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mathematics
calculus

Questions and Answers of Calculus

In Exercises 1-4, find a polynomial function that has the given zeros. (There are many correct answers.) 1. 0, 7 2. -2, 5 3. 0, -2, -4 4. 0, 1, 6
In Exercises 1-4, find a polynomial of degree n that has the given zero(s). (There are many correct answers.) Zero(s) Degree 1. x = - 3 n = 2 2. x = - √2, √2 n = 2 3. x = −5, 0, 1
In Exercises 1-2, sketch the graph of the function by a. applying the Leading Coefficient Test, b. finding the real zeros of the polynomial, plotting sufficient solution points, and d. drawing
In Exercises 1-4, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine whether the multiplicity of each zero is
In Exercises 1-2, (a) use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a
In Exercises 1-6, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]a.b. c. d. e. f. 1. f (x) = ˆ’2x2 ˆ’ 5x 2. f (x) = 2x3
You construct an open box from a square piece of material, 36inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure).(a) Write a
You construct an open box with locking tabs from a square piece of material, 24inches on a side, by cutting equal sections from the corners and folding along the dashed lines (see figure).(a) Write a
The revenue R (in millions of dollars) for a software company from 2003 through 2016 can be modeled by R = 6.212t3 - 152.87t2 + 990.2t - 414, 3 ≤ t ≤ 16 where t represents the year, with t = 3
The revenue R (in millions of dollars) for a construction company from 2003 through 2010 can be modeled by R = 0.1104t4 - 4.152t3 + 88.20t2 - 654.8t + 1907, 7 ≤ t ≤ 16 where t represents the
The revenue R (in millions of dollars) for a beverage company is related to its advertising expense by the functionR = 1/100,000 (-x3 + 600x2), 0 ‰¤ x ‰¤ 400Where x is the
The growth of a red oak tree is approximated by the function G = -0.003t3 + 0.137t2 + 0.458t - 0.839, 2 ‰¤ t ‰¤. Where G is the height of the tree (in feet) and t is its age
True or False? In Exercises 1-4, determine whether the statement is true or false. Justify your answer. 1. If the graph of a polynomial function falls to the right, then its leading coefficient is
Use long division to divide. 1. (2x2 + 10x + 12) ÷ (x + 3) 2. (5x2 − 17x − 12) ÷ (x − 4) 3. (4x3 − 7x2 − 11x + 5) ÷ (4x + 5) 4. (6x3 − 16x2 + 17x − 6) ÷ (3x − 2)
Use synthetic division to divide. 1. (2x3 − 10x2 + 14x − 24) ÷ (x − 4) 2. (5x3 + 18x2 + 7x − 6) ÷ (x + 3) 3. (6x3 + 7x2 − x + 26) ÷ (x − 3) 4. (2x3 + 12x2 + 14x − 3) ÷ (x + 4)
Write the function in the form f (x) = (x - k)q(x) + r for the given value of k, and demonstrate that f (k) = r. 1. f (x) = x3 − x2 − 10x + 7, k = 3 2. f (x) = x3 − 4x2 − 10x + 8, k = −2 3.
Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. f (x) = 2x3 − 7x + 3 (a) f (1) (b) f (−2) (c) f (3) (d) f (2)
Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. 1. x3 +
(a) verify the given factors of f (x), (b) find the remaining factor(s) of f (x), (c) use your results to write the complete factorization of f (x), (d) list all real zeros of f, and (e)
Use long division to verify that y1 = y2 1. y1 = x2/x + 2, y2 = x - 2 + 4/x + 2 2. y1 = x3 - 3x2 + 4x - 1/x + 3, y2 = x2 - 6x + 22 - 67/x + 3
(a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine the exact value of one of the zeros, and (c) use
Simplify the rational expression by using long division or synthetic division. 1. x3 + x2 - 64x - 64/x + 8 2. 4x3 - 8x2 + x + 3/2x - 3 3. x4 + 6x3 + 11x2 + 6x/x2 +3x + 2
A company that produces calculators estimates that the profit P (in dollars) from selling a specific model of calculator is given by P = - 152x3 + 7545x2 - 169,625, 0 ≤ x ≤ 45 where x is the
The numbers N of confirmed cases of Lyme disease in Maryland from 2007 through 2014 are shown in the table, where t represents the year, with t = 7 corresponding to 2007. (Source: Centers for Disease
Determine whether the statement is true or false. Justify your answer. 1. If (7x + 4) is a factor of some polynomial function f (x), then 4/7 is a zero of f. 2. (2x − 1) is a factor of the
Perform the division. Assume that n is a positive integer. 1. x3n + 9x2n + 27xn + 27/xn + 3 2. x3n - 3x2n +5xn - 6/xn - 2
Use synthetic division to find the remainder when x2 + 3x ˆ’ 5 is divided by x + 1.
(a) Use a graphing utility to graph the two equations in the same viewing window, (b) use the graphs to verify that the expressions are equivalent, and (c) use long division to verify the results
Find the constant c such that the denominator will divide evenly into the numerator. 1. x3 + 4x2 - 3x + c/x - 5 2. x5 - 2x2 + x +c/x + 2
Find the value of k such that x - 4 is a factor of x3 − kx2 + 2kx − 8.
You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let x represent the side
A rectangular package to be sent by a delivery service (see figure) has a combined length and girth (perimeter of a cross section) of 120 inches.a) Use the diagram to write the volume V of the
A bulk food storage bin with dimensions 2feet by 3feet by 4feet needs to be increased in size to hold five times as much food as the current bin.(a) Assume each dimension is increased by the same
The ordering and transportation cost C (in thousands of dollars) for machine parts is given by C (x) = 100(200/x2 + x/x + 30), x ≤ 1 where x is the order size (in hundreds). In calculus, it can be
Decide whether the statement is true or false. Justify your answer. 1. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros. 2. If x = −i is a zero
Determine (if possible) the zeros of the function g when the function f has zeros at x = r1, x = r2, and x = r3. 1. g(x) = −f (x) 2. g(x) = 3f (x) 3. g(x) = f (x − 5) 4. g(x) = f (2x)
1. Cubic polynomial function f has real zeros −2, 1/2, and 3, and its leading coefficient is negative. Write an equation for f and sketch its graph. How many different polynomial functions are
The graph of a cubic polynomial function y = f (x) is shown. One of the zeros is 1 + i. Write an equation for f.1.2.
Describe the error. The graph of a quartic (fourth-degree) polynomial y = f (x) is shown. One of the zeros is i.
Let y = f (x) be a quartic (fourth-degree) polynomial with leading coefficient a = 1 and f (i) = f (2i) = 0. Write an equation for f.
Let y = f (x) be a cubic polynomial with leading coefficient a = −1 and f (2) = f (i) = 0. Write an equation for f.
Write the equation for a quadratic function f (with integer coefficients) that has the given zeros. Assume that b is a positive integer. (a) ±√bi (b) a ±√bi
Use the Rational Zero Test to list the possible rational zeros of f. Verify that the zeros of f shown in the graph are contained in the list.1. f (x) = x3 + 2x2 ˆ’ x - 22. f (x) = x3
Find (if possible) the rational zeros of the function. 1. f (x) = x3 − 7x - 6 2. f (x) = x3 − 13x + 12 3. g(t) = t3 − 4t2 + 4
Find all real solutions of the polynomial equation. −5x3 + 11x2 − 4x − 2 = 0
(a) list the possible rational zeros of f, (b) sketch the graph of f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. 1. f (x) = x3
(a) list the possible rational zeros of f, (b) use a graphing utility to graph f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of
Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 1. 1, 5i 2. 4, -3i 3. 2, 2, 1 + i
Find the polynomial function f with real coefficients that has the given degree, zeros, and solution point.
Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in
Use the given zero to find all the zeros of the function. Function Zero 1. f (x) = x3 - x= + 4x - 4 2i 2. f (x) = 2x3 + 3x= + 18x + 27 3i
Write the polynomial as the product of linear factors and list all the zeros of the function. 1. f (x) = x2 + 36 2. f (x) = x2 + 49 3. h(x) = x2 − 2x + 17 4. g(x) = x2 + 10x + 17
Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros
Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. 1. g(x) = 2x3 − 3x2 - 3 2. h(x) = 4x2 − 8x + 3 3. h(x) = 2x3 + 3x2 + 1 4. h(x)
Use synthetic division to verify the upper and lower bounds of the real zeros of f. 1. f (x) = x3 + 3x2 − 2x + 1 a) Upper: x = 1 b) Lower: x = - 4 2. f (x) = x3 − 4x2 + 1 a) Upper: x = 4 b)
Determine the number of zeros of the polynomial function. 1. f (x) = x3 + 2x2 + 1 2. f (x) = x4 − 3x 3. g (x) = x4 − x5 4. f (x) = x3 − x6 5. f (x) = (x + 5)2 6. h (t) = (t − 1)2 − (t + 1)2
Find all real zeros of the function. f (x) = 16x3 − 12x2 − 4x + 3
Find the rational zeros of the polynomial function. 1. P(x) = x4 - 25/4x2 + 9 = 1/4 (4x4 − 25x2 + 36) 2. f (x) = x3 - 3/2x2 - 23/2 x + 6 = 1/2(2x3 − 3x2 − 23x + 12)
Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: 0; irrational zeros: 1 (b) Rational zeros: 3; irrational zeros: 0 (c) Rational zeros: 1; irrational
1. Two techniques for fitting models to data are direct and inverse ________ and least squares ________. 2. Statisticians use a measure called the ________ of the ________ ________ to find a model
The ordered pairs below give the revenues y (in billions of dollars) for Activision Blizzard, Inc., from 2008 through 2014. (Spreadsheet at LarsonPrecalculus.com) (2008, 3.03) (2009, 4.28) (2010,
Sketch the line that you think best approximates the data in the scatter plot. Then find an equation of the line. To print an enlarged copy of the graph, go to MathGraphs.com.1.2. 3.
The ordered pairs below give the winning times (in seconds) of the women's 100-meter freestyle in the Olympics from 1984 through 2012. (Spreadsheet at LarsonPrecalculus.com) (Source: International
The ordered pairs below give the starting year and gross ticket sales S (in millions of dollars) for each Broadway season in New York City from 1997 through 2014. (Spreadsheet at (Source: The
Find a direct variation model that relates y and x. 1. x = 2, y = 14 2. x = 5, y = 1 3. x = 5, y = 1 4. x = -24, y = 3 5. x = 4, y = 8π 6. x = π, y = -1
Use the given values of k and n to complete the table for the direct variation model y = kxn. Plot the points in a rectangular coordinate system.
Use the given values of k and n to complete the table for the inverse variation model y = k/xn. Plot the points in a rectangular coordinate system.
Use the graph to determine whether y varies directly as some power of x or inversely as some power of x. Explain.1.2. Determine whether the variation model represented by the ordered pairs (x, y) is
Find a mathematical model for the verbal statement. 1. A varies directly as the square of r. 2. V varies directly as the cube of l. 3. Y varies inversely as the square of x. 4. H varies inversely as
Use variation terminology to describe the formula. 1. y = 2x2 2. t = 72 / r 3. A = 1 / 2 bh 4. K = 1 / 2mv2
Find a mathematical model that represents the statement. (Determine the constant of proportionality.) 1. y is directly proportional to x. (y = 54 when x = 3.) 2. A varies directly as r2. (A = 9π
1. The simple interest on an investment is directly proportional to the amount of the investment. An investment of $3250 earns $113.75 after 1 year. Find a mathematical model that gives the interest
1. A force of 220 newtons stretches a spring 0.12 meter. What force stretches the spring 0.16 meter? 2. A force of 265 newtons stretches a spring 0.15 meter. (a) What force stretches the spring
1. The coiled spring of a toy supports the weight of a child. The weight of a 25-pound child compresses the spring a distance of 1.9 inches. The toy does not work properly when a weight compresses
1. The diameter of the largest particle that a stream can move is approximately directly proportional to the square of the velocity of the stream. When the velocity is 1/4 mile per hour, the stream
The ordered pairs below give the average water temperatures C (in degrees Celsius) at several depths d (in meters) in the Indian Ocean. (Spreadsheet at LarsonPrecalculus.com) (1000, 4.85) (1500,
The ordered pairs below give the intensities y (in microwatts per square centimeter) of the light measured by a light probe located x centimeters from a light source. (Spreadsheet at
The fundamental frequency (in hertz) of a piano string is directly proportional to the square root of its tension and inversely proportional to its length and the square root of its mass density. A
The maximum load that a horizontal beam can safely support varies jointly as the width of the beam and the square of its depth and inversely as the length of the beam. Determine how each change
Decide whether the statement is true or false. Justify your answer. 1. If y is directly proportional to x and x is directly proportional to z, then y is directly proportional to z. 2. If y is
78. Discuss how well a linear model approximates the data shown in each scatter plot.(a)(b) (c) (d) 79. Let y = 2x + 2 and t = x + 1.What kind of variation do y and t have? Explain.
1. The ordered pairs below give the civilian noninstitutional U.S. populations y (in millions of people) 16 years of age and over not in the civilian labor force from 2006 through 2014. (Spreadsheet
Sketch the graph of each quadratic function and compare it with the graph of y = x2. 1. (a) f (x) = 2x2 (b) g (x) = −2x2 (c) h (x) = x2 + 2 (d) k (x) = (x + 2)2 2. (a) f (x) = x2 − 4
The table shows the amount B (in millions of pounds) of beef produced on private farms each year from 2007 through 2014.(a) Use a graphing utility to create a scatter plot of the data. Let t
A billboard says that it is 12.5miles or 20 kilometers to the next gas station. Use this information to find a mathematical model that relates miles x to kilometers y. Then use the model to find
The travel time between two cities is inversely proportional to the average speed. A train travels between the cities in 3 hours at an average speed of 65 miles per hour. How long does it take to
Explain how to determine the maximum or minimum value of a quadratic function.
Explain the connections between factors of a polynomial, zeros of a polynomial function, and solutions of a polynomial equation.
Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.1.2. 3. Vertex: (6, 0); point: (3, ˆ’9)
A rectangle is inscribed in the region bounded by the x-axis, the y-axis, and the graph ofx + 2y ˆ’ 8 = 0, as shown in the figure.(a) Write the area A of the rectangle as a function of
The perimeter of a rectangle is 200meters. (a) Draw a diagram that gives a visual representation of the problem. Let x and y represent the length and width of the rectangle, respectively. (b) Write
The total revenue R earned (in dollars) from producing a gift box of tea is given by R (p) = −10p2 + 800p where p is the price per box (in dollars). (a) Find the revenues when the prices per box
A real estate office handles an apartment building that has 50 units. When the rent is $540 per month, all units are occupied. However, for each $30 increase in rent, one unit becomes vacant. Each
A soft-drink manufacturer has a daily production cost of C = 70,000 − 120x + 0.055x2 where C is the total cost (in dollars) and x is the number of units produced. How many units should they produce
A small theater has a seating capacity of 2000. When the ticket price is $20, attendance is 1500. For each $1 decrease in price, attendance increases by 100. (a) Write the revenue R of the theater as
Sketch the graphs of y = xn and the transformation. 1. y = x3, f (x) = (x − 2)3 2. y = x3, f (x) = 4x3 3. y = x4, f (x) = 6 − x4
Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and x-intercept(s). 1. g (x) = x2 − 2x 2. f (x) = 6x − x2
Describe the left-hand and right-hand behavior of the graph of the polynomial function. 1. f (x) = −2x2 − 5x + 12 2. f (x) = 4x - ½ x3 3. g (x) = −3x3 − 8x4 + x5 4. h (x) = 5 + 9x6 − 6x5
(a) Find all real zeros of the polynomial function, (b) Determine whether the multiplicity of each zero is even or odd, (c)determine the maximum possible number of turning points of the graph of the
Sketch the graph of the function by (a) Applying the Leading Coefficient Test, (b) Finding the real zeros of the polynomial, (c) Plotting sufficient solution points, and (d) Drawing a continuous

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