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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
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)3Factor each expression. (a) 4x2 25 (b) 2x2 + 5x 12 (c) x3 3x2 4x + 12 (d) x4 + 27x (e) 3x3/2 9x1/2 + 6x1/2 (f) x3y 4xySimplify the rational expression. (a) x2+3x+2x2x2 (b) 2x2x1x29x+32x+1 (c) x2x24x+1x+2 (d) yxxy1y1xRationalize the expression and simplify. (a) 1052 (b) 4+h2hRewrite by completing the square. (a) x2 + x + 1 (b) 2x2 12x + 11Solve 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=1ab1BDT2BDT3BDT4BDTSketch 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.Find the domain of the function. (a) f(x)=2x+1x2+x2 (b) g(x)=x3x2+1 (c) h(x)=4x+x21How 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 + x1Let f(x)={1x2ifx02x+1ifx0 (a) Evaluate f(2) and f(1). (b) Sketch the graph of f.If f(x) = x2 + 2x 1 and g(x) = 2x 3, find each of the following functions. (a) f g (b) g f (c) g g g1DDT2DDTFind 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).Prove the identities. (a) tan sin + cos = sec (b) 2tanx1+tan2x=sin2xFind 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.(a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?Discuss four ways of representing a function. Illustrate your discussion with examples.(a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function.4RCC5RCC6RCC7RCC8RCCSuppose that f has domain A and g has domain B. (a) What is the domain of f + g? (b) What is the domain of fg? (c) What is the domain of f/g?10RCC11RCC12RCC13RCC1RQ2RQ3RQ4RQ5RQ6RQ7RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. You can always divide by ex.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If 0 a b, then In a In b.10RQ11RQ12RQ13RQ14RQLet f be the function whose graph is given. (a) Estimate the value of f(2). (b) Estimate the values of x such that f(x) = 3. (c) State the domain of f (d) State the range of f. (e) On what interval is f increasing? (f) Is f one-to-one? Explain. (g) Is f even, odd, or neither even nor odd? Explain.2RE3RE4RE5RE6RE7RE8RESuppose that the graph of .f is given. Describe how the graphs of the following functions can be obtained from the graph of .f. (a) y =.f(x) + 8 (b) y = f(x + 8) (c) y = 1 + 2.f(x) (d) y = f(x 2) 2 (c) y = f(x) (f) y = f1(x)10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26REThe half-life of palladium-100, 100Pd, is four days. (So half of any given quantity of 100Pd will disintegrate in four days.) The initial mass of a sample is one gram. (a) Find the mass that remains after 16 days. (b) Find the mass m(t) that remains after t days. (c) Find the inverse of this function and explain its meaning. (d) When will the mass be reduced to 0.01g?The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is P(t)=100,000100+900et where t is measured in years. (a) Graph this function and estimated how long it takes for the population to reach 900. (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for the population to reach 900. Compare with the result of part (a).One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.2P3P4P5P6P7P8PThe notation max{a, b, } means the largest of the numbers a, b. Sketch the graph of each function. (a) f(x) = max{x, 1/x} (b) f(x) = max{sin x, cos x} (c) f(x) = max{x2, 2 + x, 2 x}10P11P12P13P14P15P16P17PProve that 1 + 3 + 5 + + (2n l ) = n2.19P20P1. 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?The graphs of f and g are given. (a) State the values of f(4) and g(3). (b) For what values of x is f(x) = g(x)? (c) Estimate the solution of the equation .f(x) = 1. (d) On what interval is f decreasing? (c) State the domain and range of f. (f) State the domain and range of g.Figure 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.7EDetermine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 8.9E10E11E12E13E14EThe graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; t is measured in hours starting at midnight.) (a) What was the power consumption at 6 AM? At 6 PM? (b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?16E17E18ESketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.You 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.21E22E23E24E25E26E27E28E29E30E31EFind the domain of the function. 32. f(x)=2x35x2+x633E34E35E36E37EFind the domain and range and sketch the graph of the function h(x)=4x2.39E40E41E42E43EEvaluate f(3), f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. 44. f(x)={1ifx172xifx145E46E47E48E49ESketch the graph of the function. 50. g(x) = ||x| 1|51E52EFind an expression for the function whose graph is the given curve. 53. The bottom half of the parabola x + (y 1)2 = 054E55E56E57E58E59E60E61EA Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.63E64E65EAn electricity company charges its customers a base rate of 10 a month, plus 6 cents per kilowatt-hour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 x 2000.In a certain country, income tax is assessed as follows. There is no tax on income up to10,000. Any income over 10,000 is taxed at a rate of 10%, up to an income of 20,000. Any income over 20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of 14,000? On 26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I.68E69E70E71E72E73E74E75E76E77E78E79E80EClassify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. 1. (a) f(x)=log2x (b) g(x)=x4 (c) h(x)=2x31x2 (d) u(t)=11.1t+2.54t2 (e) v(t)=5t (f) w()=sincos22E3E4E5E6E7E8E9E10E11E12E13EThe manager of a weekend flea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equationy=2004x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can't be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent?15E16EBiologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70F and 173 chirps per minute at 80F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.18E19E20E21E22E23E24E25E26E27E28E29E30E31E32ESuppose 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 3.2EThe graph of y=f(x) is given. Match each equation with its graph and give reasons for your choices. (a) y=f(x4) (b) y=f(x)+3 (c) y=13f(x) (d) y=f(x+4) (e) y=2f(x+6)4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24EThe city of New Orleans is located at latitude 30N. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 AM and sets at 6:18 PM in New Orleans.26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46EExpress the function in the form f g. 47. v(t) = sec(t2) tan(t2)48E49E50E51E52E53EUse 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.55E56EA ship is at a speed of 30km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s =f(d). (b) Express d as a function of t, the time elapsed since noon; that is, find g so that d = q(t). (c) Find f g. What does this function represent?58E59EThe 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 32.61E62E63E64E65E66EUse the Law of Exponents to rewrite and simplify the expression. 1. (a) 4328 (b) 1x43Use the Law of Exponents to rewrite and simplify the expression. 2. (a) 84/3 (b) x(3x2)33EUse the Law of Exponents to rewrite and simplify the expression. 4. (a) x2nx3n1xn+2 (b) abab35E6E7E8E9E10E11E12E13E14EMake a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 15. y=112ex16E17E