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All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I

39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E A model for the concentration at time t of a drug injected into the bloodstream is C(t)=K(eatebt) where a, b, and K are positive constants and b a. Sketch the graph of the concentration function. What does the graph tell us about how the concentration varies as time passes?59E 60E 61E 62E 63E 64E Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y=x2x1 66E 67E 68E 69E Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y = 1 x + e1+x/3 71E 72E 73E 74E 75E 76E 1E 2E 3E 4E 5E 6E 9E 10E 13E 14E 27E 28E 29E 30E 33E 34E 35E 38E 39E 40E 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 your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a). First number Second number Product 1 22 22 2 21 42 3 20 60 Find two numbers whose difference is 100 and whose product is a minimum.3E The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for 1 x 2?6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his bam. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs 20 per linear foot to install and the farmer is not willing to spend more than 5000, find the dimensions for the plot that would enclose the most area.19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.35E A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?37E 38E If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice?40E 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.42E 43E 44E 45E For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u (u v), then the time required to swim a distance L is L/(v u) and the total energy E required to swim the distance is given by E(v)=av3Lvu where a is the proportionality constant. (a) Determine the value of v that minimizes E. (b) Sketch the graph of E. Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.47E 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?49E 50E 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/km over land to a point P on the north bank and 800,000/km under the river to the tanks. To minimize the cost of the pipeline, where should P be located?52E 53E 54E 55E At which points on the curve y = 1 + 40x3 3x5 does the tangent line have the largest slope?57E 58E 59E 60E 61E 62E 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?64E 65E 66E 67E 68E 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 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.71E 72E 73E 74E 75E 76E Where should the point P be chosen on the line segment AB so as to maximize the angle ?78E Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W. [Hint: Express the area as a function of an angle .]The 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?82E 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.2E 3E 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 = 5 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E (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.30E 32E 33E 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.35E 36E 37E Use 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.40E 41E 42E Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 4x + 7 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E Find the most general antiderivative of the function. (Check your answer by differentiation.) r() = sec tan 2e 17E 18E Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =2x + 4 sinh x 20E 21E 22E 25E 26E 27E 28E 29E 30E Find f. f(x)=1+3x, f(4) = 25 32E 33E Find f. f(t) = t + 1/t3, t 0, f(1) = 6 35E 36E 37E 38E 39E 40E 41E 42E 43E Find f. f(x) = x3 + sinh x, f(0) = 1, f(2) = 2.6 45E 46E 47E 48E 49E 50E 51E The graph of a function f is shown. Which graph is an antiderivative of f and why?53E The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of 48 ft/s and the other is thrown a second later with a speed of 24 ft/s. Do the balls ever pass each other?69E If a diver of mass m stands at the end of a diving board with length L and linear density , then the board takes on the shape of a curve y = f(x), where EIy=mg(Lx)+12g(Lx)2 E and I are positive constants that depend on the material of the board and g ( 0) is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use f(L) to estimate the distance below the horizontal at the end of the board.71E 72E 73E 74E 75E 76E A car is traveling at 100 km/h 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 high-speed bullet train accelerates and decelerates at the rate of 4 ft/s2. Its maximum cruising speed is 90 mi/h. (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?Explain the difference between an absolute maximum and a local maximum. Illustrate with a sketch.2RCC 3RCC 4RCC 5RCC 6RCC 7RCC 8RCC 9RCC 10RCC 11RCC 1RQ 2RQ If f is continuous on (a, b), then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in (a, b).4RQ 5RQ If f(2) = 0, then (2, f(2)) is an inflection point of the curve y = f(x).7RQ 8RQ 9RQ 10RQ 11RQ 12RQ 13RQ If f and g are positive increasing functions on an interval I, then fg is increasing on I.15RQ 16RQ 17RQ 18RQ If f(x) exists and is nonzero for all x, then f(1) f(0).20RQ 21RQ 1RE 2RE 3RE 4RE 5RE 6RE Evaluate the limit. limx0ex1tanx 8RE 9RE 10RE 11RE 12RE 13RE

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