexample. Finding a function Use the following information to find the values of a, b, and c in the formula ƒ(x) = (x + a)/ (bx2 + cx + 2) . i) The values of a, b, and c are either 0 or 1. ii) The graph of ƒ passes through the point( -1, 0) . iii) The line is an asymptote of the graph of ƒ.

Problem 12AAE In either case, if the answer is yes, give a proof. If the answer is no, give a counterexample. Horizontal tangent For what value or values of the constant k will the curve have exactly one horizontal tangent?

Problem 12PE Extreme Values What are the maximum and minimum values of the function on the interval Notice that the interval is not closed. Is this consistent with the Extreme Value Theorem for continuous functions? Why?

Problem 12QGY What do the derivatives of a function tell you about the shape of its graph?

Problem 30PE Graphs and Graphing Graph the curves

Problem 31AAE Theory and Examples A particle moves along the x-axis. Its acceleration is At t = 0,the particle is at the origin. In the course of its motion, it reaches the point x = b,where b>0 but no point beyond b. Determine its velocity at t = 0.

Problem 31PE Graphs and Graphing Graph the curves

Problem 76PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 77PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 78PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 127PE Applications and Examples The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve What dimensions give the rectangle its largest area, and what is that area?

Problem 128PE Applications and Examples The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve What dimensions give the rectangle its largest area, and what is that area?

Problem 129PE Applications and Examples Find the absolute maximum and minimum values of each function on the given interval.

Problem 16AAE Optimization Kicking a field goal An American football player wants to kick a field goal with the ball being on a right hash mark. Assume that the goal posts are b feet apart and that the hash mark line is a distance a > 0 feet from the right goal post. (See the accompanying figure.) Find the distance h from the goal post line that gives the kicker his largest angle Assume that the football field is flat.

Problem 16QGY Outline a general strategy for solving max-min problems. Give examples.

Problem 37PE Graphs and Graphing Graph the curves

Problem 3AAE Functions and Derivatives Can you conclude anything about the extreme values of a continuous function on an open interval? On a half-open interval? Give reasons for your answer.

Problem 3PE Extreme Values Does have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.

Problem 3QGY How do you find the absolute extrema of a continuous function on a closed interval? Give examples.

Problem 20AAE Limits L’Hôpital’s Rule does not help with the following limits. Find them some other way.

Problem 20PE The Mean Value Theorem The formula gives a different function for each value of C. All of these functions, however, have the same derivative with respect to x, namely Are these the only differentiable functions whose derivative is 3? Could there be any others? Give reasons for your answers.

Problem 20QGY Describe Newton’s method for solving equations. Give an example. What is the theory behind the method? What are some of the things to watch out for when you use the method?

Problem 49PE Graphs and Graphing Graph each function. Then use the function’s first derivative to explain what you see.

Problem 50PE Graphs and Graphing Graph each function. Then use the function’s first derivative to explain what you see.

Problem 51PE Graphs and Graphing Graph each function. Then use the function’s first derivative to explain what you see.

Problem 100PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation. .

Problem 101PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 102PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 13AAE Optimization Largest inscribed triangle Points A and B lie at the ends of a diameter of a unit circle and point C lies on the circumference. Is it true that the area of triangle ABC is largest when the triangle is isosceles? How do you know?

Problem 13PE Extreme Values A graph that is large enough to show a function’s global behavior may fail to reveal important local features. The graph of is a case in point. a. Graph ƒ over the interval .Where does the graph appear to have local extreme values or points of inflection? b. Now factor and show that ƒ has a local maximum at and local minima at c. Zoom in on the graph to find a viewing window that shows the presence of the extreme values at The moral here is that without calculus the existence of two of the three extreme values would probably have gone unnoticed. On any normal graph of the function, the values would lie close enough together to fall within the dimensions of a single pixel on the screen.

Problem 13QGY List the steps you would take to graph a polynomial function. Illustrate with an example.

Problem 32AAE Theory and Examples A particle moves with acceleration Assuming that the velocity v = 4/3 and the position s = – 4/15 when t = 0, find a. the velocity y in terms of t. b. the position s in terms of t.

Problem 32PE Graphs and Graphing Graph the curves

Problem 33AAE Theory and Examples Given By considering the minimum, prove that for all real x if and only if

Problem 79PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 80PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 81PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 130PE Applications and Examples Find the absolute maximum and minimum values of each function on the given interval.

Problem 131PE Applications and Examples Find the absolute maxima and minima of the functions and say where they are assumed.

Problem 132PE Applications and Examples Find the absolute maxima and minima of the functions and say where they are assumed.

Problem 14AAE Optimization Proving the second derivative test The Second Derivative Test for Local Maxima and Minima (Section 4.4) says:

Problem 14PE Extreme Values (Continuation of Exercise 13.) a.Graph over the interval .Where does the graph appear to have local extreme values or points of inflection? b. Show that ƒ has a local maximum value at and a local minimum value at c. Zoom in to find a viewing window that shows the presence of the extreme values at Reference: Exercise 13 A graph that is large enough to show a function’s global behavior may fail to reveal important local features. The graph of is a case in point. a. Graph ƒ over the interval .Where does the graph appear to have local extreme values or points of inflection? b. Now factor and local minima at c. Zoom in on the graph to find a viewing window that shows the presence of the extreme values at The moral here is that without calculus the existence of two of the three extreme values would probably have gone unnoticed. On any normal graph of the function, the values would lie close enough together to fall within the dimensions of a single pixel on the screen.

Problem 14QGY What is a cusp? Give examples.

Problem 33PE Graphs and Graphing Graph the curves

Problem 34AAE Theory and Examples Schwarz’s inequality a. In Exercise 33, let and deduce Schwarz’s inequality: b. Show that equality holds in Schwarz’s inequality only if there exists a real number x that makes equal for every value of i from 1 to n. Reference: Exercise 33 Given By considering the minimum, prove that for all real x if and only if

Problem 82PE Checking Antiderivative Formulas Verify the formulas by differentiation

Problem 83PE Checking Antiderivative Formulas Right, or wrong? Say which for each formula and give a brief reason for each answer.

Problem 34PE Graphs and Graphing Graph the curves

Problem 84PE Checking Antiderivative Formulas Right, or wrong? Say which for each formula and give a brief reason for each answer.

Problem 133PE Applications and Examples Graph the following functions and use what you see to locate and estimate the extreme values, identify the coordinates of the inflection points, and identify the intervals on which the graphs are concave up and concave down. Then confirm your estimates by working with the functions’ derivatives.

Problem 134PE Applications and Examples Graph Does the function appear to have an absolute minimum value? Confirm your answer with calculus.

Problem 135PE Applications and Examples Graph Explain what you see.

Problem 38PE Graphs and Graphing Graph the curves

Problem 39PE Graphs and Graphing Graph the curves

Problem 88PE Optimization A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

Problem 89PE Optimization Find the height and radius of the largest right circular cylinder that can be put in a sphere of radius

Problem 90PE Optimization The figure here shows two right circular cones, one upside down inside the other. The two bases are parallel, and the vertex of the smaller cone lies at the center of the larger cone’s base. What values of r and h will give the smaller cone the largest possible volume?

Problem 15PE The Mean Value Theorem a. Show that decreases on every interval in its domain. b. How many solutions does the equation have? Give reasons for your answer.

Problem 15AAE Optimization Hole in a water tank You want to bore a hole in the side of the tank shown here at a height that will make the stream of water coming out hit the ground as far from the tank as possible. If you drill the hole near the top, where the pressure is low, the water will exit slowly but spend a relatively long time in the air. If you drill the hole near the bottom, the water will exit at a higher velocity but have only a short time to fall. Where is the best place, if any, for the hole? (Hint: How long will it take an exiting particle of water to fall from height y to the ground?)

Problem 15QGY List the steps you would take to graph a rational function. Illustrate with an example.

Problem 35AAE Theory and Examples The best branching angles for blood vessels and pipes When a smaller pipe branches off from a larger one in a flow system, we may want it to run off at an angle that is best from some energy saving point of view. We might require, for instance, that energy loss due to friction be minimized along the section AOB shown in the accompanying figure. In this diagram, B is a given point to be reached by the smaller pipe, A is a point in the larger pipe upstream from B, and O is the point where the branching occurs. A law due to Poiseuille states that the loss of energy due to friction in non-turbulent flow is proportional to the length of the path and inversely proportional to the fourth power of the radius. Thus, the loss along AO is and along OB is where k is a constant, d1 is the length of AO, d2 is the length of OB, R is the radius of the larger pipe, and r is the radius of the smaller pipe. The angle is to be chosen to minimize the sum of these two losses: In our model, we assume that AC = a and BC = bare fixed. Thus we have the relations so that We can express the total loss L as a function of : a. Show that the critical value of for which equals zero is b. If the ratio of the pipe radii is r/R = 5/6, estimate to the nearest degree the optimal branching angle given in part (a).

Problem 35PE Graphs and Graphing Graph the curves

Problem 36PE Graphs and Graphing Graph the curves

Problem 85PE Optimization The sum of two nonnegative numbers is 36. Find the numbers if a. the difference of their square roots is to be as large as possible. b. the sum of their square roots is to be as large as possible.

Problem 86PE Optimization The sum of two nonnegative numbers is 20. Find the numbers a. if the product of one number and the square root of the other is to be as large as possible. b. if one number plus the square root of the other is to be as large as possible.

Problem 87PE Optimization An isosceles triangle has its vertex at the origin and its base parallel to the x-axis with the vertices above the axis on the curve Find the largest area the triangle can have.

Problem 136PE Applications and Examples A round underwater transmission cable consists of a core of copper wires surrounded by nonconducting insulation. If x denotes the ratio of the radius of the core to the thickness of the insulation, it is known that the speed of the transmission signal is given by the equation If the radius of the core is 1 cm, what insulation thickness h will allow the greatest transmission speed?

Problem 5AAE Local extrema a. Suppose that the first derivative of is At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection? b. Suppose that the first derivative of is At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection?

Problem 5PE Extreme Values Does have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of g.

Problem 5QGY What are the hypotheses and conclusion of the Mean Value Theorem? What physical interpretations might the theorem have?

Problem 22AAE Theory and Examples Estimating reciprocals without division You can estimate the value of the reciprocal of a number a without ever dividing by a if you apply Newton’s method to the function For example, if a = 3,the function involved is a. Graph Where does the graph cross the x-axis? b. Show that the recursion formula in this case is so there is no need for division.

Problem 22PE The Mean Value Theorem Calculate the first derivatives of and What can you conclude about the graphs of these functions?

Problem 22QGY What is an indefinite integral? How do you evaluate one? What general formulas do you know for finding indefinite integrals?

Problem 55PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 56PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 57PE Finding Indefinite Integrals Find the most general antiderivative or indefinite integral. Check your answers by differentiation.

Problem 106PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 107PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 108PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 16PE The Mean Value Theorem a. Show that increases on every interval in its domain. b. If the conclusion in part (a) is really correct, how do you explain the fact that is less than

Problem 17AAE Optimization A max-min problem with a variable answer Sometimes the solution of a max-min problem depends on the proportions of the shapes involved. As a case in point, suppose that a right circular cylinder of radius r and height h is inscribed in a right circular cone of radius R and height H, as shown here. Find the value of r (in terms of R and H) that maximizes the total surface area of the cylinder (including top and bottom). As you will see, the solution depends on whether or2R H > 2R

Problem 17PE The Mean Value Theorem a. Show that the equation has exactly one solution on [0, 1]. b. Find the solution to as many decimal places as you can.

Problem 17QGY Describe l’Hôpital’s Rule. How do you know when to use the rule and when to stop? Give an example.

Problem 40PE Graphs and Graphing Graph the curves

Problem 41PE Graphs and Graphing Graph the curves

Problem 91PE Optimization Manufacturing tires Your company can manufacture x hundred grade A tires and y hundred grade B tires a day, where Your profit on a grade A tire is twice your profit on a grade B tire. What is the most profitable number of each kind to make?

Problem 42PE Graphs and Graphing Graph the curves

Problem 92PE Optimization Particle motion The positions of two particles on the s-axis are a. What is the farthest apart the particles ever get? b. When do the particles collide?

Problem 93PE Optimization Open-top box An open-top rectangular box is constructed from a 10-in.-by-16-in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume. Support your answers graphically.

Problem 1AAE Functions and Derivatives What can you say about a function whose maximum and minimum values on an interval are equal? Give reasons for your answer.

Problem 1PE Extreme Values Does have any local maximum or minimum values? Give reasons for your answer.

Problem 1QGY What can be said about the extreme values of a function that is continuous on a closed interval?

Problem 18QGY How can you sometimes handle limits that lead to indeterminate forms Give examples.

Problem 43PE Graphs and Graphing Each of exercise gives the first derivative of a function (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

Problem 18AAE Optimization Minimizing a parameter Find the smallest value of the positive constant m that will make greater than or equal to zero for all positive values of x.

Problem 18PE The Mean Value Theorem a. Show that increases on every interval in its domain. b. Show that has no local maximum or minimum values.

Problem 44PE Graphs and Graphing Each of exercise gives the first derivative of a function (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

Problem 45PE Graphs and Graphing Each of exercise gives the first derivative of a function (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

Problem 95PE Newton’s Method Let Show that the equation ƒ(x) = -4 has a solution in the interval [2, 3] and use Newton’s method to find it.

Problem 94PE Optimization The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.

Problem 96PE Newton’s Method Let Show that the equation ƒ(x) = 75 has a solution in the interval [3, 4] and use Newton’s method to find it.

Problem 2AAE Functions and Derivatives Is it true that a discontinuous function cannot have both an absolute maximum and an absolute minimum value on a closed interval? Give reasons for your answer.

Problem 2PE Extreme Values Does have any local maximum values? Give reasons for your answer.

Problem 2QGY What does it mean for a function to have a local extreme value on its domain? An absolute extreme value? How are local and absolute extreme values related, if at all? Give examples.

Problem 19AAE Limits Evaluate the following limits.

Problem 19PE The Mean Value Theorem Water in a reservoir As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot 43,560 ft3is the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.)

Problem 19QGY How can you sometimes handle limits that lead to indeterminate forms ? Give examples.

Problem 46PE Graphs and Graphing Each of exercise gives the first derivative of a function (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

Problem 47PE Graphs and Graphing Each of exercise gives the first derivative of a function (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

Problem 48PE Graphs and Graphing Each of exercise gives the first derivative of a function (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

Problem 97PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 98PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.

Problem 99PE Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives). Check your answers by differentiation.