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All Textbook Solutions for Single Variable Calculus: Concepts and Contexts, Enhanced Edition
53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36EGiven that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxaf(x)g(x) (b) limxaf(x)p(x) (c) limxah(x)p(x) (d) limxap(x)f(x) (e) limxap(x)q(x)Given that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxa[f(x)p(x)] (b) limxa[h(x)p(x)] (c) limxa[p(x)q(x)]3EGiven that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxa[f(x)]g(x) (b) limxa[f(x)]p(x) (c) limxa[h(x)]p(x) (d) limxa[p(x)]f(x) (e) limxa[p(x)]q(x) (f) limxap(x)q(x)5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76EConsider 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 60Find two numbers whose difference is 100 and whose product is a minimum.Find two positive numbers whose product is 100 and whose sum is a minimum.The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?5E6E7EThe rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P=100II2+I+4 where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62EThe 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 = 55E6E7E8EUse 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).11E12E13E14E15E16E17E18E19E20E21E22E(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.25E(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.28E29E30E31E32E33E34E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38EThe graph of f is shown in the figure. Sketch the graph of f if f is continuous on [0, 3] and f(0) = 1.40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC9RCC10RCC11RCC12RCC13RCC1RQ