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All Textbook Solutions for Single Variable Calculus: Concepts and Contexts, Enhanced Edition
2RQ3RQ4RQ5RQ6RQ7RQ8RQ9RQ10RQ11RQ12RQ13RQ14RQ15RQ16RQ17RQ18RQ19RQ20RQ21RQ22RQ23RQ1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE1P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16P17P18P19P20P21P22P23P24P1E(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.4E5E6E7E8E9E10EThe 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.12E13E14E15EThe 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.17E18E19E20E21E22E23E24E25E26E27E28EEvaluate the Riemann sum for f(x) = x 1, 6 x 4, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.2E3E(a) Find the Riemann sum for f(x) = 1/x, 1 x 2, with four terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a) with midpoints as the sample points.5E6EA table of values of an increasing function f is shown. Use the table to find lower and upper estimates for 1030f(x)dx.8EUse the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 08sinxdx,n=4Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 01x3+1dx,n=5Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 02xx+1dx,n=5Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 0xsin2xdx,n=413EWith a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function f(x) = x/(x + 1) on the interval [0, 2] with n = 100. Explain why these estimates show that 0.894602xx+1dx0.908115EUse a calculator or computer to make a table of values of left and right Riemann sums Ln and Rn for the integral 02ex2dx with n = 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you make a similar statement for the integral 12ex2dx? Explain.17E18E19E20E21E22E23E24E25E26E27E28E29E30E31EThe graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral. (a) 02g(x)dx (b) 26g(x)dx (c) 07g(x)dx33E34E35E36E37E38E39E40E41E42E43E44E45E46E47EIf , F(x)=2xf(t)dt, where f is the function whose graph is given, which of the following values is largest? (A) F(0) (B) F(1) (C) F(2) (D) F(3) (E) F(4)Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of 42[f(x)+2x+5]dx50E51E52E53E54E55E56E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76EExplain exactly what is meant by the statement that differentiation and integration are inverse processes.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) Evaluate g(x) for x = 0, 1, 2, 3, 4, 5, and 6. (b) Estimate g(7). (c) Where does g have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of g.3E4E5ESketch the area represented by g(x). Then find g(x) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. g(x)=0x(2+sint)dt7E8E9E10E11E12E13E14E15E16E17E18E19E20E