Question: Consider a rectangular corral with two partitions,
Consider a rectangular corral with two partitions, as in Fig. 15. Assign letters to the outside dimensions of the corral. Write an equation expressing the fact that the corral has a total area of 2500 square feet. Write an expression for the amount of fencing needed to construct the corral (including both partitions).
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Derivative of Linear Function If f(x) = mx + b, then we have f'(x) = m. (1)
> 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
> Consider the corral of Exercise 16. If the fencing for the boundary of the corral costs $10 per foot and the fencing for the inner partitions costs $8 per foot, write an expression for the total cost of the fencing. Corral of Exercise 16: Derivative
> Consider a rectangular corral with a partition down the middle, as shown in Fig. 14. Assign letters to the outside dimensions of the corral. Write an equation expressing the fact that 5000 feet of fencing is needed to construct the corral (including the
> Consider the cylinder of Exercise 6. Write an equation expressing the fact that the surface area is 30Ï€ square inches. Write an expression for the volume. 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(f (x))
> Consider the cylinder of Exercise 5. Write an equation expressing the fact that the volume is 100 cubic inches. Suppose that the material to construct the left end costs $5 per square inch, the material to construct the right end costs $6 per square inch
> Consider the closed rectangular box in Exercise 4. Write an expression for the surface area. Write an equation expressing the fact that the volume is 10 cubic feet. P
> In Exercises 47–50, find the zeros of the function. (Use the specified viewing window.) f (x) = x3 - 3x + 2; [-3, 3] [-10, 10]
> Relate to the function whose graph is sketched in Fig. 12. Find f (2) and f (-1). Y1= +6) X=2.01 Y=2.000833 [1.2899, 2.5399] by [1.3679, 2.6179] When x = 2, y=2. Find a second point on the line using value: x = 2.01, y= 2.000833 2.000833 – 2 m = =
> Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. (-8y9)2/3
> 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 (x) / g(x)
> Consider the rectangular box in Exercise 3, and suppose that it has no top. Write an expression for the volume. Write an equation expressing the fact that the surface area is 65 square inches. Y1=CX-1/0*+1 X=1.01 Y=.00497512 [.93251, 1.08876] by [-.
> Consider the Norman window of Exercise 2. Write an expression for the perimeter. Write an equation expressing the fact that the area is 2.5 square meters. M1=2x2-3%+2 X=.05 Iy=i.855
> Decide which curves are graphs of functions. y = f(x) 3 3 3 + h Figure 17 Geometric representation of values.
> Consider a circle of radius r. Write an expression for the area. Write an equation expressing the fact that the circumference is 15 centimeters.
> Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. (25xy)3/2 / x2y
> Consider the rectangle in Exercise 1. Write an expression for the area. Write an equation expressing the fact that the perimeter is 30 centimeters. y = x³| (x, y) Slope is 3a Figure 15 Slope of tangent line to y = x³.
> Consider the rectangle in Exercise 1. Write an expression for the perimeter. If the area is 25 square feet, write this fact as an equation. y = -x - y = x² a
> Assign variables to the dimensions of the geometric object. y = x2 y = 2x – 1 a
> 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 (x + 2) + g(x + 2)
> Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. (16x8)-3/4
> An office supply firm finds that the number of laptop computers sold in year x is given approximately by the function f (x) = 150 + 2x + x2, where x = 0 corresponds to 2015. (a) What does f (0) represent? (b) Find the number of laptops sold in 2020.
> Compute the numbers. 1100
> 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)h(x)
> Let f (x) = x2 - 2x + 4, g(x) = 1/x2, and h(x) = 1/(√x - 1). Determine the following functions. f (g(x))
> Compute the numbers. (-2)3
> Compute the numbers. 33
> Convert the numbers from graphing calculator form to standard form (that is, without E). 8.23E-6
> Convert the numbers from graphing calculator form to standard form (that is, without E). 1.35E13
> Refer to the cost function in Fig. 18. Translate the task “find C(400)” into a task involving the graph. Vx for 0<x < 2 f (x) = - |1+x for 2 <x<5 f(1) = Vĩ = 1 f(2) = 1+2 = 3 f (3) = 1+3 = 4
> Convert the numbers from graphing calculator form to standard form (that is, without E). 8.103E-4
> Convert the numbers from graphing calculator form to standard form (that is, without E). 5E-5
> Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. 1 / yx-5
> Velocity When a car’s brakes are slammed on at a speed of x miles per hour, the stopping distance is 1 20x2 feet. Show that when the speed is doubled the stopping distance increases fourfold.
> Let f (x) = x6, g(x) = x / 1 - x, and h(x) = x3 - 5x2 + 1. Calculate the following functions. h (f (t))
> Draw the following intervals on the number line. (4, 3π)
> Semiannual Compound Assume that a $1000 investment earns interest compounded semiannually. Express the value of the investment after 2 years as a polynomial in the annual rate of interest r.
> Assume that a $500 investment earns interest compounded quarterly. Express the value of the investment after 1 year as a polynomial in the annual rate of interest r.
> Use intervals to describe the real numbers satisfying the inequalities. x ≥ 12
> Assume that a couple invests $4000 each year for 4 years in an investment that earns 8% compounded annually. What will the value of the investment be 8 years after the first amount is invested?
> Assume that a couple invests $1000 upon the birth of their daughter. Assume that the investment earns 6.8% compounded annually. What will the investment be worth on the daughter’s 18th birthday?
> Refer to the cost function in Fig. 18. Translate the task “solve C(x) = 3500 for x” into a task involving the graph of the function. C D B E A Figure 12
> If the cylinder in Exercise 6 has a volume of 54p cubic inches, find the surface area of the cylinder. Cylinder in Exercise 6: C D B E A Figure 12
> Calculate the compound amount from the given data. principal = $1500, compounded daily, 3 years, annual rate = 6%
> Calculate the compound amount from the given data. principal = $1500, compounded daily,1 year, annual rate = 6%
> Describe the domain of the function. f (x) =8x / (x - 1)(x - 2)
> Decide which curves are graphs of functions. 8 7 6 3 2 1 2 3 4 5 6 7 8 4) 1.
> Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. 2x / √x
> Calculate the compound amount from the given data. principal = $500, compounded monthly,1 year, annual rate = 4.5%
> Calculate the compound amount from the given data. principal = $100, compounded monthly, 10 years, annual rate = 5%
> In Exercises 47–50, find the zeros of the function. (Use the specified viewing window.) f (x) = x2 - x - 2; [-4, 5] [-4, 10]
> Calculate the compound amount from the given data. principal = $20,000, compounded quarterly, 3 years, annual rate = 12%
> Calculate the compound amount from the given data. principal = $50,000, compounded quarterly, 10 years, annual rate = 9.5%
> Calculate the compound amount from the given data. principal = $700, compounded annually, 8 years, annual rate = 8%
> In Exercises 51–54, find the points of intersection of the graphs of the functions. (Use the specified viewing window.) f (x) = 2x - 1; g(x) = x2 - 2; [-4, 4] by [-6, 10]
> 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 (x)g(x)
> Let f (x) = x/(x2 - 1), g(x) = (1 - x)/(1 + x), and h(x) = 2/(3x + 1). Express the following as rational functions. f (x) + g(x)
> A catering company estimates that, if it has x customers in a typical week, its expenses will be approximately C(x) = 550x + 6500 dollars, and its revenue will be approximately R(x) = 1200x dollars. (a) How much profit will the company earn in 1 week whe
