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All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I
Evaluate each expression without using a calculator. (a) (3)4 (b) 34 (c) 34 (d) 523521 (e) (23)2 (f) 163/4Simplify each expression. Write your answer without negative exponents. (a) 20032 (b) (3a3b3)(4ab2)2 (c) (3x3/2y3x2y1/2)2Expand and simplify. (a) 3(x + 6) + 4(2x 5) (b) (x + 3)(4x 5) (c) (a+b)(ab) (d) (2x + 3)2 (e) (x + 2)34ADTSimplify the rational expression. (a) x2+3x+2x2x2 (b) 2x2x1x29x+32x+1 (c) x2x24x+1x+2 (d) yxxy1y1x6ADT7ADTSolve the equation. (Find only the real solutions.) (a) x+5=1412x (b) 2xx+1=2x1x (c) x2 x 12 = 0 (d) 2x2 + 4x + 1 = 0 (e) x4 3x2 + 2 = 0 (f) 3|x 4| = 10 (g) 2x(4x)1/234x=0Solve each inequality. Write your answer using interval notation. (a) 4 5 3x 17 (b) x2 2x + 8 (c) x(x 1)(x + 2) 0 (d) |x 4| 3 (e) 2x3x+11State whether each equation is true or false. (a) (p + q)2 = p2 + q2 (b) ab=ab (c) a2+b2=a+b (d) 1+TCC=1+T (e) 1xy=1x1y (f) 1/xa/xb/x=1abFind an equation for the line that passes through the point (2, 5) and (a) has slope 3 (b) is parallel to the x-axis (c) is parallel to the y-axis (d) is parallel to the line 2x 4y = 32BDT3BDTLet A(7, 4) and B(5, 12) be points in the plane. (a) Find the slope of the line that contains A and B. (b) Find an equation of the line that passes through A and B. What are the intercepts? (c) Find the midpoint of the segment AB. (d) Find the length of the segment AB. (e) Find an equation of the perpendicular bisector of AB. (f) Find an equation of the circle for which AB is a diameter.Sketch the region in the xy-plane defined by the equation or inequalities. (a) 1 y 3 (b) |x| 4 and |y| 2 (c) y112x (d) y x2 1 (e) x2 + y2 4 (f) 9x2 + 16y2 = 144FIGURE FOR PROBLEM 1 1. The graph of a function f is given at the left (a) State the value of f(1) (b) Estimate the value of f(2) (c) For what values of x is f(x) = 2? (d) Estimate the values of x such that f(x) = 0. (e) State the domain and range of f.If f(x) = x3, evaluate the difference quotient f(2+h)f(2)h and simplify your answer.3CDTHow are graphs of the functions obtained from the graph of f? (a) y = f(x) (b) y = 2f(x) 1 (c) y = f(x 3) + 2Without using a calculator, make a rough sketch of the graph. (a) y = x3 (b) y = (x + 1)3 (c) y = (x 2)3 + 3 (d) y = 4 x2 (e) y=x (f) y=2x (g) y = 2x (h) y = 1 + x16CDT7CDT1DDT2DDTFind the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of 30.4DDTExpress the lengths a and b in the figure in terms of . FIGURE FOR PROBLEM 5If sinx=13 and secy=54, where x and y lie between 0 and /2, evaluate sin(x + y).7DDTFind all values of x such that sin 2x = sin x and 0 x 2.Sketch the graph of the function y = 1 + sin 2x without using a calculator.1. If f(x)=x+2x and g(u)=u+2u, is it true that f = g?If f(x)=x2xx1andg(x)=x is it true that f = g?The graph of a function f is given. (a) State the value of f(1). (b) Estimate the value of f(1). (c) For what values of x is f(x) = 1? (d) Estimate the value of x such that f(x) = 0. (e) State the domain and range of f. (f) On what interval is f increasing?4EFigure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.7E8E9E10EShown is a graph of the global average temperature T during the 20th century. Estimate the following. (a) The global average temperature in 1950 (b) The year when the average temperature was 14.2C (c) The year when the temperature was smallest? Largest? (d) The range of T Source: Adapted from Globe and Mail [Toronto]. 5 Dec. 2009. Print.Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000. (a) What is the range of the ring width function? (b) What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century?13E14E15E16E17E18E19EYou place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.21EAn airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane. (a) Sketch a possible graph of x(t). (b) Sketch a possible graph of y(t). (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity.23E24E25E26E27EEvaluate the difference quotient for the given function. Simplify your answer. 28. f(x)=x3,f(a+h)f(a)hEvaluate the difference quotient for the given function. Simplify your answer. 29. f(x)=1x, f(x)f(a)xa30E31E32E33E34E35E36E37E38E39E40E41E42E43E44ESketch the graph of the function. 45. f(x) = x + |x|46ESketch the graph of the function. 47. g(t) = |1 3t|48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19EThe monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her 380 to drive 480 mi and in June it cost her 460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d. assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the C-intercept represent? (e) Why does a linear function give a suitable model in this situation?21EFor each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. 22.Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32EFind the functions (a) fg, (b) gf, (c) ff, and (d) gg and their domains. 33. f(x)=3x+5,g(x)=x2+x34E35E36E37E38E39E40E41E42E43E44EExpress the function in the form f g. 45. F(x)=x31+x346E47E48E49E50E51EUse the table to evaluate each expression. (a) f(g(1)) (b) g(f(1)) (c) f(f(1)) (d) g(g(1)) (e) (g f)(3) (f) (f g)(6)53EUse the given graphs of f and g to estimate the value of f(g(x)) for x = 5, 4, 3,. . . , 5. Use these estimates to sketch a rough graph of f g.55E56E57E58E59EThe Heaviside function defined in Exercise 59 can also be used to define the ramp function y = ctH(t), which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y = tH(t). (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for V(t) in terms of H(t) for t 60. (c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V(t) in terms of H(t) for t 32.Let f and g be linear functions with equations f(x) = m1x + b1, and g(x) = m2x + b2. Is f g also a linear function? If so, what is the slope of its graph?62E63E64E65E66E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28EA bacteria culture starts with 500 bacteria and doubles in size every half hour. (a) How many bacteria are there after 3 hours? (b) How many bacteria are there after t hours? (c) How many bacteria are there after 40 minutes? (d) Graph the population function and estimate the time for the population to reach 100,000.31EAn isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. (a) Find the amount remaining after 60 hours. (b) Find the amount remaining after t hours. (c) Estimate the amount remaining after 4 days. (d) Use a graph to estimate the time required for the mass to he reduced to 0.01 g.33EAfter alcohol is fully absorbed into the body, it is metabolized with a half-life of about 1.5 hours. Suppose you have had three alcoholic drinks and an hour later, at midnight, your blood alcohol concentration (BAC) is 0.6 mg/ mL. (a) Find an exponential decay model for your BAC t hours after midnight. (b) Graph your BAC and use the graph to determine when your BAC is 0.08 mg/mL. Source: Adapted from P. Wilkinson et al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 20724.35E36E37E38E1E2E3E4E5E6E7E8E9E10E11E12E13E14EAssume that f is a one-to-one function. (a) If f(6) = 17, what is f1(17)? (b) If f1(3) = 2, what is f(2)?16E17E18EThe formula C=59(F32), where F 459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?20E21EFind a formula for the inverse of the function. 22. f(x)=4x12x+323E24E25E26E27E28E29E30ELet f(x)=1x2,0x1. (a) Find f1. How is it related to f? (b) Identify the graph of f and explain your answer to part (a).32E(a) How is the logarithmic function y = logb x defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the. general shape of the graph of the function y = logb x if b 1.34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E(a) What are the values of eln 300 and ln(e300)? (b) Use your calculator to evaluate eln 300 and ln(e300). What do you notice? Can you explain why the calculator has trouble?If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n = f(t) = 1002t/3. (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50.000?62E63E64E65E66E67E