Explain the relationships and differences among real numbers, rational numbers, and irrational numbers.
> Exercises 43–46 relate to Fig. 13. When a drug is injected into a person’s muscle tissue, the concentration y of the drug in the blood is a function of the time elapsed since the injection. The graph of a typical time&
> Suppose that $7000 is deposited in a savings account that pays a rate of interest r per annum, compounded annually, for 20 years. (a) Express the account balance A(r) as a function of r. (b) Calculate the account balance for r = .07 and r = .12.
> Solve the equations in Exercises 39–44. x2 + 14x + 49 / x2 + 1 = 0
> Use the quadratic formula to find the zeros of the functions in Exercises 1–6. f (x) = 2x2 - 7x + 6
> Suppose that $15,000 is deposited in a savings account that pays a rate of interest r per annum, compounded annually, for 10 years. (a) Express the account balance A(r) as a function of r. (b) Calculate the account balance for r = 0.04 and r = 0.06
> Suppose that $7000 is deposited in a savings account that pays 9% per annum, compounded biannually, for t years. (a) Express the account balance A(t) as a function of t, the number of years that the principal has been in the account. (b) Calculate the ac
> Refer to the function f (r), which gives the cost (in cents) of constructing a 100-cubic-inch cylinder of radius r inches. The graph of f (r) is shown in Fig. 16. Interpret the fact that the point (3, 162) is on the graph of the function. 1 lim lim
> Suppose that $15,000 is deposited in a savings account that pays 4% per annum, compounded monthly, for t years. (a) Express the account balance A(t) as a function of t, the number of years that the principal has been in the account. (b) Calculate the acc
> Use the laws of exponents to simplify the algebraic expressions. 3√x (8x2/3)
> Use the laws of exponents to simplify the algebraic expressions. x3/2/ √x
> Let f (x) = 3√x and g(x) = 1 / x2. Calculate the following functions. Take x > 0. √ (f (x)/ g(x))
> Use the laws of exponents to simplify the algebraic expressions. xy3 / x-5y6
> Use the laws of exponents to simplify the algebraic expressions. (√(x + 1))4
> Let f (x) = x2 - 2x, g(x) = 3x - 1, and h(x) = √x. Find the following functions. f (x)h(x)
> The revenue R(x) (in thousands of dollars) that a company receives from the sale of x thousand units is given by R(x) = 5x - x2. The sales level x is in turn a function f (d) of the number d of dollars spent on advertising, where Express the revenue as
> Let f (x) = x6, g(x) = x / 1 - x, and h(x) = x3 - 5x2 + 1. Calculate the following functions. h(g(x))
> The population of a city is estimated to be 750 + 25t + .1t2 thousand people t years from the present. Ecologists estimate that the average level of carbon monoxide in the air above the city will be 1 + .4x ppm (parts per million) when the population is
> Find a good window setting for the graph of the function. The graph should show all the zeros of the polynomial. f (x) = 2x5 - 24x4 - 24x + 2
> Refer to the function f (r), which gives the cost (in cents) of constructing a 100-cubic-inch cylinder of radius r inches. The graph of f (r) is shown in Fig. 16. For what value(s) of r is the cost 330 cents? Vx2 – 5x – 36 lim x-5 8- 3х 1/2 lim x? -
> Find the points of intersection of the pairs of curves in Exercises 31–38. y = ½ x3 + x2 + 5, y = 3x2 - 12x + 5
> Relate to the function whose graph is sketched in Fig. 12. For what values of x is f (x) ≤ 0? lim (x + Vx - 6)(x² - 2x +1) 6) (x² – 2x + 1) = lim (x + Vx - 6)(x – 1)? x→7 lim x+ lim Vr – 6 | lim x – lim 1 x→7 x→7 = (7+1)(7 – 1)²
> Find the points of intersection of the curves y = -x2 + x + 1 and y = x - 5.
> Find the points of intersection of the curves y = 5x2 - 3x – 2 and y = 2x - 1.
> Find the zeros of the quadratic function y = -2x2 - x + 2.
> Find the zeros of the quadratic function y = 5x2 - 3x - 2.
> Factor the polynomials. x5 - x4 - 2x3
> Find a good window setting for the graph of the function. The graph should show all the zeros of the polynomial. f (x) = x4 - 200x3 - 100x2
> Sketch the graph of the function. f (x) = 1 / f(x) = x + 1
> Factor the polynomials. 18 + 3x - x2
> An average sale at a small florist shop is $21, so the shop’s weekly revenue function is R(x) = 21x, where x is the number of sales in 1 week. The corresponding weekly cost is C(x) = 9x + 800 dollars. (a) What is the florist shop’s weekly profit function
> Exercises 43–46 relate to Fig. 13. When a drug is injected into a person’s muscle tissue, the concentration y of the drug in the blood is a function of the time elapsed since the injection. The graph of a typical time&
> Draw the following intervals on the number line. [ -1, 4]
> Let f (x) = x6, g(x) = x / 1 - x, and h(x) = x3 - 5x2 + 1. Calculate the following functions. f (g(x))
> Factor the polynomials. 3x2 - 3x - 60
> Factor the polynomials. 5x3 + 15x2 - 20x
> Is the point (1, -2) on the graph of the function k(x) = x2 + (2/x)?
> Is the point (1/2, - 3/5) on the graph of the function h(x) = (x2 - 1)/(x2 + 1)?
> Relate to the function whose graph is sketched in Fig. 12. For what values of x does f (x) = 0? nDeriv(2"X,X,0) 6931472361
> Draw the following intervals on the number line. [1, 3/2]
> Determine the domains of the following functions. f (x) = 1 /√(3x)
> Determine the domains of the following functions. f (x) = √(x2 + 1)
> Simplify (100)3/2 and (.001)1/3.
> Let f (x) = 3√x and g(x) = 1 / x2. Calculate the following functions. Take x > 0. f (x) / g(x)
> A cellular telephone company estimates that, if it has x thousand subscribers, its monthly profit is P(x) thousand dollars, where P(x) = 12x - 200. (a) How many subscribers are needed for a monthly profit of 160 thousand dollars? (b) How many new subscri
> Find a good window setting for the graph of the function. The graph should show all the zeros of the polynomial. f (x) = x3 - 22x2 + 17x + 19
> Determine the domains of the following functions. f (x) = √(x – 1)
> What is the difference between an open interval and a closed interval from a to b?
> Let f (x) = x / x - 2, g(x) = 5 – x / 5 + x, and h(x) = x + 1 / 3x - 1. Express the following as rational functions. h(1/x2)
> What are the four types of inequalities, and what do they each mean?
> Explain how to find f (a) geometrically from the graph of y = f (x).
> Explain how to solve f (x) = b geometrically from the graph of y = f (x).
> Simplify (81)3/4, 85/3, and (.25)-1.
> In the formula A = P(1 + i)n, what do A, P, i, and n represent?
> State the six laws of exponents.
> A frozen yogurt stand makes a profit of P(x) = .40x – 80 dollars when selling x scoops of yogurt per day. (a) Find the breakeven sales level, that is, the level at which P(x) = 0. (b) What sales level generates a daily profit of $30? (c) How many more s
> In Exercises 7–12, express f (x) + g(x) as a rational function. Carry out all multiplications. f (x) = x + 6 / x – 6, g(x) = x – 6 / x + 6
> Relate to the function whose graph is sketched in Fig. 12. What is the range of f? y == a 2 Figure 16
> In Exercises 51–54, find the points of intersection of the graphs of the functions. (Use the specified viewing window.) f (x) = 1 / x; g(x) = √(x2 – 1); [0, 4] by [-1, 3]
> Give two methods for finding the zeros of a quadratic function.
> What is a zero of a function?
> What five operations on functions are discussed in this chapter? Give an example of each.
> Let f (x) = x / x - 2, g(x) = 5 – x / 5 + x, and h(x) = x + 1 / 3x - 1. Express the following as rational functions. g(1 / u)
> Let f (x) = 3√x and g(x) = 1 / x2. Calculate the following functions. Take x > 0. [ f (x)g(x)]3
> Let f (x) = x2 - 2x, g(x) = 3x - 1, and h(x) = √x. Find the following functions. f (x) - g(x)
> What is meant by the absolute value of a number?
> Define and give an example of each of the following types of functions. (a) quadratic function (b) polynomial function (c) rational function (d) power function
> What is a quadratic function? What shape does its graph have?
> A college student earns income by typing term papers on a computer, which she leases (along with a printer). The student charges $4 per page for her work, and she estimates that her monthly cost when typing x pages is C(x) = .10x + 75 dollars. (a) What
> What are the x- and y-intercepts of a function, and how are they found?
> Use intervals to describe the real numbers satisfying the inequalities. x ≥ -1 and x < 8
> What is a linear function? Constant function? Give examples.
> What is the graph of a function, and how is it related to vertical lines?
> Graph the following equations. y = - ½ x - 4
> What is meant by the domain and range of a function?
> Let f (x) = x2 - 2x + 4, g(x) = 1/x2, and h(x) = 1/(√x - 1). Determine the following functions. h(f (x))
> What is meant by “the value of a function at x”?
> What is a function?
> Let f (x) = x / x - 2, g(x) = 5 – x / 5 + x, and h(x) = x + 1 / 3x - 1. Express the following as rational functions. f(1 / t)
> A specialty shop prints custom slogans and designs on T-shirts. The shop’s total cost at a daily sales level of x T-shirts is C(x) = 73 + 4x dollars. (a) At what sales level will the cost be $225? (b) If the sales level is at 40 T-shirts, how much will t
> Assign variables to the dimensions of the geometric object. y = f(x) f(a + h)): (a + h, f(a + h)) + + a + h a
> Assign variables to the dimensions of the geometric object. 1 f (x) = ² → s"(x)= - -3/2 7/1- = f'(x): When x = 1, f (x)= =1. The slope of the 1 tangent at x = 1 is ƒ'(1 = -÷(1)-*2 2 Thus, the equation of the tangent at (1,1) in point-slope form is y
> Let f (x) = 3√x and g(x) = 1 / x2. Calculate the following functions. Take x > 0. f (x)g(x)
> Assign variables to the dimensions of the geometric object. f (x) = \x = xV? = f" (x) =² 1 f (x) = Vx = x2 = f" (x): -1/2 2Vx When x-. S(x)= = 1 ) = : 1 The slope of 3 9 1 the tangent at x =- is 1 1(1 -1/2 1 (9)'2 3 Thus, the 2 equation of the tange
> Let f (x) = x2 - 2x + 4, g(x) = 1/x2, and h(x) = 1/(√x - 1). Determine the following functions. f (h(x))
> A store estimates that the total revenue (in dollars) from the sale of x bicycles per year is given by the function R(x) = 250x - .2x2. (a) Graph R(x) in the window [200, 500] by [42000, 75000]. (b) What sales level produces a revenue of $63,000? (c) Wha
> In Exercises 51–54, find the points of intersection of the graphs of the functions. (Use the specified viewing window.) f (x) = 3x4 - 14x3 + 24x - 3; g(x) = 2x - 30; [-3, 5] by [-80, 30]
> The daily cost (in dollars) of producing x units of a certain product is given by the function C(x) = 225 + 36.5x - .9x2 + .01x3. (a) Graph C(x) in the window [0, 70] by [ -400, 2000]. (b) What is the cost of producing 50 units of goods? (c) Consider the
> A ball thrown straight up into the air has height -16x2 + 80x feet after x seconds. (a) Graph the function in the window [0, 6] by [ -30, 120]. (b) What is the height of the ball after 3 seconds? (c) At what times will the height be 64 feet? (d) At what
> A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. In Exercises 45–50, translate the task into both a statement involving the function and a statement involving the graph of the function.
> Let f (x) = x / x - 2, g(x) = 5 – x / 5 + x, and h(x) = x + 1 / 3x - 1. Express the following as rational functions. h(s) / f (s)
> A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. In Exercises 45–50, translate the task into both a statement involving the function and a statement involving the graph of the function.
> Let f (x) = x / x - 2, g(x) = 5 – x / 5 + x, and h(x) = x + 1 / 3x - 1. Express the following as rational functions. g(x + 5) / f (x + 5)
> Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. (-32y-5)3/5
> Let f (x) = x2 - 2x + 4, g(x) = 1/x2, and h(x) = 1/(√x - 1). Determine the following functions. h(g(x))
> Evaluate each of the functions in Exercises 37–42 at the given value of x. f (x) = |x|, x = -2.5
> A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. In Exercises 45–50, translate the task into both a statement involving the function and a statement involving the graph of the function.
> A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. In Exercises 45–50, translate the task into both a statement involving the function and a statement involving the graph of the function.
> A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. In Exercises 45–50, translate the task into both a statement involving the function and a statement involving the graph of the function.
> In Exercises 51–54, find the points of intersection of the graphs of the functions. (Use the specified viewing window.) f (x) = -x - 2; g(x) = -4x2 + x + 1; [-2, 2] by [-5, 2]
> A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. In Exercises 45–50, translate the task into both a statement involving the function and a statement involving the graph of the function.
