MML PRECALCULUS ENHANCED
7th Edition
ISBN: 9780134119250
Author: Sullivan
Publisher: INTER PEAR
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Textbook Question
Chapter 1.4, Problem 98SB
In Problems 93—102, (a) find the intercepts of the graph of each equation and (b) graph the equation.
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A 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…
Total 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]
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]
Chapter 1 Solutions
MML PRECALCULUS ENHANCED
Ch. 1.1 - 1. On a real number line the origin is assigned...Ch. 1.1 - 2. If 3 and 5 are the coordinates of two points on...Ch. 1.1 - 3. If 3 and 4 are the legs of a right triangle,...Ch. 1.1 - 4. Use the converse of the Pythagorean Theorem to...Ch. 1.1 - 5. The area of a triangle whose base is b and...Ch. 1.1 - 6. True or False Two triangles are congruent if...Ch. 1.1 - 7. If ( x,y ) are the coordinates of a point P in...Ch. 1.1 - 8. The coordinate axes divide the xy-plane into...Ch. 1.1 - 9. If three distinct points P , Q and R all lie on...Ch. 1.1 - 10. True or False The distance between two points...
Ch. 1.1 - 11. True or False The point (1,4) lies in quadrant...Ch. 1.1 - 12. True or False The midpoint of a line segment...Ch. 1.1 - 13. Given that the intercepts of a graph are ( 4,0...Ch. 1.1 - 14. Which of the following points does not satisfy...Ch. 1.1 - In Problems 15 and 16, plot each point in the...Ch. 1.1 - In Problems 15 and 16, plot each point in the...Ch. 1.1 - 17. Plot the points ( 2,0 ),( 2,3 ),( 2,4 ),(2,1)...Ch. 1.1 - 18. Plot the points ( 0,3 ),( 1,3 ),( 2,3 ),(5,3)...Ch. 1.1 - In Problems 19-22, determine the coordinates of...Ch. 1.1 - In Problems 19-22, determine the coordinates of...Ch. 1.1 - In Problems 19-22, determine the coordinates of...Ch. 1.1 - In Problems 19-22, determine the coordinates of...Ch. 1.1 - In Problems 23-28, select a setting so that each...Ch. 1.1 - In Problems 23-28, select a setting so that each...Ch. 1.1 - In Problems 23-28, select a setting so that each...Ch. 1.1 - In Problems 23-28, select a setting so that each...Ch. 1.1 - In Problems 23-28, select a setting so that each...Ch. 1.1 - In Problems 23-28, select a setting so that each...Ch. 1.1 - In Problems 29-34, determine the viewing window...Ch. 1.1 - In Problems 29-34, determine the viewing window...Ch. 1.1 - In Problems 29-34, determine the viewing window...Ch. 1.1 - In Problems 29-34, determine the viewing window...Ch. 1.1 - In Problems 29-34, determine the viewing window...Ch. 1.1 - In Problems 29-34, determine the viewing window...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 35-46, find the distance d( P 1 , P 2...Ch. 1.1 - In Problems 47-50, find the length of the line...Ch. 1.1 - In Problems 47-50, find the length of the line...Ch. 1.1 - In Problems 47-50, find the length of the line...Ch. 1.1 - In Problems 47-50, find the length of the line...Ch. 1.1 - In Problems 51-56, plot each point and form the...Ch. 1.1 - In Problems 51-56, plot each point and form the...Ch. 1.1 - In Problems 51-56, plot each point and form the...Ch. 1.1 - In Problems 51-56, plot each point and form the...Ch. 1.1 - In Problems 51-56, plot each point and form the...Ch. 1.1 - In Problems 51-56, plot each point and form the...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 57-64, find the midpoint of the line...Ch. 1.1 - In Problems 65-70, tell whether the given points...Ch. 1.1 - In Problems 65-70, tell whether the given points...Ch. 1.1 - In Problems 65-70, tell whether the given points...Ch. 1.1 - In Problems 65-70, tell whether the given points...Ch. 1.1 - In Problems 65-70, tell whether the given points...Ch. 1.1 - In Problems 65-70, tell whether the given points...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 71-78, the graph of an equation is...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 79-90, graph each equation by plotting...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - In Problems 91-98, graph each equation using a...Ch. 1.1 - If the point ( 2,5 ) is shifted 3 units right and...Ch. 1.1 - If the point ( 1,6 ) is shifted 2 units left and 4...Ch. 1.1 - The medians of a triangle are the line segments...Ch. 1.1 - An equilateral triangle is one in which all three...Ch. 1.1 - In Problems 103-106, find the length of each side...Ch. 1.1 - In Problems 103-106, find the length of each side...Ch. 1.1 - In Problems 103-106, find the length of each side...Ch. 1.1 - In Problems 103-106, find the length of each side...Ch. 1.1 - Completing a Line Segment Plot the points A=( 1,8...Ch. 1.1 - Completing a Line Segment Plot the points M=( 5,4...Ch. 1.1 - Baseball A major league baseball “diamond� is...Ch. 1.1 - Little league Baseball The layout of a Little...Ch. 1.1 - Baseball Refer to Problem 109. Overlay a...Ch. 1.1 - Little League Baseball Refer to Problem 110....Ch. 1.1 - Distance between Moving Objects A Ford Focus and a...Ch. 1.1 - Distance of a Moving Object from a Fixed Point A...Ch. 1.1 - Drafting Error When a draftsman draws three lines...Ch. 1.1 - Net Sales The figure illustrates how net sales of...Ch. 1.1 - Poverty Threshold Poverty thresholds arc...Ch. 1.1 - In Problem 118, you may use a graphing utility,...Ch. 1.1 - Make up an equation satisfied by the ordered pairs...Ch. 1.1 - Draw a graph that contains the points ( 2,1 ),(...Ch. 1.1 - Explain what is meant by a complete graph.Ch. 1.1 - Write a paragraph that describes a Cartesian...Ch. 1.2 - Solve: 2( x+3 )1=7 (pp. A44-A46)Ch. 1.2 - Solve: x 2 4x12=0 (pp. A46-A52)Ch. 1.2 - The points, if any, at which a graph crosses or...Ch. 1.2 - Prob. 4CVCh. 1.2 - Prob. 5CVCh. 1.2 - Prob. 6CVCh. 1.2 - True or False To find the y-intercepts of the...Ch. 1.2 - True or False If a graph is symmetric with respect...Ch. 1.2 - To find the x-intercept( s ) , if any, of the...Ch. 1.2 - To test whether the graph of an equation is...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 11-22, find the intercepts and graph...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 23-32, plot each point. Then plot the...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - In Problems 33-44, the graph of an equation is...Ch. 1.2 - Prob. 45SBCh. 1.2 - Prob. 46SBCh. 1.2 - Prob. 47SBCh. 1.2 - Prob. 48SBCh. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 49-64, list the intercepts and test...Ch. 1.2 - In Problems 65-68, draw a quick sketch of each...Ch. 1.2 - In Problems 65-68, draw a quick sketch of each...Ch. 1.2 - In Problems 65-68, draw a quick sketch of each...Ch. 1.2 - In Problems 65-68, draw a quick sketch of each...Ch. 1.2 - If (3,b) is a point on the graph of y=4x+1 , what...Ch. 1.2 - If (2,b) is a point on the graph of 2x+3y=2 , what...Ch. 1.2 - If (a,4) is a point on the graph of y= x 2 +3x ,...Ch. 1.2 - If (a,5) is a point on the graph of y= x 2 +6x ,...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - In Problems 73-80, (a) find the intercepts of each...Ch. 1.2 - Prob. 81AECh. 1.2 - If the graph of an equation is symmetric with...Ch. 1.2 - If the graph of an equation is symmetric with...Ch. 1.2 - Prob. 84AECh. 1.2 - Prob. 85AECh. 1.2 - Prob. 86AECh. 1.2 - Draw a graph of an equation that contains two...Ch. 1.2 - Prob. 88DWCh. 1.2 - Draw a graph that contains the points ( 2,1 ) , (...Ch. 1.2 - Draw a graph that contains the points ( 2,5 ) , (...Ch. 1.3 - Solve the equation 2 x 2 +5x+2=0 . (pp. A46-A52)Ch. 1.3 - Solve the equation 2x+3=4(x1)+1 . (pp. A44-A46)Ch. 1.3 - To solve an equation of the form { expressioninx...Ch. 1.3 - True or False In using a graphing utility to solve...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - Prob. 11SBCh. 1.3 - Prob. 12SBCh. 1.3 - Prob. 13SBCh. 1.3 - Prob. 14SBCh. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 5-16, use a graphing utility to...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - Prob. 26SBCh. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - Prob. 30SBCh. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.3 - In Problems 17-36, solve each equation...Ch. 1.4 - The slope of a vertical line is ______; the slope...Ch. 1.4 - For the line 2x+3y=6 , the x-intercept is ______...Ch. 1.4 - True or False The equation 3x+4y=6 is written in...Ch. 1.4 - True or False The slope of the line 2y=3x+5 is 3.Ch. 1.4 - True or False The point ( 1,2 ) is on the line...Ch. 1.4 - Prob. 6CVCh. 1.4 - Prob. 7CVCh. 1.4 - Prob. 8CVCh. 1.4 - Prob. 9CVCh. 1.4 - Choose the formula for finding the slope m of a...Ch. 1.4 - If a line slants downward from left to right, then...Ch. 1.4 - Choose the correct statement about the graph of...Ch. 1.4 - In Problems 13-16, (a) find the slope of the line...Ch. 1.4 - In Problems 13-16, (a) find the slope of the line...Ch. 1.4 - In Problems 13-16, (a) find the slope of the line...Ch. 1.4 - In Problems 13-16, (a) find the slope of the line...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 17-24, plot each pair of points and...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 25-32, graph the line containing the...Ch. 1.4 - In Problems 33-38, the slope and a point on a line...Ch. 1.4 - In Problems 33-38, the slope and a point on a line...Ch. 1.4 - In Problems 33-38, the slope and a point on a line...Ch. 1.4 - In Problems 33-38, the slope and a point on a line...Ch. 1.4 - In Problems 33-38, the slope and a point on a line...Ch. 1.4 - In Problems 33-38, the slope and a point on a line...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 39-46, find an equation of the line L...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 47-72, find an equation for the line...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 73-92, find the slope and y-intercept...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - In Problems 93—102, (a) find the intercepts of...Ch. 1.4 - Find an equation of the x-axis .Ch. 1.4 - Find an equation of the y-axis .Ch. 1.4 - In Problems 105-108, the equations of two lines...Ch. 1.4 - In Problems 105-108, the equations of two lines...Ch. 1.4 - In Problems 105-108, the equations of two lines...Ch. 1.4 - In Problems 105-108, the equations of two lines...Ch. 1.4 - In Problems 109-112, write an equation of each...Ch. 1.4 - In Problems 109-112, write an equation of each...Ch. 1.4 - In Problems 109-112, write an equation of each...Ch. 1.4 - In Problems 109-112, write an equation of each...Ch. 1.4 - Geometry Use slopes to show that the triangle...Ch. 1.4 - Geometry Use slopes to show that the quadrilateral...Ch. 1.4 - Geometry Use slopes to show that the quadrilateral...Ch. 1.4 - Geometry Use slopes and the distance formula to...Ch. 1.4 - Geometry Truck Rentals A truck rental company...Ch. 1.4 - Cost Equation The fixed costs of operating a...Ch. 1.4 - Cost of Driving a Car The annual fixed costs far...Ch. 1.4 - Wages of a Car Salesperson Dan receives 375 per...Ch. 1.4 - Electricity Rates in Illinois Commonwealth Edison...Ch. 1.4 - Electricity Rates in Florida Florida Power 7.57...Ch. 1.4 - Electricity Rates in Florida Measuring Temperature...Ch. 1.4 - Measuring Temperature The Kelvin (K) scale for...Ch. 1.4 - Access Ramp A wooden access ramp is being built to...Ch. 1.4 - Cigarette Use A report in the Child Trends...Ch. 1.4 - Product Promotion A cereal company finds that the...Ch. 1.4 - Show that the line containing the points ( a,b...Ch. 1.4 - The equation 2xy=C defines a family of lines, one...Ch. 1.4 - Prove that if two nonvertical lines have slopes...Ch. 1.4 - Which of the following equations might have the...Ch. 1.4 - Which of the following equations might have the...Ch. 1.4 - The figure shows the graph of two parallel lines....Ch. 1.4 - The figure shows the graph of two perpendicular...Ch. 1.4 - m is for Slope The accepted symbol used to denote...Ch. 1.4 - Grade of a Road The term grade is used to describe...Ch. 1.4 - Carpentry Carpenters use the term pitch to...Ch. 1.4 - Can the equation of every line be written in...Ch. 1.4 - Does every line have exactly one x-intercept and...Ch. 1.4 - What can you say about two lines that have equal...Ch. 1.4 - What can you say about two lines with the same...Ch. 1.4 - If two distinct lines have the same slope but...Ch. 1.4 - If two distinct lines have the same y-intercept...Ch. 1.4 - Which form of the equation of a line do you prefer...Ch. 1.4 - What Went Wrong? A student is asked to find the...Ch. 1.5 - In Problems 1-4, find the following for each pair...Ch. 1.5 - In Problems 1-4, find the following for each pair...Ch. 1.5 - In Problems 1-4, find the following for each pair...Ch. 1.5 - In Problems 1-4, find the following for each pair...Ch. 1.5 - 5. List the intercepts of the following graph.Ch. 1.5 - 6. Graph y= x 2 +15 using a graphing utility....Ch. 1.5 - Prob. 7CVCh. 1.5 - Prob. 8CVCh. 1.5 - Prob. 9SBCh. 1.5 - Prob. 10SBCh. 1.5 - Prob. 11SBCh. 1.5 - Prob. 12SBCh. 1.5 - Prob. 13SBCh. 1.5 - Prob. 14SBCh. 1.5 - Prob. 15SBCh. 1.5 - Prob. 16SBCh. 1.5 - Prob. 17SBCh. 1.5 - Prob. 18SBCh. 1.5 - Prob. 19SBCh. 1.5 - Prob. 20SBCh. 1.5 - Prob. 21SBCh. 1.5 - Prob. 22SBCh. 1.5 - Prob. 23SBCh. 1.5 - Prob. 24SBCh. 1.5 - Prob. 25SBCh. 1.5 - Prob. 26SBCh. 1.5 - Prob. 27SBCh. 1.5 - Prob. 28SBCh. 1.5 - Prob. 29SBCh. 1.5 - Prob. 30SBCh. 1.5 - Prob. 31SBCh. 1.5 - Prob. 32SBCh. 1.5 - Prob. 33SBCh. 1.5 - Prob. 34SBCh. 1.5 - Prob. 35SBCh. 1.5 - Prob. 36SBCh. 1.5 - Prob. 37SBCh. 1.5 - Prob. 38SBCh. 1.5 - Prob. 39SBCh. 1.5 - Prob. 40SBCh. 1.5 - Prob. 41SBCh. 1.5 - Prob. 42SBCh. 1.5 - In Problems 37-44, find the standard form of the...Ch. 1.5 - In Problems 37-44, find the standard form of the...Ch. 1.5 - In Problems 45-48, match each graph with the...Ch. 1.5 - In Problems 45-48, match each graph with the...Ch. 1.5 - In Problems 45-48, match each graph with the...Ch. 1.5 - In Problems 45-48, match each graph with the...Ch. 1.5 - 49. Find the area of the square in the figure.Ch. 1.5 - 50. Find the area of the blue shaded region in the...Ch. 1.5 - Prob. 51AECh. 1.5 - Prob. 52AECh. 1.5 - Prob. 53AECh. 1.5 - 54. The tangent line to a circle may be defined as...Ch. 1.5 - 55. The Greek Method the Greek method for finding...Ch. 1.5 - 56. Use the Greek method described in Problem 55...Ch. 1.5 - 57. Refer to Problem 54. The line x2y+4=0 is...Ch. 1.5 - 58. Find an equation of the line containing the...Ch. 1.5 - Prob. 59AECh. 1.5 - 60. If the circumference of a circle is 6 , what...Ch. 1.5 - 61. Which of the following equations might have...Ch. 1.5 - 62. Which of the following equations might have...Ch. 1.5 - 63. Explain how the center and radius of a circle...Ch. 1.5 - Prob. 64DWCh. 1.R - In Problems 1-4, find the following for each pair...Ch. 1.R - In Problems 1-4, find the following for each pair...Ch. 1.R - In Problems 1-4, find the following for each pair...Ch. 1.R - In Problems 1-4, find the following for each pair...Ch. 1.R - List the intercepts of the following graph.Ch. 1.R - 6. Graph y= x 2 +15 using a graphing utility....Ch. 1.R - In Problems 7-9, determine the intercepts and...Ch. 1.R - In Problems 7-9, determine the intercepts and...Ch. 1.R - In Problems 7-9, determine the intercepts and...Ch. 1.R - In Problems 10-14, test each equation for symmetry...Ch. 1.R - In Problems 10-14, test each equation for symmetry...Ch. 1.R - In Problems 10-14, test each equation for symmetry...Ch. 1.R - In Problems 10-14, test each equation for symmetry...Ch. 1.R - In Problems 10-14, test each equation for symmetry...Ch. 1.R - Sketch a graph of y= x 3 .Ch. 1.R - In Problems 16 anti 17, use a graphing utility to...Ch. 1.R - In Problems 16 anti 17, use a graphing utility to...Ch. 1.R - In Problems 18-25, find an equation of the line...Ch. 1.R - In Problems 18-25, find an equation of the line...Ch. 1.R - In Problems 18-25, find an equation of the line...Ch. 1.R - x-intercept=2 ; containing the point ( 4,5 )Ch. 1.R - y-intercept=2 ; containing the point ( 5,3 )Ch. 1.R - Containing the points ( 3,4 ) and ( 2,1 )Ch. 1.R - Parallel to the line 2x3y=4 ; containing the point...Ch. 1.R - Perpendicular to the line 3xy=4 ; containing the...Ch. 1.R - Prob. 26RECh. 1.R - Prob. 27RECh. 1.R - Prob. 28RECh. 1.R - Prob. 29RECh. 1.R - Show that the points A=( 2,0 ) , B=( 4,4 ) , and...Ch. 1.R - 31. Show that the points A=( 2,5 ) , B=( 6,1 ) ,...Ch. 1.R - Prob. 32RECh. 1.R - Prob. 33RECh. 1.R - 34. Find two numbers y such that the distance from...Ch. 1.R - 35. Graph the line with slope 2 3 containing the...Ch. 1.R - Make up four problems that you might be asked to...Ch. 1.R - 37. Describe each of the following graphs in the...
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- Total 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_forward4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, 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 n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]arrow_forward
- 3. Let f R2 R be a function. (i) Explain in your own words the relationship between the existence of all partial derivatives of f and differentiability of f at a point x = R². (ii) Consider R2 → R defined by : [5 Marks] f(x1, x2) = |2x1x2|1/2 Show that af af -(0,0) = 0 and -(0, 0) = 0, Jx1 მx2 but f is not differentiable at (0,0). [10 Marks]arrow_forward(1) Write the following quadratic equation in terms of the vertex coordinates.arrow_forwardThe final answer is 8/π(sinx) + 8/3π(sin 3x)+ 8/5π(sin5x)....arrow_forward
- Keity x२ 1. (i) Identify which of the following subsets of R2 are open and which are not. (a) A = (2,4) x (1, 2), (b) B = (2,4) x {1,2}, (c) C = (2,4) x R. Provide a sketch and a brief explanation to each of your answers. [6 Marks] (ii) Give an example of a bounded set in R2 which is not open. [2 Marks] (iii) Give an example of an open set in R2 which is not bounded. [2 Marksarrow_forward2. (i) Which of the following statements are true? Construct coun- terexamples for those that are false. (a) sequence. Every bounded sequence (x(n)) nEN C RN has a convergent sub- (b) (c) (d) Every sequence (x(n)) nEN C RN has a convergent subsequence. Every convergent sequence (x(n)) nEN C RN is bounded. Every bounded sequence (x(n)) EN CRN converges. nЄN (e) If a sequence (xn)nEN C RN has a convergent subsequence, then (xn)nEN is convergent. [10 Marks] (ii) Give an example of a sequence (x(n))nEN CR2 which is located on the parabola x2 = x², contains infinitely many different points and converges to the limit x = (2,4). [5 Marks]arrow_forward2. (i) What does it mean to say that a sequence (x(n)) nEN CR2 converges to the limit x E R²? [1 Mark] (ii) Prove that if a set ECR2 is closed then every convergent sequence (x(n))nen in E has its limit in E, that is (x(n)) CE and x() x x = E. [5 Marks] (iii) which is located on the parabola x2 = = x x4, contains a subsequence that Give an example of an unbounded sequence (r(n)) nEN CR2 (2, 16) and such that x(i) converges to the limit x = (2, 16) and such that x(i) # x() for any i j. [4 Marksarrow_forward
- 1. (i) which are not. Identify which of the following subsets of R2 are open and (a) A = (1, 3) x (1,2) (b) B = (1,3) x {1,2} (c) C = AUB (ii) Provide a sketch and a brief explanation to each of your answers. [6 Marks] Give an example of a bounded set in R2 which is not open. (iii) [2 Marks] Give an example of an open set in R2 which is not bounded. [2 Marks]arrow_forward2. if limit. Recall that a sequence (x(n)) CR2 converges to the limit x = R² lim ||x(n)x|| = 0. 818 - (i) Prove that a convergent sequence (x(n)) has at most one [4 Marks] (ii) Give an example of a bounded sequence (x(n)) CR2 that has no limit and has accumulation points (1, 0) and (0, 1) [3 Marks] (iii) Give an example of a sequence (x(n))neN CR2 which is located on the hyperbola x2 1/x1, contains infinitely many different Total marks 10 points and converges to the limit x = (2, 1/2). [3 Marks]arrow_forward3. (i) Consider a mapping F: RN Rm. Explain in your own words the relationship between the existence of all partial derivatives of F and dif- ferentiability of F at a point x = RN. (ii) [3 Marks] Calculate the gradient of the following function f: R2 → R, f(x) = ||x||3, Total marks 10 where ||x|| = √√√x² + x/2. [7 Marks]arrow_forward
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