Question: Use Newton’s method to find all

> The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that F(0) = 1. y. y = f(x)

> The graph of a function f is shown. Which graph is an antiderivative of f and why? f a X. b

> The graph of a function f is shown. Which graph is an antiderivative of f and why? y. f b a

> Find a function f such that f ,(x) = x3 and the line x + y = 0 is tangent to the graph of f .

> Given that the graph of f passes through the point (2, 5) and that the slope of its tangent line at (x, f (x)) is 3 - 4x, find f (1).

> Find f. f ’’’(x) = cos x, f (0) = 1, f ’’(0) = 2, f ’’’(0) = 3

> Find f. f ‘‘(x) = x-2, x > 0, f (1) = 0, f (2) = 0

> Find f. f ‘‘(x) = ex - 2sin x, f (0) = 3, f (π/2) = 0

> Find f. f ‘‘(x) = x3 + sinh x, f (0) = 1, f (2) = 2.6

> Find f. f ‘‘(x) = 4 + 6x + 24x2, f (0) = 3, f (1) = 10

> Find f. f ’’(t) = t2 + 1/t2, t > 0, f (2) = 3, f ‘(1) = 2

> Find f. f ’’(θ) = sin θ + cos θ, f (0) = 3, f ‘(0) = 4

> Find f. f ’’(x) = 8x3 + 5, f (1) = 0, f ‘(1) = 8

> Find f. f ’’(x)= -2 + 12x - 12x2, f (0) = 4, f ‘(0) = 12

> Find f. f 9’(t) = 3t – 3/t , f (1) = 2, f (-1) = 1

> Find f. f ‘(x) = 5x2/3, f (8) = 21

> Find f. f ‘(t) = t + 1/t3, t > 0, f (1) = 6

> Find f. f ‘(t) = 4/(1 + t2), f (1) = 0

> Find f. f ‘(x) = 5x4 - 3x2 + 4, f (-) = 2

> Small birds like finches alternate between flapping their wings and keeping them folded while gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are

> Differentiate the function. Y = ln (e-x + xe-x)

> Differentiate the function. Y = ln (csc x – cot x)

> Find the most general antiderivative of the function. (Check your answer by differentiation.) f (x) = 1/5 – 2/x

> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = e2

> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 2

> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 7x2/5 + 8x-4/5

> For which of the initial approximations x1 = a, b, c, and d do you think Newton’s method will work and lead to the root of the equation f (x) = 0? y. а b d

> For each initial approximation, determine graphically what happens if Newton’s method is used for the function whose graph is shown. (a) x1 = 0 (b) x1 = 1 (c) x1 = 3 (d) x1 = 4 (e) x1 = 5 y. + + 3 5

> Find the derivative of f (x) = (1 + 2x2)(x - x2) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?

> Suppose the tangent line to the curve y = f (x) at the point (2, 5) has the equation y = 9 - 2x. If Newton’s method is used to locate a root of the equation f (x) = 0 and the initial approximation is x1 = 2, find the second approximation x2.

> Follow the instructions for Exercise 1(a) but use x1 = 1 as the starting approximation for finding the root r. Data from Exercise 1: The figure shows the graph of a function f. Suppose that Newton’s method is used to approximate the ro

> The figure shows the graph of a function f. Suppose that Newton’s method is used to approximate the root s of the equation f (x) = 0 with initial approximation x1 = 6. (a) Draw the tangent lines that are used to find x2 and x3, and esti

> The figure shows the sun located at the origin and the earth at the point (1, 0). (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ≈ 1.496 3 108 km.) There are five locations L1, L2, L3, L4, a

> In the figure, the length of the chord AB is 4 cm and the length of the arc AB is 5 cm. Find the central angle θ, in radians, correct to four decimal places. Then give the answer to the nearest degree. 5 cm A 4 cm

> Use Newton’s method to find the coordinates, correct to six decimal places, of the point on the parabola y = (x – 1)2 that is closest to the origin.

> Of the infinitely many lines that are tangent to the curve y = 2sin x and pass through the origin, there is one that has the largest slope. Use Newton’s method to find the slope of that line correct to six decimal places.

> Use Newton’s method to find the coordinates of the inflection point of the curve y = x2 sin x, 0 ≤ x ≤ π, correct to six decimal places.

> Use Newton’s method to find the absolute maximum value of the function f (x) = x cos x, 0 ≤ x ≤ π, correct to six decimal places.

> Differentiate the function. g(t) = 2t-3/4

> (a) Use Newton’s method to find the critical numbers of the function f (x) = x6 - x4 + 3x3 - 2x correct to six decimal places. (b) Find the absolute minimum value of f correct to four decimal places.

> If then the root of the equation f (x) = 0 is x = 0. Explain why Newton’s method fails to find the root no matter which initial approximation x1 ≠ 0 is used. Illustrate your explanation with a sketch. S(1) = {VE if

> (a) Use Newton’s method with x1 = 1 to find the root of the equation x3 - x = 1 correct to six decimal places. (b) Solve the equation in part (a) using x1 = 0.6 as the initial approximation. (c) Solve the equation in part (a) using x1 = 0.57. (You defini

> Explain why Newton’s method doesn’t work for finding the root of the equation x3 - 3x + 6 = 0 if the initial approximation is chosen to be x1 = 1.

> Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. cos (x2 - x) = x4

> Differentiate the function. T(z) = 2z log 2z

> Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x5 - 3x4 + x3 - x2 - x + 6 = 0

> Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. -2x7 - 5x4 + 9x3 + 5 = 0

> Use Newton’s method to find all solutions of the equation correct to six decimal places. sin x = x2 - 2

> Use Newton’s method to find all solutions of the equation correct to six decimal places. 2x = 2 – x2

> Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. The positive root of 3 sin x = x

> Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. The negative root of ex = 4 - x2

> What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for --1 ≤ x ≤ 2?

> (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places. -2x5 + 9x4 - 7x3 - 11x = 0, [3, 4]

> The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?

> (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places. 3x4 - 8x3 + 2 = 0, [2, 3]

> Find two positive numbers whose product is 100 and whose sum is a minimum.

> Use Newton’s method to approximate the given number correct to eight decimal places. 8 500

> Find two numbers whose difference is 100 and whose product is a minimum.

> Use Newton’s method to approximate the given number correct to eight decimal places. 4 75

> Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the one at the right, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in

> Use Newton’s method with initial approximation x1 = 1 to find x2, the second approximation to the root of the equation x4 - x - 1 = 0. Explain how the method works by first graphing the function and its tangent line at (1, -1).

> Differentiate the function. F(s) = ln ln s

> Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ,l, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on , so that the i

> Use Newton’s method with initial approximation x1 = -1 to find x2, the second approximation to the root of the equation x3 + x + 3 = 0. Explain how the method works by first graphing the function and its tangent line at (-1, 1).

> Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls

> Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) x7 + 4 = 0, x1 = -1

> Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W.

> Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) 2x3 - 3x2 + 2 = 0 , x1 = -1

> A painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer

> Where should the point P be chosen on the line segment AB so as to maximize the angle θ? В 2 Po 3 А 5

> Write the composite function in the form f ( g(x) ). [Identify the inner function u = g(x) and the outer function y = f (u).] Then find the derivative dy / dx. y − tan πx

> A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water? —

> An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sig

> A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

> The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y? 12 - y 8

> Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when &Ic

> The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consump

> A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B. and C is minimized (see the figure). Express L as a function of x = |AP | and use the graphs of L and dL/dx to estimate the minimum

> The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? b a a b

> Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($400,000ykm). You may suspect that in some instances, the minimum distance possible under the river should be

> A retailer has been selling 1200 tablet computers a week at $350 each. The marketing department estimates that an additional 80 tablets will sell each week for every $10 that the price is lowered. (a) Find the demand function. (b) What should the price b

> During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that the average decreased by two sales per day. (a) Find

> A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a) Find the demand function, assuming that

> (a) Show that if the profit P(x) is a maximum, then the marginal revenue equals the marginal cost. (b) If C(x) = 16,000 + 500x - 1.6x2 + 0.004x3 is the cost function and p(x) = 1700 - 7x is the demand function, find the production level that will maximiz

> (a) If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If C(x) = 16,000 + 200x + 4x3/2, in dollars, fin

> What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y = 4 - x2 at some point?

> What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3/x at some point?

> At which points on the curve y = 1 + 40x3 - 3x5 does the tangent line have the largest slope?

> Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a, b).

> Find an equation of the line through the point (3, 5) that cuts off the least area from the first quadrant.

> The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 f

> Suppose the refinery in Exercise 51 is located 1 km north of the river. Where should P be located? Exercise 51: An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to stor

> An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000/k

> A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mi/h and row a boat

> Solve the problem in Example 4 if the river is 5 km wide and point B is only 5 km downstream from A. Example 4: A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite ba

> A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest together?

> Differentiate the function. F(t) = (ln t)2 sin t

> A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1/3 H.

> A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.

> A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. A B R C