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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S=6sh32s2cot+(3s23/2)csc where s, the length of the sides of the hexagon, and h, the height, are constants. (a) Calculate dS/d. (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of s and h). Note: Actual measurements of the angle in beehives have been made, and the measures of these angles seldom have been made, and the measures of these angles seldom differ from the calculated value by more than 2.48E49EA 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 at 2 mi/h. How should she proceed?51E52EThe 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 ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?54E55EAt which points on the curve y = 1 + 40x3 3x5 does the tangent line have the largest slope?57E58E59E(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 maximize profit.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 it is linear. (b) How should ticket prices be set to maximize revenue?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 the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry 6, what should the selling price be to maximize his profit?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 be set at in order to maximize revenue? (c) If the retailers weekly cost function is C(x)=35,000+120x what price should it choose in order to maximize its profit?64EShow that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.66E67E68EA 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 value of L.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 consumption increases. You can see that c(v) is minimized for this car when v 30 mi/h. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Lets call this consumption G. Using the graph, estimate the speed at which G has its minimum value.71E72E73EA 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?75E76E77EA 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 stand so as to maximize the angle subtended at his eye by the painting?)79EThe blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuilles Laws gives the resistance R of the blood as R=CLr4 where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood vessel with radius r1 branching at an angle into a smaller vessel with radius r2. (a) Use Poiseuilles Law to show that the total resistance of the blood along the path ABC is R=C(abcotr14+bcscr24) where a and b are the distances shown in the figure. (b) Prove that this resistance is minimized when cos=r24r14 (c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds the radius of the larger vessel.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 over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let W and L denote the energy(in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio W/L mean in terms of the birds flight? What would a small value mean? Determine the ratio W/L corresponding to the minimum expenditure of energy. (c) What should the value of W/L be in order for the bird to fly directly to its nesting area D? What should the value of W/L be for the bird to fly to B and then along the shore to D? (d) If the ornithologists observe that birds of a certain species reach the shore at a point 4 km from B, how many times more energy does it take a bird to fly over water than over land?Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line , 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 intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. (a) Find an expression for the intensity I(x) at the point P. (b) If d = 5 m, use graphs of I(x) and I(x) to show that the intensity is minimized when x = 5 m, that is, when P is at the midpoint of . (c) If d = 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint. (d) Somewhere between d = 5 m and d = 10 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d.The figure shows the graph of a function f. Suppose that Newtons 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 estimate the numerical values of x2 and x3. (b) Would x1 = 8 be a better first approximation? Explain.Follow the instructions for Exercise 1(a) but use x1 = 1 as the starting approximation for finding the root r.Suppose the tangent line to the curve y = f(x) at the point (2, 5) has the equation y = 9 2x. If Newtons 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.For each initial approximation, determine graphically what happens if Newtons method is used for the function whose graph is shown. (a) x1 = 0 (b) x1 = 1 (c) x1 = 3 (d) x1 = 4 (e) x1 = 5For which of the initial approximations x1 = a, b, c, and d do you think Newtons method will work and lead to the root of the equation f(x) = 0?Use Newtons 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 = 1Use Newtons 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.) 2xx2+1=0,x1=2Use Newtons 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 = 1Use Newtons 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).Use Newtons 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).Use Newtons method to approximate the given number correct to eight decimal places. 754Use Newtons method to approximate the given number correct to eight decimal places. 5008(a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newtons method to approximate the root correct to six decimal places. 3x4 8x3 + 2 = 0, [2, 3](a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newtons method to approximate the root correct to six decimal places. 2x5 + 9x4 7x3 11x = 0, [3, 4]Use Newtons method to approximate the indicated root of the equation correct to six decimal places. The negative root of ex = 4 x2Use Newtons method to approximate the indicated root of the equation correct to six decimal places. The positive root of 3 sin x = xUse Newtons method to find all solutions of the equation correct to six decimal places. 3 cos x = x + 1Use Newtons method to find all solutions of the equation correct to six decimal places. x+1=x2xUse Newtons method to find all solutions of the equation correct to six decimal places. 2x = 2 x2Use Newtons method to find all solutions of the equation correct to six decimal places. lnx=1x3Use Newtons method to find all solutions of the equation correct to six decimal places. x3 = tan1xUse Newtons method to find all solutions of the equation correct to six decimal places. sin x = x2 2Use Newtons 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 = 0Use Newtons method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x3 3x4 + x3 x2 x + 6 = 0Use Newtons method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. xx2+1=1xUse Newtons 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) = x4Use Newtons method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. 4ex2sinx=x2x+128E(a) Apply Newtons method to the equation x2 a = 0 to derive the following square-root algorithm (used by the ancient Babylonians to compute a): xn+1=12(xn+axn) (b) Use part (a) to compute 1000 correct to six decimal places.(a) Apply Newtons method to the equation 1/x a = 0 to derive the following reciprocal algorithm: xn+1=2xnaxn2 (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute 1/1.6984 correct to six decimal places.31E(a) Use Newtons 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 definitely need a programmable calculator for this part.) (d) Graph f(x) = x3 x 1 and its tangent lines at x1 = 1, 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 x3=0 with any initial approximation x1 0. Illustrate your explanation with a sketch.If f(x)={xifx0xifx0 then the root of the equation f(x) = 0 is x = 0. Explain why Newtons method fails to find the root no matter which initial approximation x1 0 is used. Illustrate your explanation with a sketch.(a) Use Newtons method to find the critical numbers of the function f(x)=x6x4+3x32x correct to six decimal places. (b) Find the absolute minimum value of f correct to four decimal places.Use Newtons method to find the absolute maximum value of the function f(x) = x cos x, 0 x , correct to six decimal places.Use Newtons method to find the coordinates of the inflection point of the curve y = x2 sin x, 0 x , correct to six decimal places.38EUse Newtons 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.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 , in radians, correct to four decimal places. Then give the answer to the nearest degree.A car dealer sells a new car for 18,000. He also offers to sell the same car for payments of 375 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 A of an annuity consisting of n equal payments of size R with interest rate i per time period: A=Ri[1(1+i)n] Replacing i by x, show that 48x(1+x)60(1+x)60+1=0 Use Newtons method to solve this equation.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 1.496 108 km.) There are five locations L1, L2, L3, L4, and L5 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 m1 is the mass of the sun, m2 is the mass of the earth, and r = m2/(m1 + m2), it turns out that the x-coordinate of L1 is the unique root of the fifth-degree equation p(x)=x5(2+r)x4+(1+2r)x3(1r)x2+2(1r)x+r1=0 and the x-coordinate of L2 is the root of the equation p(x)2rx2=0 Using the value r 3.04042 106, find the locations of the libration points (a) L1 and (b) L2.Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 4x + 7Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = x2 3x + 23E4E5EFind the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = (x 5)2Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 7x2/5 + 8x4/5Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x)=x3,42x21Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x)=2Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = e211EFind the most general antiderivative of the function. (Check your answer by differentiation.) f(x)=x23+xx13E14EFind the most general antiderivative of the function. (Check your answer by differentiation.) g(t)=1+t+t2tFind the most general antiderivative of the function. (Check your answer by differentiation.) r() = sec tan 2eFind the most general antiderivative of the function. (Check your answer by differentiation.) h() = 2 sin sec218EFind the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =2x + 4 sinh x20E21E22E23E24EFind f. f(x) = 20x3 12x2 + 6xFind f. f(x) = x6 4x4 + x + 1Find f. f(x) = 2x + 3exFind f. f(x) = 1/x2Find f. f(t) = 12 + sin t30E31EFind f. f(x) = 5x4 3x2 + 4, f(1) = 233EFind f. f(t) = t + 1/t3, t 0, f(1) = 6Find f. f(x) = 5x2/3, f(8) = 2136E37E38E39EFind f. f(x) = 8x3 + 5, f(1) = 0, f(1) = 841E42E43EFind f. f(x) = x3 + sinh x, f(0) = 1, f(2) = 2.6Find f. f(x) = ex 2 sin x, f(0) = 3, f(/2) = 0Find f. f(t)=t3cost, f(0) = 2, f(1) = 247E48E49E50E51EThe graph of a function f is shown. Which graph is an antiderivative of f and why?53E54EThe graph of f is shown in the figure. Sketch the graph of f if f is continuous on [0, 3] and f(0) = 1.56E57E58EA particle is moving with the given data. Find the position of the particle. v(t) = sin t cos t, s(0) = 0A particle is moving with the given data. Find the position of the particle. v(t)=t23t, s(4) = 861EA particle is moving with the given data. Find the position of the particle. a(t) 3 cos t 2 sin t, s(0) = 0, v(0) = 463EA particle is moving with the given data. Find the position of the particle. a(t) = t2 4t + 6, s(0) = 0, s(1) = 2065E66E67E68E69E70E71E72ESince 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 m/s and its downward acceleration is a={90.9tif0t100ift10 If the raindrop is initially 500 m above the ground, how long does it take to fall?74E75E76E77E78E79E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC9RCC1RQ2RQ3RQ4RQ5RQ6RQ7RQ8RQ9RQ10RQ11RQ12RQ13RQ14RQ15RQ16RQ17RQ18RQ1RE2RE3RE4RE5RE6RE7RE8REThe graph of f consists of the three line segments shown. If g(x)=0xf(t)dt, find g(4) and g(4).10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50REUse Property 8 of integrals to estimate the value of the integral. 13x2+3dx52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69REFind limh01h22+h1+t3dt71RE72RE73RE1P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16P17PThe figure shows a region consisting of all points inside a square that are closer to the center than to the sides of the square. Find the area of the region. FIGURE FOR PROBLEM 1819P20P1E(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12. (i) L6 (sample points are left endpoints) (ii) R6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L6 an underestimate or overestimate of the true area? (c) Is R6 an underestimate or overestimate of the true area? (d) Which of the numbers L6, R6, or M6 gives the best estimate? Explain.(a) Estimate the area under the graph of f(x) = 1/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.4E5E6E7E8E9E10E11E12EThe speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.The table shows speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida. (a) Estimate the distance the race car traveled during this time period using the velocities at the beginning of the time intervals. Time (s) Velocity (mi/h) 0 182.9 10 168.0 20 106.6 30 99.8 40 124.5 50 176.1 60 175.6 (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.15E16E17EThe velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.19EThe table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003. Date Deaths per day March 1 0.0079 March 15 0.0638 March 29 0.1944 April 12 0.4435 April 26 0.5620 May 10 0.4630 May 24 0.2897 (a) By using an argument similar to that in Example 4, estimate the number of people who died of SARS in Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints. (b) How would you interpret the number of SARS deaths as an area under a curve? Source: A. Gumel et al., Modelling Strategies for Controlling SARS Outbreaks, Proceedings of the Royal Society of London: Series B 271 (2004): 222332.21EUse Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=x2+1+2x,4x723E