(188) Hw 3.1 (Homework)

(188) Hw 3.2 (Homework)

' g A soft-drink vendor at a popular beach analyzes his sales records and nds that if he sells x cans of soda pop in one day, his prot {in dollars} is given by P{x} = 0.001x2 + 4x 1375' What is his maximum prot per day? $|:| How many cans must he sell for maximum prot? Need Help? 13. [-/0.62 Points] DETAILS SALGTRIG4 3.1.062. A wire 10 cm long is cut into two pieces, one of length x and the other of length 10 - x, as shown in the figure. Each piece is bent into the shape of a square. -10 cm 10 - x (a) Find a function that models the total area enclosed by the two squares. A(x) = (b) Find the value of x that minimizes the total area of the two squares. X = cm Need Help? Read ItCarol has 1200 ft of fencing to fence in a rectangular horse corral. 600 - X (a) Find a function that models the area A of the corral in terms of the width x of the corral. A(X) = (b) Find the dimensions of the rectangle that maximize the area of the corral. width ft length ft Need Help? Read It Watch It15. [-/0.62 Points] DETAILS SALGTRIG4 3.1.064. A rain gutter is formed by bending up the sides of a 34-in.-wide rectangular metal sheet as shown in the figure. width (a) Find a function that models the cross-sectional area of the gutter in terms of x. A(X) = (b) Find the value of x that maximizes the cross-sectional area of the gutter. X = in (c) What is the maximum cross-sectional area for the gutter? in2 Need Help? Read ItOnly one of the following graphs could be the graph of a polynomial function. Which one? Why are the others not graphs of polynomials? (Select all that apply.) y X The graph could be that of a polynomial function. The graph could not be that of a polynomial function because it has a cusp. O The graph could not be that of a polynomial function because it has a break. The graph could not be that of a polynomial function because it does not pass the horizontal line test. The graph could not be that of a polynomial function because it is not smooth. II y 4 X O The graph could be that of a polynomial function. O The graph could not be that of a polynomial function because it has a cusp. O The graph could not be that of a polynomial function because it has a break. O The graph could not be that of a polynomial function because it does not pass the horizontal line test. The graph could not be that of a polynomial function because it is not smooth.III The graph could be that of a polynomial function. The graph could not be that of a polynomial function because it has a cusp. The graph could not be that of a polynomial function because it has a break. O The graph could not be that of a polynomial function because it does not pass the horizontal line test. O The graph could not be that of a polynomial function because it is not smooth. IV O The graph could be that of a polynomial function. O The graph could not be that of a polynomial function because it has a cusp. The graph could not be that of a polynomial function because it has a break. O The graph could not be that of a polynomial function because it does not pass the horizontal line test. O The graph could not be that of a polynomial function because it is not smooth. Need Help? Read It3. [-/0.37 Points] DETAILS HPRECALC5 4.2.003.MC. Determine whether the given algebraic expression is a polynomial. If it is, list its degree, leading coefficient, and constant term. (x - 1)(x + 2) Opolynomial of degree 5; leading coefficient 1; constant term 2 Opolynomial of degree 4; leading coefficient 1; constant term - 2 Opolynomial of degree 4; leading coefficient 1; constant term 2 ONot a polynomial. Opolynomial of degree 5; leading coefficient 1; constant term - 2Determine whether the given algebraic expression is a polynomial. If it is, list its degree, leading coefficient, and constant term. 8 + 84 Opolynomial of degree 1; leading coefficient 2; constant term - 84 Opolynomial of degree -1; leading coefficient 2; constant term - 84 Opolynomial of degree -2; leading coefficient 8; constant term - 84 ONot a polynomial. Opolynomial of degree 2; leading coefficient 8; constant term - 84A graphing calculator is recommended. Determine the end behavior of P. P(X) = - 2 + 13x 4 6 as x - 00 as x - -00 Need Help? Read ItA polynomial function is given. (x) = -x (x - 9) (=) Describe the end behavior of the polynomial function. End behavior: y = as x - 00 as x - -00 (b) Match the polynomial function with one of the following graphs. 20- 20- 10 LL 3 -2 -1 -3 -2 2 3 -10 -10 -20 -20 O O 20- 20- 10 10 -2 1 2 -3 -2 1 2 -10 - 10 -20 20 O O 20- 10 10 -3 -2 2 2 -10 -10A graphing calculator is recommended. Determine the end behavior of P. P(x) = xil - 7x9 35 x - 00 as x - -00 Need Help? Read It Watch It 9. [-/0.37 Points] DETAILS SALGTRIG4 3.2.002. Describe the end behavior of each polynomial. (a) y = x3 - 4x2 + 3x - 17 End behavior: y 35 x + 00 35 x- -00 (b) y= -3x*+ 13x + 700 End behavior: y - V 35 x- -00 Need Help? Read ItS( x) = 2x6 - 2x4 (a) Describe the end behavior of the polynomial function. End behavior: y - as x + 00 as x - -00 (b) Match the polynomial function with one of the following graphs. y 10- 10- JUL X UL X -3 -2 2 -5 -5 10 O -10- O 10- 10- 12 -3 -2 -1 2 -5 -5 -10 -10- O 10 10 5 -3 -2 -1 -3 2 -5 -5 -10 O O -10- d Help? Read It1. [-70.2/ Points] DETAILS If c is a zero of the polynomial P, then consider the following. (a) P(c) = ? v (b) x - c is a ---Select--. v of P(x). (c) c is a(n) ?v -intercept of the graph of P. Need Help? Read ItFactor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) P(x) = -2x3 - x2 + x * = Sketch the graph. 2- 1- 1 LX X -2 1 -1 -2 -1 -1 -21 -2- O O N 1 -T X -2 IN -2 2 -1 - 1- O O Need Help? Read It Master It13. [-/0.37 Points] DETAILS SALGTRIG4 3.2.038.MI. Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) F(x) = x3 + 4x2 - 9x - 36 * = Sketch the graph. 40 20- 20 - - X -5 -5 5 -20 - 20 -40 O O 40 40 20 20- - X -5 5 -20 20 -40 -40 O O Need Help? Read It Watch It Master It14. [-/0.37 Points] DETAILS LARPCALC9 2.2.055. MY NOTES A Find a polynomial function that has the given zeros. (There are many correct answers.) 0, 2 F( x ) = 15. [-/0.37 Points] DETAILS LARPCALC9 2.2.006. Fill in the blanks. A factor (x - a)*, k > 1, yields [--Select-.. v x = a of ---Select... 16. [-/0.37 Points] DETAILS LARPCALC9 2.2.005. Fill in the blanks. When a real zero of a polynomial function fis of even multiplicity, the graph of f [--Select--- the x-axis at x = a, and when it is of odd multiplicity, the graph of f [---Select--- V the x-axis at x = a.Consider the following. g (t) = > - 10t- + 25t (a) Find all real zeros of the polynomial function. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) t = (b) Determine the multiplicity of each zero. smallest t-value ---Select--- ---Select--- largest t-value ---Select--- (c) Determine the maximum possible number of turning points of the graph of the function. turning point (s) (d) Use a graphing utility to graph the function and verify your answers. g(t) g(t) 20 20 10 -10 -5 5 10 -10 -5 5 10 - 10 20 -20 O O g(t) g(t) 20 20 10- -t - 10 -5 5 10 -10 -5 5 10 -10 -20/ -20 O OConsider the following. g(x) = x5 + 4x2 - 9x - 36 (a) Find all real zeros of the polynomial function. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) X = (b) Determine the multiplicity of each zero. smallest x-value ---Select--- ---Select--- largest x-value ---Select--- (c) Determine the maximum possible number of turning points of the graph of the function. | turning point(s) (d) Use a graphing utility to graph the function and verify your answers. g(X) g(x) 50 50 L X -10 -5 X 5 10 - 10 5 10 -50 -50 O O g(x) g(x) 50 50 - X -10 5 10 -10 - X -5 5 10 -50 -50 O19. [-/0.37 Points] DETAILS LARPCALC9 2.2.091. Use a graphing utility to graph the function. g (x ) = =(x + 2)2(x - 3)(2x - 9) g(x) g(x) 100 100 50- 50- -6 L -2 2 4 6 -6 LX -4 -2 2 4 6 -50 .50 O -100 O -100 g(x g(x) 100 100 50- -6 X -4 -2 2 -6 -4 -2 X 2 4 -50 -50 O -100 O - 100 Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero. (Round each zero to one decimal place.) * = ---Select-- (smallest x-value) ---Select--- ---Select-- (largest x-value)The graph of a polynomial function is given. F(x) = -x-+ 6x 101 -10 -5 10 -10 (a) From the graph, find the x- and y-intercepts. (If an answer does not exist, enter DNE.) x-intercepts (x , y ) = (smaller x-value) (x , V ) = (larger x-value) y-intercept (x, y ) = (b) Find the coordinates of all local extrema. (If an answer does not exist, enter DNE.) local minimum ( x , y ) = local maximum (x, y) = Need Help? Read It Watch ItA graphing calculator is recommended. Graph the polynomial in the given viewing rectangle. y = x3 - 12x + 3, [-5, 5] by [-20, 20] 20- 20 10 - x -2 -4 -2 A -10 - 10 -20 -20 O O 20 20 10 10 - X -4 -2 2 -4 -2 -10 -20 -20 Find the coordinates of all local extrema. State each answer rounded to two decimal places. (If an answer does not exist, enter DNE.) local minimum (x, y) = local maximum (x, y) = State the domain and range. (Enter your answers using interval notation. Round your answers to two decimal places.) domain rangeA graphing calculator is recommended. Graph the polynomial, and determine how many local maxima and minima it has. y = 5x5 - x2 - 3x The polynomial has ---Select--- and ---Select-- Need Help? Read It Watch It(a) On the same coordinate axes, sketch graphs (as accurately as possible) of the functions y = x3 - 2x2 - x + 2 and y = -x2 + 11x + 2. 50 - X 5 10 -10 -5 10 -50 O O 50 50 LX -10 -5 10 -10 -5 5 10 -50 -50 O O (b) On the basis of your sketch in part (a), at how many points do the two graphs appear to intersect? (c) Find the coordinates of all intersection points. (Select all that apply.) O (0, -2) O (4, -30) O (-3, -40) O (-3, 40) O (4, 30) O (0, 2)4. [-/0.37 Points] DETAILS SALGTRIG4 3.2.083. This exercise involves local maxima and minima of polynomial functions. A graphing calculator is recommended. (a) Graph the function P(x) = (x - 3)(x - 5)(x - 6) and find all local extrema, correct to the nearest tenth. local minimum (x, y) = local maximum (x, y ) = (b) Graph the function Q(x) = (x - 3)(x - 5)(x - 6) + 7 and use your answers to part (a) to find all local extrema, correct to the nearest tenth. local minimum (x, y) = local maximum (x, y) = Need Help? Read ItA graphing calculator is recommended. A market analyst working for a small-appliance manufacturer finds that if the firm produces and sells x blenders annually, the total profit (in dollars) is P(x) = 8x + 0.3x2 - 0.0013x5 - 374. Graph the function P in an appropriate viewing rectangle and use the graph to answer the following questions. (a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, P(10) = -265.30, so the firm loses $265.30 if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even? (Round your answer to the integer that gives the company the least positive profit.) (b) Does profit increase indefinitely as more blenders are produced and sold? O Yes O No If not, what is the largest possible profit the firm could have? (Round your answer to the nearest cent. If an answer does not exist, enter DNE.) $ Need Help? Read It Watch It26. [-/0.37 Points] DETAILS SALGTRIG4 3.2.089. A graphing calculator is recommended. An open box is to be constructed from a piece of cardboard 20 cm by 40 cm by cutting squares of side length x from each corner and folding up the sides, as shown in the figure. length width (a) Express the volume V of the box as a function of x. V( x ) = (b) What is the domain of V? (Use the fact that length and volume must be positive. Enter your answer using interval notation.) (c) Draw a graph of the function V. 4 3000 3000 2500 2500 2000 2000 1500 1500 1000 1000 500 500 O 2 6 8 12 O 2 6 8 10 12 3000 3000 2500 2500 2000 2000 1500 1500 1000 1000 500 500 6 8 12 2 6 8 10 12\flJ A graphing calculator is recommended. A quadratic function is given. x) : 1 + X x2 (a) Use a graphing device to nd the maximum or minimum value of the quadratic function f, rounded to two decimal places. The rm = :- (b) Find the exact maximum or minimum value of f, and compare it with your answer to part (a). Z Need rue-p? Find the maximum or minimum value of the function. h[x]|=%x2+4x m = E 15 this a maximum or minimum value? 0 maximum value 0 minimum value Need Help? Find a function f whose graph is a parabola with the given vertex and that passes through the given point. vertex (1, -1); point (2, 5) f(x ) Need Help? Read ItA ball is thrown across a playing field from a height of h = 5 ft above the ground at an angle of 450 to the horizontal at the speed of 20 ft/s. It can be deduced from physical principles that the path of the ball is modeled by the function y = - (20)2 x2 + x + 5 where x is the distance in feet that the ball has traveled horizontally. (a) Find the maximum height attained by the ball. (Round your answer to three decimal places.) ft (b) Find the horizontal distance the ball has traveled when it hits the ground. (Round your answer to one decimal place.) ft