Let f and t be linear functions with equations f(x) = m1x + b1 and g(x) = m2x + b2. Is f 0 t also a linear function? If so, what is the slope of its graph?
> Graph several members of the family of functions f(x) = 1/ 1 + aebx where a > 0. How does the graph change when b changes? How does it change when a changes?
> If you graph the function you’ll see that f appears to be an odd function. Prove it. 1- e 1/1 f(x) 1+ e ,1/1
> The table gives the population of the United States, in millions, for the years 1900–2010. Use a graphing calculator with exponential regression capability to model the US population since 1900. Use the model to estimate the population
> A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
> Use a graphing calculator with exponential regression capability to model the population of the world with the data from 1950 to 2010 in Table 1 on page 49. Use the model to estimate the population in 1993 and to predict the population in the year 2020.
> After 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 exponentia
> Use the graph of V in Figure 11 to estimate the half-life of the viral load of patient 303 during the first month of treatment. From Figure 11: 60 40+ 20 i (days) 10 20 30 RNA copies / mL
> An 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
> The half-life of bismuth-210, 210Bi, is 5 days. a. If a sample has a mass of 200 mg, find the amount remaining after 15 days. b. Find the amount remaining after t days. c. Estimate the amount remaining after 3 weeks. d. Use a graph to estimate th
> A 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 func
> A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. a. Make a scatter pl
> Use a graph to estimate the values of x such that ex > 1,000,000,000.
> Compare the functions f(x) = x10 and g(x) = ex by graphing both f and g in several viewing rectangles. When does the graph of g finally surpass the graph of f ?
> Compare the functions f(x) = x5 and g(x) = 5x by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when x is large?
> 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 tim
> If is it true that f = g? x² – x S(x)· X - 1 х) — х and
> Suppose the graphs of f(x) = x2 and g(x) = 2x are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that, at a distance 2 ft to the right of the origin, the height of the graph off is 48 ft but the height of the graph of t is about
> Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? I. One million dollars at the end of the month. II. One cent on the first day of the month, two cents on the second day, four cents on the thir
> If f(x) = 5x, show that f(x + h) – f(x) 5* – 1 5* h h
> Find the exponential function f(x) = Cbx whose graph is given. y. (-1, 3) jen
> Find the exponential function f(x) = Cbx whose graph is given. (3, 24), (1, 6)
> Find the domain of each function. a. g(t) = 10t − 100 b. g(t) = sin(et - 1)
> Find the domain of each function. a. f(x) = 1 - ex2 / 1 - e1-x2 b. f(x) = 1 + x / ecos x
> Starting with the graph of y = ex, find the equation of the graph that results from a. reflecting about the line y = 4. b. reflecting about the line x = 2.
> Starting with the graph of y = ex, write the equation of the graph that results from a. shifting 2 units downward. b. shifting 2 units to the right. c. reflecting about the x-axis. d. reflecting about the y-axis. e. reflecting about the x-axis and t
> Make 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. Y = 2(1-ex) From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=
> Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.
> Make 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. y = 1 – 1/2e-x From Figure 3: From figure 15: yt 10
> Make 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. y = e|x| From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=1
> Make 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. y = -2-x From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=1
> Make 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. y = (0.5)x-1 From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m
> Make 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. y = 4x - 1 From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=1
> Graph the given functions on a common screen. How are these graphs related? y = 0.9x, y = 0.6x, y = 0.3x, y = 0.1x
> Graph the given functions on a common screen. How are these graphs related? y = 3x, y = 10x, y = (1/3)x, y = (1/10)x
> Graph the given functions on a common screen. How are these graphs related? y = ex, y = e2x, y = 8x, y = 82x
> Graph the given functions on a common screen. How are these graphs related? y = 2x, y = ex, y = 5x, y = 20x
> a. How is the number e defined? b. What is an approximate value for e? c. What is the natural exponential function?
> Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
> a. Write an equation that defines the exponential function with base b>0. b. What is the domain of this function? c. If b ≠ 1, what is the range of this function? d. Sketch the general shape of the graph of the exponential function for each of the fol
> The graph off is given. Draw the graphs of the following functions. a. y = f(x) - 3 b. y = f(x + 1) c. y = 1/2 f(x) d. y = -f (x) 0 1 2.
> The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices. a. y = f(x - 4) b. y = f(x) + 3 c. y = 1/3 f(x) d. y = -f(x - 4) e. y = 2f(x + 6) 3 -3 3 6 -3 5)
> Explain how each graph is obtained from the graph of y = f(x). a. y = f(x) + 8 b. y = f(x + 8) c. y = 8f(x) d. y = f(8x) e. y = -f(x) - 1 f. y = 8f(1/8 x)
> Suppose the graph off is given. Write equations for the graphs that are obtained from the graph off 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-a
> Suppose g is an odd function and let h = f 0 g. Is h always an odd function? What if f is odd? What if f is even?
> Suppose g is an even function and let h = f 0 g. Is h always an even function?
> If f(x) = x + 4 and h(x) = 4x - 1, find a function g such that g 0 f = h.
> a. If g(x) = 2x + 1 and h(x) = 4x2 + 4x + 7, find a function f such that f 0 g = h. (Think about what operations you would have to perform on the formula for g to end up with the formula for h.) b. If f(x) = 3x + 5 and h(x) = 3x2 + 3x + 2, find a functi
> If you invest x dollars at 4% interest compounded annually, then the amount A(x) of the investment after one year is A(x) = 1.04x. Find A 0 A, A 0 A 0 A, and A 0 A 0 A 0 A. What do these compositions represent? Find a formula for the composition of n cop
> Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
> The 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 th
> The Heaviside function H is defined by It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. a. Sketch the graph of the Heaviside function. b. Sketch th
> An airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time t = 0. a. Express the horizontal distance d (in miles) that the plane has flown as a function of t. b. Express the distance s betwe
> A ship is moving at a speed of 30 km/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
> A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s. a. Express the radius r of the balloon as a function of the time t (in seconds). b. If V is the volume of the balloon as a function of the radius, fi
> A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. a. Express the radius r of this circle as a function of the time t (in seconds). b. If A is the area of this circle as a function of the radius, find
> Use the given graphs of f and g to estimate the value of f(g(x)) for x = 25, 24, 23,..., 5. Use these estimates to sketch a rough graph of f 0 g. 1 f
> Use the given graphs of f and g to evaluate each expression, or explain why it is undefined. a. f(g(2)) b. g(f(0)) c. (f 0 g)(0) d. (g 0 f)(6) e. (g 0 g)(-2) f. (f 0 f)(4) 19 f 2 2.
> Use the table to evaluate each expression. a. f(g(1)) b. g(f(1)) c. f(f(1)) d. g(g(1)) e. (g 0 f)(3) f. (f 0 g)(6) 1 3 4 5 f(x) 3 1 4 2 2 5 g(x) 3 2 1 2 3
> Sketch a rough graph of the number of hours of daylight as a function of the time of year.
> Express the function in the form f 0 g 0 h. S(t) = sin2(cos t)
> Express the function in the form f 0 g 0 h. H(x) = 8 2 +|x|
> Express the function in the form f 0 g 0 h. R(x) =√ x − 1
> Express the function in the form f 0 g. u(t) =tan t / 1 + tan t
> Express the function in the form f 0 g. v(t) = sec(t2) tan(t2)
> Express the function in the form f 0 g. G(x) = 3 x/ 1+x
> Express the function in the form f 0 g. F(x) = ∛x / 1 + ∛x
> Express the function in the form f 0 g. F(x) = cos2x
> Express the function in the form f 0 g. F(x) = (2x + x2)4
> Find f 0 g 0 h. f(x) = tan x, g(x) = x/x - 1, h(x) = 3√ x
> The 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?
> Find f 0 g 0 h. f(x) =
> Find f 0 g 0 h. f(x) =|x - 4|, g(x) = 2x, h(x) = √x
> Find f 0 g 0 h. f(x) = 3x - 2
> Find the functions a. f 0 g, b. g 0 f, c. f 0 f, and d. g 0 g and their domains. F(x) = x / 1 + x , g(x) = sin 2x
> Find the functions a. f 0 g, b. g 0 f, c. f 0 f, and d. g 0 g and their domains. f(x) = x + 1 / x , g(x) = x + 1 / x + 2
> Find the functions a. f 0 g, b. g 0 f, c. f 0 f, and d. g 0 g and their domains. f(x) = sin x, g(x) = x2 + 1
> Find the functions a. f 0 g, b. g 0 f, c. f 0 f, and d. g 0 g and their domains. f(x) = x + 1 , g(x) = 4x - 3
> Find the functions a. f 0 g, b. g 0 f, c. f 0 f, and d. g 0 g and their domains. f(x) = x3 - 2, g(x) = 1 - 4x
> Find the functions a. f 0 g, b. g 0 f, c. f 0 f, and d. g 0 g and their domains. f(x) = 3x + 5, g(x) = x2 + x
> Find a. f + t, b. f - t, c. ft, and d. f/t and state their domains. f(x) = 3 − x , g(x) = x2 − 1
> Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race? y4 (meters) AB C 100
> Find a. f + t, b. f - t, c. ft, and d. f/t and state their domains. f(x) = x3 + 2x2, g(x) = 3x2 - 1
> Use the given graph off to sketch the graph of y = 1/f(x). Which features of f are the most important in sketching y = 1/f(x)? Explain how they are used. yA 1
> a. How is the graph of y = f(|x|) related to the graph off ? b. Sketch the graph of y = sin|x|. c. Sketch the graph of y = √|x|.
> In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 s
> Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on
> A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its b
> The city of New Orleans is located at latitude 30°N. 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
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y =| cosπ x|
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y =|√x - 1|
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = ¼ tan (x – π/4)
> You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the ela
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = |x - 2|
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y =|x|- 2
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = sin(1/2x)
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = 3 - 2 cos x
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = 2 - √x
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = 1 + sinπ x
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = x2 - 4x + 5
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = 2 x + 1
