Solved: Water is flowing at a constant rate into a spherical tank. Let be the volume of

The figure shows the graph of a function . Suppose that Newtons method is used to approximate the root of the equation with initial approximation . (a) Draw the tangent lines that are used to find and , and esti mate the numerical values of and . (b) Would be a better first approximation? Explain.

Follow the instructions for Exercise 1(a) but use as the starting approximation for finding the root .

Suppose the tangent line to the curve at the point has the equation . If Newtons method is used to locate a root of the equation and the initial approximation is , find the second approximation .

For each initial approximation, determine graphically what happens if Newtons method is used for the function whose graph is shown

For which of the initial approximations and do you think Newtons method will work and lead to the root of the equation

68 Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) 3 x 3 1 2 x 2 3 0 x1 3 x1

68 Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) x 1 0 4.

68 Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.)x1 1 x

Use Newtons method with initial approximation to find , the second approximation to the root of the equation . Explain how the method works by first graphing the function and its tangent line at .

Use Newtons method with initial approximation to find , the second approximation to the root of the equation . Explain how the method works by first graphing the function and its tangent line at .

1112 Use Newtons method to approximate the given number correct to eight decimal places.

1112 Use Newtons method to approximate the given number correct to eight decimal places.

1316 Use Newtons method to approximate the indicated root of the equation correct to six decimal places.13. The root of in the interval

1316 Use Newtons method to approximate the indicated root of the equation correct to six decimal places.2.2x 5 4.4x 3 1.3x 2 0.9x 4.0 0 x 1,

1316 Use Newtons method to approximate the indicated root of the equation correct to six decimal places.The negative root of

1316 Use Newtons method to approximate the indicated root of the equation correct to six decimal places.The positive root of

1722 Use Newtons method to find all roots of the equation correct to six decimal places.3 cos x x 1 2

1722 Use Newtons method to find all roots of the equation correct to six decimal places.sx 1 x 3

1722 Use Newtons method to find all roots of the equation correct to six decimal places.x 2 2 ln x sx

1722 Use Newtons method to find all roots of the equation correct to six decimal places.1 x 1 x 3

1722 Use Newtons method to find all roots of the equation correct to six decimal places.3 tan1 x 1

1722 Use Newtons method to find all roots of the equation correct to six decimal places.sin x x x

2328 Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.x 6 x 5 6x 4 x 2 x 10 0 sin x

2328 Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.x 5 3x 4 x 3 x 2 x 6 0 x 6 x

2328 Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.x x 2 1 s1 x x

2328 Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.cosx 2 x x 4 x x

2328 Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.x 2 sin x x 2 x 1 cos

2328 Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.earctan x sx 4earctan x sx 4

(a) Apply Newtons method to the equation to derive the following square-root algorithm used by the ancient Babylonians to compute : xn1 1 2 xn a xn  sa (b) Use part (a) to compute correct to six decimal places.

(a) Apply Newtons method to the equation to derive the following reciprocal algorithm: (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute correct to six decimal places.

Explain why Newtons method doesnt work for finding the root of the equation if the initial approximation is chosen to be .

(a) Use Newtons method with to find the root of the equation correct to six decimal places. (b) Solve the equation in part (a) using as the initial approximation. (c) Solve the equation in part (a) using . (You defi- nitely need a programmable calculator for this part.) ; (d) Graph and its tangent lines at , 0.6, and 0.57 to explain why Newtons method is so sensitive to the value of the initial approximation.

Explain why Newtons method fails when applied to the equation with any initial approximation . Illustrate your explanation with a sketch.

If then the root of the equation is . Explain why Newtons method fails to find the root no matter which initial approximation is used. Illustrate your explanation with a sketch.

(a) Use Newtons method to find the critical numbers of the function correct to six decimal places. (b) Find the absolute minimum value of correct to four decimal places.

Use Newtons method to find the absolute maximum value of the function , correct to six decimal places.

Use Newtons method to find the coordinates of the inflection point of the curve , , correct to six decimal places

Of the infinitely many lines that are tangent to the curve and pass through the origin, there is one that has the largest slope. Use Newtons method to find the slope of that line correct to six decimal places.

Use Newtons method to find the coordinates, correct to six decimal places, of the point on the parabola that is closest to the origin.

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

A car dealer sells a new car for . He also offers to sell the same car for payments of per month for five years. What monthly interest rate is this dealer charging? To solve this problem you will need to use the formula for the present value of an annuity consisting of equal payments of size with interest rate per time period: Replacing by , show that Use Newtons method to solve this equation.

The figure shows the sun located at the origin and the earth at the point . (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU km.) There are five locations , , , , and in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravitational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed at one of these libration points.) If is the mass of the sun, is the mass of the earth, and , it turns out that the -coordinate of is the unique root of the fifth-degree equation and the -coordinate of is the root of the equation Using the value , find the locations of the libration points (a) and (b) . AB  AB L L L L L

25 48 Find .f x 4 6x 24x f0 3 f1 10 2 f t

25 48 Find .f x x f0 1 f2 2.6 3 sin

25 48 Find .f x 2 cos x f 0 1 f 2 0 f x

25 48 Find .f t 2e f0 0 f  0 t 3 s

25 48 Find .f x x x  0 f 1 0 f 2 0 2 f

25 48 Find .f x cos x f0 1 f0 2 f 0 3 f x

Given that the graph of passes through the point and that the slope of its tangent line at is , find .

Find a function such that and the line is tangent to the graph of .

5152 The graph of a function is shown. Which graph is an antiderivative of and why?

5152 The graph of a function is shown. Which graph is an antiderivative of and why?

The graph of a function is shown in the figure. Make a rough sketch of an antiderivative , given that .

The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.

The graph of is shown in the figure. Sketch the graph of if is continuous and .

(a) Use a graphing device to graph . (b) Starting with the graph in part (a), sketch a rough graph of the antiderivative that satisfies . (c) Use the rules of this section to find an expression for . (d) Graph using the expression in part (c). Compare with your sketch in part (b).

5758 Draw a graph of and use it to make a rough sketch of the antiderivative that passes through the origin.fx sin x 1 x 2  2  x  2

5758 Draw a graph of and use it to make a rough sketch of the antiderivative that passes through the origin.fx sx 4  2x 2 2  2 3  x  3 fx

5964 A particle is moving with the given data. Find the position of the particle.vt sin t  cos t s0 0 fx

5964 A particle is moving with the given data. Find the position of the particle.vt 1.5st , s4 10 vt

5964 A particle is moving with the given data. Find the position of the particle.at 2t 1 s0 3 v0 2 vt 1

5964 A particle is moving with the given data. Find the position of the particle.at 3 cos t  2 sin t s0 0 v0 4 at

5964 A particle is moving with the given data. Find the position of the particle.at 10 sin t 3 cos t s0 0 s2  12 at 3

5964 A particle is moving with the given data. Find the position of the particle.at t 2  4t 6 s0 0 s1 20 at 1

A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, m above the ground. (a) Find the distance of the stone above ground level at time . (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 ms, how long does it take to reach the ground?

Show that for motion in a straight line with constant acceleration , initial velocity , and initial displacement , the dis - placement after time is

An object is projected upward with initial velocity meters per second from a point meters above the ground. Show that

Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of fts and the other is thrown a second later with a speed of fts. Do the balls ever pass each other?

A stone was dropped off a cliff and hit the ground with a speed of 120 fts. What is the height of the cliff?

. If a diver of mass stands at the end of a diving board with length and linear density , then the board takes on the shape of a curve , where and are positive constants that depend on the material of the board and is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use to estimate the distance below the horizontal at the end of the board.

A company estimates that the marginal cost ( in dollars per item) of producing items is . If the cost of producing one item is , find the cost of producing items.

The linear density of a rod of length m is given by , in grams per centimeter, where is measured in centimeters from one end of the rod. Find the mass of the rod.

. Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 ms and its downward acceleration is If the raindrop is initially m above the ground, how long does it take to fall?

A car is traveling at 50 mih when the brakes are fully applied, producing a constant deceleration of 22 fts . What is the distance traveled before the car comes to a stop?

What constant acceleration is required to increase the speed of a car from 30 mih to 50 mih in 5 s?

A car braked with a constant deceleration of 16 fts , producing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

A car is traveling at when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is , at which time the fuel is exhausted and it becomes a freely falling body. Fourteen seconds later, the rockets parachute opens, and the (downward) velocity slows linearly to fts in 5 s. The rocket then floats to the ground at that rate. (a) Determine the position function and the velocity function (for all times ). Sketch the graphs of and . (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land?

A high-speed bullet train accelerates and decelerates at the rate of . Its maximum cruising speed is 90 mih. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?

If a projectile is fired with an initial velocity at an angle of inclination from the horizontal, then its trajectory, neglecting air resistance, is the parabola (a) Suppose the projectile is fired from the base of a plane that is inclined at an angle , , from the horizontal,as shown in the figure. Show that the range of the projectile, measured up the slope, is given by (b) Determine so that is a maximum. (c) Suppose the plane is at an angle below the horizontal. Determine the range in this case, and determine the angle at which the projectile should be fired to maxi mize .

Show that, for ,

Sketch the graph of a function such that for all for for , an

A light is to be placed atop a pole of height feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination at any point on the circle is directly proportional to the cosine of the angle (see the figure) and inversely proportional to the square of the distance from the source. (a) How tall should the light pole be to maximize ? (b) Suppose that the light pole is feet tall and that a woman is walking away from the base of the pole at the rate of 4 fts. At what rate is the intensity of the light at the point on her back 4 ft above the ground decreasing when she reaches the outer edge of the traffic circle? 8

Water is flowing at a constant rate into a spherical tank. Let be the volume of water in the tank and be the height of the water in the tank at time . (a) What are the meanings of and ? Are these derivatives positive, negative, or zero? (b) Is positive, negative, or zero? Explain. (c) Let , , and be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values , , and positive, negative, or zero? Why?