Mylab Math With Pearson Etext -- Standalone Access Card -- For Precalculus (11th Edition)
11th Edition
ISBN: 9780135189795
Author: Michael Sullivan
Publisher: PEARSON
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Textbook Question
Chapter 4.7, Problem 60AYU
Problems
Find the difference equation of
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1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
i need help please
6.
(i)
Sketch the trace of the following curve on R²,
(t) = (sin(t), 3 sin(t)),
tЄ [0, π].
[3 Marks]
Total marks 10
(ii)
Find the length of this curve.
[7 Marks]
Chapter 4 Solutions
Mylab Math With Pearson Etext -- Standalone Access Card -- For Precalculus (11th Edition)
Ch. 4.1 - The intercepts of the equation 9 x 2 +4y=36 are...Ch. 4.1 - Is the expression 4 x 3 3.6 x 2 2 a polynomial?...Ch. 4.1 - To graph y= x 2 4 , you would shift the graph of...Ch. 4.1 - True or False The x-intercepts of the graph of a...Ch. 4.1 - Multiple Choice The cube function f(x)=x3 is ....Ch. 4.1 - The graph of every polynomial function is both...Ch. 4.1 - If r is a real zero of even multiplicity of a...Ch. 4.1 - The graphs of power functions of the form f(x)= x...Ch. 4.1 - If r is a solution to the equation f(x)=0 , name...Ch. 4.1 - The points at which a graph changes direction...
Ch. 4.1 - The graph of the function f(x)=3x4x3+5x22x7...Ch. 4.1 - If f( x )=2 x 5 + x 3 5 x 2 +7 , then lim x f( x...Ch. 4.1 - The _______ of a zero is the number of times its...Ch. 4.1 - The graph of y=5 x 6 3 x 4 +2x9 has at most how...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 1526, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 17-28, determine which functions are...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 29-42, use transformations of the...Ch. 4.1 - In Problems 43-50, form a polynomial function...Ch. 4.1 - In Problems 43-50, form a polynomial function...Ch. 4.1 - In Problems 4148, find a polynomial function whose...Ch. 4.1 - In Problems 43-50, form a polynomial function...Ch. 4.1 - In Problems 4148, find a polynomial function whose...Ch. 4.1 - In Problems 43-50, form a polynomial function...Ch. 4.1 - In Problems 43-50, form a polynomial function...Ch. 4.1 - In Problems 43-50, form a polynomial function...Ch. 4.1 - In Problems, find a polynomial function with the...Ch. 4.1 - In Problems 51-56, find the polynomial function...Ch. 4.1 - In Problems, find a polynomial function with the...Ch. 4.1 - In Problems 51-56, find the polynomial function...Ch. 4.1 - In Problems 115-118, construct a polynomial...Ch. 4.1 - In Problems 115-118, construct a polynomial...Ch. 4.1 - In Problems 51-56, find the polynomial function...Ch. 4.1 - In Problems 51-56, find the polynomial function...Ch. 4.1 - In Problems 115-118, construct a polynomial...Ch. 4.1 - In Problems 115-118, construct a polynomial...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 5970, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 5970, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 57-68, for each polynomial function:...Ch. 4.1 - In Problems 69-72, identify which of the graphs...Ch. 4.1 - In Problems 69-72, identify which of the graphs...Ch. 4.1 - In Problems 69-72, identify which of the graphs...Ch. 4.1 - In Problems 69-72, identify which of the graphs...Ch. 4.1 - In Problems 73-76, construct a polynomial function...Ch. 4.1 - In Problems 73-76, construct a polynomial function...Ch. 4.1 - In Problems 73-76, construct a polynomial function...Ch. 4.1 - In Problems 73-76, construct a polynomial function...Ch. 4.1 - In Problems 77-80, write a polynomial function...Ch. 4.1 - In Problems 77-80, write a polynomial function...Ch. 4.1 - In Problems 77-80, write a polynomial function...Ch. 4.1 - In Problems 77-80, write a polynomial function...Ch. 4.1 - G( x )= (x+3) 2 (x2) a. Identify the x-intercepts...Ch. 4.1 - h( x )=( x+2 ) ( x4 ) 3 a. Identify the...Ch. 4.1 - 85. Challenge Problem Find the real zero of
...Ch. 4.1 - 86. Challenge Problem Determine the power function...Ch. 4.1 - Can the graph of a polynomial function have no...Ch. 4.1 - The illustration shows the graph of a polynomial...Ch. 4.1 - Which of the following statements are true...Ch. 4.1 - Which of the following statements are true...Ch. 4.1 - Make up two polynomials, not of the same degree,...Ch. 4.1 - Make up a polynomial that has the following...Ch. 4.1 - Write a few paragraphs that provide a general...Ch. 4.1 - Design a polynomial function with the following...Ch. 4.1 - Find the equation of the line that contains the...Ch. 4.1 - Find the domain of the function h( x )= x3 x+5 .Ch. 4.1 - Problems are based on material learned earlier in...Ch. 4.1 - Problems are based on material learned earlier in...Ch. 4.1 - Problems 95104 are based on material learned...Ch. 4.1 - Problems are based on material learned earlier in...Ch. 4.1 - Problems are based on material learned earlier in...Ch. 4.1 - Problems 95104 are based on material learned...Ch. 4.1 - Problems are based on material learned earlier in...Ch. 4.1 - Problems 95104 are based on material learned...Ch. 4.2 - What are the intercepts of y=5x+10?Ch. 4.2 - Determine the leading term of ?
Ch. 4.2 - Use a graphing utility to approximate (rounded to...Ch. 4.2 - Use a graphing utility to find the quadratic...Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems 522, graph each polynomial function....Ch. 4.2 - In Problems graph each polynomial function.
Ch. 4.2 - In Problems 2330, use a graphing utility to graph...Ch. 4.2 - In Problems use a graphing utility to graph each...Ch. 4.2 - In Problems use a graphing utility to graph each...Ch. 4.2 - In Problems 2330, use a graphing utility to graph...Ch. 4.2 - In Problems 2330, use a graphing utility to graph...Ch. 4.2 - In Problems use a graphing utility to graph each...Ch. 4.2 - In Problems 2330, use a graphing utility to graph...Ch. 4.2 - In Problems 2330, use a graphing utility to graph...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems graph each polynomial function f.
...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems graph each polynomial function f.
...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems graph each polynomial function f.
...Ch. 4.2 - In Problems 3142, graph each polynomial function...Ch. 4.2 - In Problems graph each polynomial function f.
...Ch. 4.2 - Hurricanes In 2012. Hurricane Sandy struck the...Ch. 4.2 - Poverty Rates The data (top, right) represent the...Ch. 4.2 - Temperature The following data reprint the...Ch. 4.2 - Future Value of Money Suppose that you make...Ch. 4.2 - Challenge Problem A Geometric Series In calculus...Ch. 4.2 - Challenge Problem Tennis Anyone? Assume that the...Ch. 4.2 - Challenge Problem Suppose f(x)=ax2(xb)(x+c)2,...Ch. 4.2 - Challenge Problem Suppose , where .
Graph.
In...Ch. 4.2 - Determine the function that is graphed if the...Ch. 4.2 - Find the vertex of the graph of f(x)=2x2+7x3.Ch. 4.2 - The strain S on a solid object depends on the...Ch. 4.2 - Given f(x)=2x37x+1, find f(12).Ch. 4.2 - Find the domain of f(x)=9x4+1.Ch. 4.2 - Find the average rate of change of f(x)=x2+4x3...Ch. 4.2 - Prob. 57AYUCh. 4.2 - Determine if the function g(x)=x3x3x is even, odd,...Ch. 4.2 - How long will it take to grow to at a simple...Ch. 4.3 - True or False The quotient of two polynomial...Ch. 4.3 - What are the quotient and remainder when 3 x 4 x...Ch. 4.3 - Graph y= 1 x .(pp.22-23)Ch. 4.3 - Graph y=2 ( x+1 ) 2 3 using...Ch. 4.3 - True or False The domain of every rational...Ch. 4.3 - If, as x or as x , the values of R( x ) approach...Ch. 4.3 - If, as x approaches some number c , the values of...Ch. 4.3 - For a rational function R , if the degree of the...Ch. 4.3 - Prob. 9AYUCh. 4.3 - True or False The graph of a rational function may...Ch. 4.3 - True or False If the degree of the numerator of a...Ch. 4.3 - If a rational function is proper, then _____ is a...Ch. 4.3 - If R( x )= p( x ) q( x ) is a rational function...Ch. 4.3 - Which type of asymptote, when it occurs, describes...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 15-26, find the domain of each...Ch. 4.3 - In Problems 27-32, use the graph shown to find a....Ch. 4.3 - In Problems 27-32, use the graph shown to find a....Ch. 4.3 - In Problems 27-32, use the graph shown to find a....Ch. 4.3 - In Problems 27-32, use the graph shown to find a....Ch. 4.3 - In Problems 27-32, use the graph shown to find a....Ch. 4.3 - In Problems 27-32, use the graph shown to find a....Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 33-44, (a) graph the rational function...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In problems , (a) graph the rational function...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - In Problems 45-56, find the vertical, horizontal,...Ch. 4.3 - Population Model A rare species of insect was...Ch. 4.3 - Resistance in Parallel Circuits From Ohm’s Law...Ch. 4.3 - Exploration The standard form of the rational...Ch. 4.3 - Exploration Repeat Problem 61 for the rational...Ch. 4.3 - Prob. 63AYUCh. 4.3 - If the graph of a rational function R has the...Ch. 4.3 - The graph of a rational function cannot have both...Ch. 4.3 - If the graph of a rational function R has the...Ch. 4.3 - Problems 67-70 are based on material learned...Ch. 4.3 - Prob. 68AYUCh. 4.3 - Problems 67-70 are based on material learned...Ch. 4.3 - Prob. 70AYUCh. 4.3 - Prob. 71AYUCh. 4.3 - Prob. 72AYUCh. 4.3 - Problems are based on material learned earlier in...Ch. 4.3 - Prob. 74AYUCh. 4.3 - Problems 6776 are based on material learned...Ch. 4.3 - Prob. 76AYUCh. 4.4 - True or False The quotient of two polynomial...Ch. 4.4 - Prob. 2AYUCh. 4.4 - Prob. 3AYUCh. 4.4 - Prob. 4AYUCh. 4.4 - R( x )= x ( x2 ) 2 x2 (a) Find the domain of R ....Ch. 4.4 - Prob. 6AYUCh. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 7-50, follow Steps 1 through 7 on page...Ch. 4.4 - In Problems 51-54, find a rational function that...Ch. 4.4 - In Problems 51-54, find a rational function that...Ch. 4.4 - In Problems 51-54, find a rational function that...Ch. 4.4 - In Problems 51-54, find a rational function that...Ch. 4.4 - Probability Suppose you attend a fundraiser where...Ch. 4.4 - Waiting in Line Suppose that two employees at a...Ch. 4.4 - Drug Concentration The concentration C of a...Ch. 4.4 - Drug Concentration The concentration C of a...Ch. 4.4 - Minimum Cost A rectangular area adjacent to a...Ch. 4.4 - Doppler Effect The Doppler effect (named after...Ch. 4.4 - Minimizing Surface Area United Parcel Service has...Ch. 4.4 - Minimizing Surface Area United Parcel Service has...Ch. 4.4 - Cost of a Can A can in the shape of a right...Ch. 4.4 - Material Needed to Make a Drum A steel drum in the...Ch. 4.4 - Tennis Anyone? To win a game in tennis, a player...Ch. 4.4 - 66. Taxiing Speed A study of a new keyboard layout...Ch. 4.4 - Challenge Problem Removing a Discontinuity In...Ch. 4.4 - Challenge Problem Removing a Discontinuity Refer...Ch. 4.4 - Graph each of the following functions: y= x 2 1 x1...Ch. 4.4 - Graph each of the following functions: y= x 2 x1...Ch. 4.4 - Create a rational function that has the following...Ch. 4.4 - Create a rational function that has the following...Ch. 4.4 - Write a few paragraphs that provide a general...Ch. 4.4 - Create a rational function with the following...Ch. 4.4 - Explain the circumstances under which the graph of...Ch. 4.4 - Problems 7685 are based on material learned...Ch. 4.4 - Problems are based on material learned earlier...Ch. 4.4 - Problems 72-75 are based on material learned...Ch. 4.4 - Problems 72-75 are based on material learned...Ch. 4.4 - Problems 72-75 are based on material learned...Ch. 4.4 - Problems 72-75 are based on material learned...Ch. 4.4 - Problems are based on material learned earlier in...Ch. 4.4 - Problems 7685 are based on material learned...Ch. 4.4 - Problems are based on material learned earlier in...Ch. 4.4 - Problems are based on material learned earlier in...Ch. 4.5 - Solve the inequality 34x5 . Graph the solution...Ch. 4.5 - Solve the inequality x 2 5x24 . Graph the solution...Ch. 4.5 - Which of the following could be a test number for...Ch. 4.5 - True or False The graph of f( x )= x x3 is above...Ch. 4.5 - In Problems 5-8, use the graph of the function f...Ch. 4.5 - In Problems 5-8, use the graph of the function f...Ch. 4.5 - In Problems 5-8, use the graph of the function f...Ch. 4.5 - In Problems 5-8, use the graph of the function f...Ch. 4.5 - In Problems 9-14, solve the inequality by using...Ch. 4.5 - In Problems 9-14, solve the inequality by using...Ch. 4.5 - In Problems 9-14, solve the inequality by using...Ch. 4.5 - In Problems 9-14, solve the inequality by using...Ch. 4.5 - In Problems 9-14, solve the inequality by using...Ch. 4.5 - In Problems 9-14, solve the inequality by using...Ch. 4.5 - In Problems 15-18, solve the inequality by using...Ch. 4.5 - In Problems 15-18, solve the inequality by using...Ch. 4.5 - In Problems 15-18, solve the inequality by using...Ch. 4.5 - In Problems 15-18, solve the inequality by using...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In Problems 19-48, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In problems 1954, solve each inequality...Ch. 4.5 - In Problems 63-66, (a) graph each function by...Ch. 4.5 - In Problems 63-66, (a) graph each function by...Ch. 4.5 - In Problems 63-66, (a) graph each function by...Ch. 4.5 - In Problems 63-66, (a) graph each function by...Ch. 4.5 - For what positive numbers will the cube of a...Ch. 4.5 - For what positive numbers will the cube of a...Ch. 4.5 - What is the domain of the function f( x )= x 4 -16...Ch. 4.5 - What is the domain of the function f( x )= x 3 -3...Ch. 4.5 - What is the domain of the function f( x )= x-2 x+4...Ch. 4.5 - What is the domain of the function f( x )= x-1 x+4...Ch. 4.5 - In Problems 73-76, determine where the graph of f...Ch. 4.5 - In Problems 73-76, determine where the graph of f...Ch. 4.5 - In Problems 73-76, determine where the graph of f...Ch. 4.5 - In Problems 73-76, determine where the graph of f...Ch. 4.5 - Where is the graph of R(x)=x416x29 above the...Ch. 4.5 - Where is the graph of R(x)=x38x225 above the...Ch. 4.5 - Average Cost Suppose that the daily cost C of...Ch. 4.5 - Average Cost See Problem 77. Suppose that the...Ch. 4.5 - Bungee Jumping Originating on Pentecost Island in...Ch. 4.5 - Gravitational Force According to Newtons Law of...Ch. 4.5 - The inequality x 4 +15 has no solution. Explain...Ch. 4.5 - A student attempted to solve the inequality x+4 x3...Ch. 4.5 - Write a rational inequality whose solution set is...Ch. 4.5 - Make up an inequality that has no solution. Make...Ch. 4.5 - Problems 7988 are based on material framed earlier...Ch. 4.5 - Problems are based on material framed earlier in...Ch. 4.5 - Problems 7988 are based on material framed earlier...Ch. 4.5 - Problems are based on material framed earlier in...Ch. 4.5 - Problems 7988 are based on material framed earlier...Ch. 4.5 - Problems 7988 are based on material framed earlier...Ch. 4.5 - Problems 7988 are based on material framed earlier...Ch. 4.5 - Problems 86-89 are based on material learned...Ch. 4.5 - Problems are based on material framed earlier in...Ch. 4.5 - Problems are based on material framed earlier in...Ch. 4.6 - 1. Find f( 1 ) if f( x )=2 x 2 xCh. 4.6 - 2. Factor the expression 6 x 2 +x-2Ch. 4.6 - 3. Find the quotient and remainder if 3 x 4 -5 x 3...Ch. 4.6 - 4. Solve x 2 =3-x .Ch. 4.6 - 5. f( x )=q(x)g( x )+r(x) , the function r( x ) is...Ch. 4.6 - 6. When a polynomial function f is divided by x-c...Ch. 4.6 - 7. Given f( x )=3 x 4 -2 x 3 +7x-2 , how many sign...Ch. 4.6 - 8. True or False Every polynomial function of...Ch. 4.6 - 9. If f is a polynomial function and x4 is a...Ch. 4.6 - 10. True or False If f is a polynomial function of...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems , use the Remainder Theorem to find...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems , use the Remainder Theorem to find...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems 11-20, use the Remainder Theorem to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems, determine the maximum number of real...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 2132, determine the maximum number of...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 21-32, use Descartes' Rule of Signs to...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 3344, list the potential rational...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 3344, list the potential rational...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 33-44, determine the maximum number of...Ch. 4.6 - In Problems 51-68, find the real zeros of f. Use...Ch. 4.6 - In Problems 51-68, find the real zeros of f. Use...Ch. 4.6 - In Problems 4556, use the Rational Zeros Theorem...Ch. 4.6 - In Problems 4556, use the Rational Zeros Theorem...Ch. 4.6 - In Problems 4556, use the Rational Zeros Theorem...Ch. 4.6 - In Problems 4556, use the Rational Zeros Theorem...Ch. 4.6 - In Problems, use the Rational Zeros Theorem to...Ch. 4.6 - In Problems, use the Rational Zeros Theorem to...Ch. 4.6 - In Problems 51-68, find the real zeros of f. Use...Ch. 4.6 - In Problems 51-68, find the real zeros of f. Use...Ch. 4.6 - In Problems 4556, use the Rational Zeros Theorem...Ch. 4.6 - In Problems 4556, use the Rational Zeros Theorem...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 75-84, find the real solutions of each...Ch. 4.6 - In Problems 5768, solve each equation in the real...Ch. 4.6 - Prob. 68AYUCh. 4.6 - In Problems 6978, find bounds on the real zeros of...Ch. 4.6 - In Problems, find bounds on the real zeros of each...Ch. 4.6 - In Problems 6978, find bounds on the real zeros of...Ch. 4.6 - In Problems 6978, find bounds on the real zeros of...Ch. 4.6 - In Problems 6978, find bounds on the real zeros of...Ch. 4.6 - In Problems 6978, find bounds on the real zeros of...Ch. 4.6 - In Problems, find bounds on the real zeros of each...Ch. 4.6 - In Problems, find bounds on the real zeros of each...Ch. 4.6 - In Problems, find bounds on the real zeros of each...Ch. 4.6 - In Problems, find bounds on the real zeros of each...Ch. 4.6 - In Problems 85-90, use the Intermediate Value...Ch. 4.6 - In Problems 85-90, use the Intermediate Value...Ch. 4.6 - In Problems 85-90, use the Intermediate Value...Ch. 4.6 - In Problems 85-90, use the Intermediate Value...Ch. 4.6 - In Problems 85-90, use the Intermediate Value...Ch. 4.6 - In Problems 85-90, use the Intermediate Value...Ch. 4.6 - In Problems, each equation has a solution in the...Ch. 4.6 - In Problems, each equation has a solution in the...Ch. 4.6 - In Problems 8588, each equation has a solution r...Ch. 4.6 - In Problems, each equation has a solution in the...Ch. 4.6 - In Problems 8992, each polynomial function has...Ch. 4.6 - In Problems, each polynomial function has exactly...Ch. 4.6 - In Problems, each polynomial function has exactly...Ch. 4.6 - In Problems 8992, each polynomial function has...Ch. 4.6 - In Problems 91-98, analyze each polynomial...Ch. 4.6 - In Problems 91-98, analyze each polynomial...Ch. 4.6 - Mixed Practice In Problems 93104, grapheach...Ch. 4.6 - Mixed Practice In Problems 93104, grapheach...Ch. 4.6 - Mixed Practice In Problems, graph each polynomial...Ch. 4.6 - Mixed Practice In Problems 93104, grapheach...Ch. 4.6 - Mixed Practice In Problems 93104, grapheach...Ch. 4.6 - Mixed Practice In Problems 93104, grapheach...Ch. 4.6 - In Problems 91-98, analyze each polynomial...Ch. 4.6 - In Problems 91-98, analyze each polynomial...Ch. 4.6 - In Problems 91-98, analyze each polynomial...Ch. 4.6 - In Problems 91-98, analyze each polynomial...Ch. 4.6 - 105. Suppose that. Find the zeros of.
Ch. 4.6 - Suppose that f(x)=4x311x226x+24. Find the zeros of...Ch. 4.6 - Find k such that f( x )= x 3 k x 2 +kx+2 has the...Ch. 4.6 - Find k such that f( x )= x 4 k x 3 +k x 2 +1 has...Ch. 4.6 - What is the remainder when f( x )=2 x 20 8 x 10...Ch. 4.6 - Prob. 110AYUCh. 4.6 - Prob. 111AYUCh. 4.6 - Prob. 112AYUCh. 4.6 - Prob. 113AYUCh. 4.6 - One solution of the equation x 3 +5 x 2 +5x2=0 is...Ch. 4.6 - Prob. 115AYUCh. 4.6 - Prob. 116AYUCh. 4.6 - Prob. 117AYUCh. 4.6 - Prob. 118AYUCh. 4.6 - Prob. 119AYUCh. 4.6 - Prob. 120AYUCh. 4.6 - 121. Challenge Problem Use the Intermediate Value...Ch. 4.6 - Prob. 122AYUCh. 4.6 - Prob. 123AYUCh. 4.6 - Prob. 124AYUCh. 4.6 - Prob. 125AYUCh. 4.6 - Prob. 126AYUCh. 4.6 - Prob. 127AYUCh. 4.6 - Prob. 128AYUCh. 4.6 - Prob. 129AYUCh. 4.6 - Prob. 130AYUCh. 4.6 - Prob. 131AYUCh. 4.6 - Prob. 132AYUCh. 4.6 - Prob. 133AYUCh. 4.6 - Prob. 134AYUCh. 4.6 - Prob. 135AYUCh. 4.7 - 1. Find the sum and the product of the complex...Ch. 4.7 - 2. In the complex number system, find the complex...Ch. 4.7 - What is the conjugate of 3+4i?Ch. 4.7 - Given, find the product.
Ch. 4.7 - 3. Every polynomial function of odd degree with...Ch. 4.7 - 4. If 3+4i is a zero of a polynomial function of...Ch. 4.7 - 5. True or False A polynomial function of degree n...Ch. 4.7 - 6. True or False A polynomial function of degree 4...Ch. 4.7 - In Problems 7-16, information is given about a...Ch. 4.7 - In Problems 7-16, information is given about a...Ch. 4.7 - In Problems, information is given about a...Ch. 4.7 - In Problems 7-16, information is given about a...Ch. 4.7 - In Problems, information is given about a...Ch. 4.7 - In Problems 7-16, information is given about a...Ch. 4.7 - In Problems 918, information is given about a...Ch. 4.7 - In Problems 7-16, information is given about a...Ch. 4.7 - In Problems, information is given about a...Ch. 4.7 - In Problems 7-16, information is given about a...Ch. 4.7 - In Problems 1924, find a polynomial function f...Ch. 4.7 - In Problems 1924, find a polynomial function f...Ch. 4.7 - In Problems 1924, find a polynomial function f...Ch. 4.7 - In Problems, find a polynomial function with real...Ch. 4.7 - In Problems, find a polynomial function with real...Ch. 4.7 - In Problems, find a polynomial function with real...Ch. 4.7 - In Problems, use the given zero to find the...Ch. 4.7 - In Problems 23-30, use the given zero to find the...Ch. 4.7 - In Problems 2532, use the given zero to find the...Ch. 4.7 - In Problems 23-30, use the given zero to find the...Ch. 4.7 - In Problems 2532, use the given zero to find the...Ch. 4.7 - In Problems 23-30, use the given zero to find the...Ch. 4.7 - In Problems 2532, use the given zero to find the...Ch. 4.7 - In Problems 23-30, use the given zero to find the...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - In Problems 31-40, find the complex zeros of each...Ch. 4.7 - Given f( x )=2 x 3 14 x 2 +bx3 with f( 2 )=0 , g(...Ch. 4.7 - Let f be the polynomial function of degree 4 with...Ch. 4.7 - The complex zeros of f( x )= x 4 +1 . For the...Ch. 4.7 - In Problems 44 and 45, explain why the facts given...Ch. 4.7 - In Problems 44 and 45, explain why the facts given...Ch. 4.7 - f is a polynomial function of degree 4 whose...Ch. 4.7 - f is a polynomial function of degree 4 whose...Ch. 4.7 - For the polynomial function f( x )= x 2 +2ix-10 :...Ch. 4.7 - Draw a scatter diagram for the given data.Ch. 4.7 - Given f( x )= 3-x , find x so that f( x )=5 .Ch. 4.7 - Given f( x )=2x-5 and g( x )=3 x 2 +x-4 , find (...Ch. 4.7 - Prob. 54AYUCh. 4.7 - Problems 5160 are based on material learned...Ch. 4.7 - Problems 5160 are based on material learned...Ch. 4.7 - Problems 5160 are based on material learned...Ch. 4.7 - Problems 5160 are based on material learned...Ch. 4.7 - Problems 5160 are based on material learned...Ch. 4.7 - Problemsare based on material learned earlier in...Ch. 4 - In Problems 14, determine whether the function is...Ch. 4 - In Problems 14, determine whether the function is...Ch. 4 - In Problems 14, determine whether the function is...Ch. 4 - In Problems 14, determine whether the function is...Ch. 4 - In Problems , graph each function using...Ch. 4 - In Problems 57, graph each function using...Ch. 4 - In Problems 57, graph each function using...Ch. 4 - In Problems 811, graph each function by following...Ch. 4 - In Problems 811, graph each function by following...Ch. 4 - In Problems 811, graph each function by following...Ch. 4 - In Problems 811, graph each function by following...Ch. 4 - In Problems 1214, find the domain of each rational...Ch. 4 - In Problems 1214, find the domain of each rational...Ch. 4 - In Problems , find the domain of each rational...Ch. 4 - In Problems 1520, graph each rational function...Ch. 4 - In Problems , graph each rational function...Ch. 4 - In Problems , graph each rational function...Ch. 4 - In Problems , graph each rational function...Ch. 4 - In Problems 1520, graph each rational function...Ch. 4 - In Problems 1520, graph each rational function...Ch. 4 - In Problems 2125, solve each inequality. Graph the...Ch. 4 - In Problems , solve each inequality. Graph the...Ch. 4 - In Problems , solve each inequality. Graph the...Ch. 4 - In Problems 2125, solve each inequality. Graph the...Ch. 4 - In Problems , solve each inequality. Graph the...Ch. 4 - In Problems 26 and 27, find the remainder R when...Ch. 4 - In Problems and , find the remainder when is...Ch. 4 - Find the value of f(x)=12x68x4+1 at x=4.Ch. 4 - In the Problems 29 and 30, use Descartess Rule of...Ch. 4 - In the Problems, use Descartes’s Rule of Signs to...Ch. 4 - List all the potential rational zeros of...Ch. 4 - In the Problems , use the Rational Zeros Theorem...Ch. 4 - In the Problems , use the Rational Zeros Theorem...Ch. 4 - In the Problems , use the Rational Zeros Theorem...Ch. 4 - In the Problems and , solve each equation in the...Ch. 4 - In the Problems and , solve each equation in the...Ch. 4 - In the Problems 37 and 38, find bounds on the real...Ch. 4 - In the Problems 37 and 38, find bounds on the real...Ch. 4 - In the Problems and , use the Intermediate Value...Ch. 4 - In the Problems 39 and 40, use the Intermediate...Ch. 4 - In the Problems 41 and 42, each polynomial...Ch. 4 - In the Problems and , each polynomial function...Ch. 4 - In the Problems and , information is given about...Ch. 4 - In the Problems 43 and 44, information is given...Ch. 4 - In the Problems , find the complex zeros of each...Ch. 4 - In the Problems 4548, find the complex zeros of...Ch. 4 - In the Problems , find the complex zeros of each...Ch. 4 - In the Problems 4548, find the complex zeros of...Ch. 4 - 49. Graph the polynomial function by following...Ch. 4 - Making a Can A can in the shape of a right...Ch. 4 - Housing Prices The data in the table on the right...Ch. 4 - Graph f(x)=(x3)42 using transformations.Ch. 4 - For the polynomial function ,
Determine the...Ch. 4 - Find the complex zeros of f(x)=x34x2+25x100.Ch. 4 - Solve in the complex number system.
Ch. 4 - In problems 5 and 6, find the domain of each...Ch. 4 - In problems and , find the domain of each...Ch. 4 - Graph the function in Problem 6. Label all...Ch. 4 - In Problems 8 and 9, write a function that meets...Ch. 4 - Rational function; asymptotes: , ; domain :
Ch. 4 - Use the Intermediate Value Theorem to show that...Ch. 4 - Solve:
Ch. 4 - Solve: x3+7x22x26x.Ch. 4 - Find the distance between the points and .
Ch. 4 - Solve the inequality x2x and graph the solution...Ch. 4 - Solve the inequality x23x4 and graph the solution...Ch. 4 - Find a linear function with slope 3 that contains...Ch. 4 - Find the equation of the line parallel to the...Ch. 4 - Graph the equation.
Ch. 4 - Does the relation {(3, 6), (1, 3), (2, 5), (3, 8)}...Ch. 4 - Solve the equation x36x2+8x=0.Ch. 4 - Solve the inequality 3x+25x1 and graph the...Ch. 4 - Prob. 10CRCh. 4 - For the equation y=x39x, determine the intercepts...Ch. 4 - Find the equation of the line perpendicular to...Ch. 4 - Is the following the graph of a function? Why or...Ch. 4 - For the function f(x)=x2+5x2, find f(3) f(x) f(x)...Ch. 4 - Given the function f(x)=x+5x1 What is the domain...Ch. 4 - Graph the function f(x)=3x+7.Ch. 4 - Graph f(x)=2x24x+1 by determining whether its...Ch. 4 - Find the average rate of change of from to ....Ch. 4 - In parts (a) to (f), use the following graph, Find...Ch. 4 - Determine algebraically whether the function...Ch. 4 - For the function
Find the domain of .
Locate...Ch. 4 - Graph the function f(x)=3(x+1)2+5 using...Ch. 4 - Suppose that and .
Find and state its...Ch. 4 - Demand Equation The price (in dollars) and the...
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- helppparrow_forward7. Let F(x1, x2) (F₁(x1, x2), F2(x1, x2)), where = X2 F1(x1, x2) X1 F2(x1, x2) x+x (i) Using the definition, calculate the integral LF.dy, where (t) = (cos(t), sin(t)) and t = [0,2]. [5 Marks] (ii) Explain why Green's Theorem cannot be used to find the integral in part (i). [5 Marks]arrow_forward6. Sketch the trace of the following curve on R², п 3п (t) = (t2 sin(t), t2 cos(t)), tЄ 22 [3 Marks] Find the length of this curve. [7 Marks]arrow_forward
- Total marks 10 Total marks on naner: 80 7. Let DCR2 be a bounded domain with the boundary OD which can be represented as a smooth closed curve : [a, b] R2, oriented in the anticlock- wise direction. Use Green's Theorem to justify that the area of the domain D can be computed by the formula 1 Area(D) = ½ (−y, x) · dy. [5 Marks] (ii) Use the area formula in (i) to find the area of the domain D enclosed by the ellipse y(t) = (10 cos(t), 5 sin(t)), t = [0,2π]. [5 Marks]arrow_forwardTotal marks 15 Total marks on paper: 80 6. Let DCR2 be a bounded domain with the boundary ǝD which can be represented as a smooth closed curve : [a, b] → R², oriented in the anticlockwise direction. (i) Use Green's Theorem to justify that the area of the domain D can be computed by the formula 1 Area(D) = . [5 Marks] (ii) Use the area formula in (i) to find the area of the domain D enclosed by the ellipse (t) = (5 cos(t), 10 sin(t)), t = [0,2π]. [5 Marks] (iii) Explain in your own words why Green's Theorem can not be applied to the vector field У x F(x,y) = ( - x² + y²²x² + y² ). [5 Marks]arrow_forwardTotal marks 15 པ་ (i) Sketch the trace of the following curve on R2, (t) = (t2 cos(t), t² sin(t)), t = [0,2π]. [3 Marks] (ii) Find the length of this curve. (iii) [7 Marks] Give a parametric representation of a curve : [0, that has initial point (1,0), final point (0, 1) and the length √2. → R² [5 Marks] Turn over. MA-201: Page 4 of 5arrow_forward
- Total marks 15 5. (i) Let f R2 R be defined by f(x1, x2) = x² - 4x1x2 + 2x3. Find all local minima of f on R². (ii) [10 Marks] Give an example of a function f: R2 R which is not bounded above and has exactly one critical point, which is a minimum. Justify briefly your answer. [5 Marks] 6. (i) Sketch the trace of the following curve on R2, y(t) = (sin(t), 3 sin(t)), t = [0,π]. [3 Marks]arrow_forwardA ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder, its x coordinate, at time t seconds is given by x(t)=7+2t. wall y(1) 25 ft. ladder x(1) ground (a) Find the formula for the location of the top of the ladder, the y coordinate, as a function of time t. The formula for y(t)= √ 25² - (7+2t)² (b) The domain of t values for y(t) ranges from 0 (c) Calculate the average velocity of the top of the ladder on each of these time intervals (correct to three decimal places): . (Put your cursor in the box, click and a palette will come up to help you enter your symbolic answer.) time interval ave velocity [0,2] -0.766 [6,8] -3.225 time interval ave velocity -1.224 -9.798 [2,4] [8,9] (d) Find a time interval [a,9] so that the average velocity of the top of the ladder on this…arrow_forwardTotal marks 15 3. (i) Let FRN Rm be a mapping and x = RN is a given point. Which of the following statements are true? Construct counterex- amples for any that are false. (a) If F is continuous at x then F is differentiable at x. (b) If F is differentiable at x then F is continuous at x. If F is differentiable at x then F has all 1st order partial (c) derivatives at x. (d) If all 1st order partial derivatives of F exist and are con- tinuous on RN then F is differentiable at x. [5 Marks] (ii) Let mappings F= (F1, F2) R³ → R² and G=(G1, G2) R² → R² : be defined by F₁ (x1, x2, x3) = x1 + x², G1(1, 2) = 31, F2(x1, x2, x3) = x² + x3, G2(1, 2)=sin(1+ y2). By using the chain rule, calculate the Jacobian matrix of the mapping GoF R3 R², i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)? (iii) [7 Marks] Give reasons why the mapping Go F is differentiable at (0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0). [3 Marks]arrow_forward
- 5. (i) Let f R2 R be defined by f(x1, x2) = x² - 4x1x2 + 2x3. Find all local minima of f on R². (ii) [10 Marks] Give an example of a function f: R2 R which is not bounded above and has exactly one critical point, which is a minimum. Justify briefly Total marks 15 your answer. [5 Marks]arrow_forwardTotal marks 15 4. : Let f R2 R be defined by f(x1, x2) = 2x²- 8x1x2+4x+2. Find all local minima of f on R². [10 Marks] (ii) Give an example of a function f R2 R which is neither bounded below nor bounded above, and has no critical point. Justify briefly your answer. [5 Marks]arrow_forward4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 Marks]arrow_forward
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