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]
> 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 the relationships and differences among real numbers, rational numbers, and irrational numbers.
> 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
> 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.
> In Exercises 47–50, find the zeros of the function. (Use the specified viewing window.) f (x) = √(x + 2) - x + 2; [-2, 7] by [-2, 4]
> Refer to the profit function in Fig. 19. The point (1500, 42,500) is on the graph of the function. Restate this fact in terms of the function P(x). Derivative of Linear Function If f(x) = mx + b, then we have f'(x) = m. (1)
> Let f (x) = x2 - 2x + 4, g(x) = 1/x2, and h(x) = 1/(√x - 1). Determine the following functions. g(h(x))
> Refer to the profit function in Fig. 19. The point (2500, 52,500) is the highest point on the graph of the function. What does this say in terms of profit versus quantity? Derivative of Linear Function If f(x) = mx + b, then we have f'(x) = m. (1)
> Solve the equations in Exercises 39–44. x2 - 8x + 16 / 1 + √x = 0
> Refer to the cost function in Fig. 18. If 500 units of goods are produced, estimate the cost of producing 100 more units of goods? Derivative of Linear Function If f(x) = mx + b, then we have f'(x) = m. (1)
> Decide which curves are graphs of functions. Derivative of Linear Function If f(x) = mx + b, then we have f'(x) = m. (1)
> If the rectangle in Exercise 1 has a perimeter of 40 cm, find the area of the rectangle. Rectangle in Exercise 1: Derivative of Linear Function If f(x) = mx + b, then we have f'(x) = m. (1)
> Cost of an Open Box Consider the rectangular box of Exercise 3. Assume that the box has no top, the material needed to construct the base costs $5 per square foot, and the material needed to construct the sides costs $4 per square foot. Write an equation
