MML PRECALCULUS ENHANCED
7th Edition
ISBN: 9780134119250
Author: Sullivan
Publisher: INTER PEAR
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Textbook Question
Chapter 10.2, Problem 41SB
In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.
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Students have asked these similar questions
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]
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]
4.
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]
Chapter 10 Solutions
MML PRECALCULUS ENHANCED
Ch. 10.2 - The formula for the distance d from P 1 =( x 1 , y...Ch. 10.2 - To complete the square of x 2 4x , add_______...Ch. 10.2 - Use the Square Root Method to find the real...Ch. 10.2 - The point that is symmetric with respect to the...Ch. 10.2 - To graph y= ( x3 ) 2 +1 , shift the graph of y= x...Ch. 10.2 - A( n ) _____ is the collection of all points in a...Ch. 10.2 - The line through the focus and perpendicular to...Ch. 10.2 - For the parabola y 2 =4ax , the line segment...Ch. 10.2 - Answer Problems 9-12 using the figure. If a0 , the...Ch. 10.2 - Answer Problems 9-12 using the figure. The...
Ch. 10.2 - Answer Problems 9-12 using the figure. If a=4 ,...Ch. 10.2 - Answer Problems 9-12 using the figure. If a=4 ,...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 13-20, the graph of a parabola is...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 21-38, find the equation of the...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 39-56, find the vertex, focus, and...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - In Problems 57-64, write an equation for each...Ch. 10.2 - Satellite Dish A satellite dish is shaped like a...Ch. 10.2 - Constructing a TV Dish A cable TV receiving dish...Ch. 10.2 - Constructing a Flashlight The reflector of a...Ch. 10.2 - Constructing a Headlight A sealed-beam headlight...Ch. 10.2 - Suspension Bridge The cables of a suspension...Ch. 10.2 - Suspension Bridge The cables of a suspension...Ch. 10.2 - Searchlight A searchlight is shaped like a...Ch. 10.2 - Searchlight A searchlight is shaped like a...Ch. 10.2 - Solar Heat A mirror is shaped like a paraboloid of...Ch. 10.2 - Reflecting Telescope A reflecting telescope...Ch. 10.2 - Parabolic Arch Bridge A bridge is built in the...Ch. 10.2 - Parabolic Arch Bridge A bridge is to be built in...Ch. 10.2 - Gateway Arch The Gateway Arch in St. Louis is...Ch. 10.2 - Show that an equation of the form A x 2 +Ey=0 A0...Ch. 10.2 - Show that an equation of the form C y 2 +Dx=0 C0...Ch. 10.2 - Show that the graph of an equation of the form A x...Ch. 10.2 - Show that the graph of an equation of the form C y...Ch. 10.2 - Problems 82-85 are based on material learned...Ch. 10.2 - Problems 82-85 are based on material learned...Ch. 10.2 - Problems 82-85 are based on material learned...Ch. 10.2 - Problems 82-85 are based on material learned...Ch. 10.3 - The distance d from P 1 =( 2,5 ) to P 2 =( 4,2 )...Ch. 10.3 - To complete the square of x 2 3x , Add _____. (p....Ch. 10.3 - Find the intercepts of the equation y 2 =164 x 2 ....Ch. 10.3 - The point that is symmetric with respect to the...Ch. 10.3 - The point that is symmetric with respect to the...Ch. 10.3 - Prob. 6AYPCh. 10.3 - A(n) _______ is the collection of all points in a...Ch. 10.3 - For an ellipse, the foci lie on a line called the...Ch. 10.3 - For the ellipse x 2 4 + y 2 25 =1 , the vertices...Ch. 10.3 - For the ellipse x 2 25 + y 2 9 =1 , the value of a...Ch. 10.3 - If the center of an ellipse is ( 2,3 ) , the major...Ch. 10.3 - If the foci of an ellipse are ( 4,4 ) and ( 6,4 )...Ch. 10.3 - In problems 13-16, the graph of an ellipse is...Ch. 10.3 - In problems 13-16, the graph of an ellipse is...Ch. 10.3 - In problems 13-16, the graph of an ellipse is...Ch. 10.3 - In problems 13-16, the graph of an ellipse is...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In problems 17-26, find the vertices and foci of...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 27-38, find an equation for each...Ch. 10.3 - In Problems 39-42, write an equation for each...Ch. 10.3 - In Problems 39-42, write an equation for each...Ch. 10.3 - In Problems 39-42, write an equation for each...Ch. 10.3 - In Problems 39-42, write an equation for each...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 43-54, analyze each equation; that is,...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 55-64, find an equation for each...Ch. 10.3 - In Problems 65-68, graph each function. Be sure to...Ch. 10.3 - In Problems 65-68, graph each function. Be sure to...Ch. 10.3 - In Problems 65-68, graph each function. Be sure to...Ch. 10.3 - In Problems 65-68, graph each function. Be sure to...Ch. 10.3 - Semielliptical Arch Bridge An arch in the shape of...Ch. 10.3 - Semielliptical Arch Bridge The arch of a bridge is...Ch. 10.3 - Whispering Gallery A hall 100 feet in length is to...Ch. 10.3 - Whispering Gallery Jim, standing at one focus of a...Ch. 10.3 - Semielliptical Arch Bridge A bridge is built in...Ch. 10.3 - Semielliptical Arch Bridge A bridge is to be built...Ch. 10.3 - Racetrack Design Consult the figure. A racetrack...Ch. 10.3 - Semielliptical Arch Bridge An arch for a bridge...Ch. 10.3 - Installing a Vent Pipe A homeowner is putting in a...Ch. 10.3 - Semielliptical Arch Bridge An arch for a bridge...Ch. 10.3 - In Problems 79-83, use the fact that the orbit of...Ch. 10.3 - In Problems 79-83, use the fact that the orbit of...Ch. 10.3 - In Problems 79-83, use the fact that the orbit of...Ch. 10.3 - In Problems 79-83, use the fact that the orbit of...Ch. 10.3 - †In Problems 79-83, use the fact that the orbit...Ch. 10.3 - A rectangle is inscribed in an ellipse with major...Ch. 10.3 - Let D be the line given by the equation x+5=0 ....Ch. 10.3 - Show that an equation of the form A x 2 +C y 2...Ch. 10.3 - Show that the graph of an equation of the form A x...Ch. 10.3 - The eccentricity e of an ellipse is defined as the...Ch. 10.3 - Problems 89-92 are based on material learned...Ch. 10.3 - Problems 89-92 are based on material learned...Ch. 10.3 - Problems 89-92 are based on material learned...Ch. 10.3 - Prob. 92RYKCh. 10.4 - The distance d from P 1 =( 2,5 ) to P 2 =( 4,2 )...Ch. 10.4 - To complete the square of x 2 +5x , add ______....Ch. 10.4 - Find the intercepts of the equation y 2 =9+4 x 2 ....Ch. 10.4 - True or False The equation y 2 =9+ x 2 is...Ch. 10.4 - To graph y= ( x5 ) 3 4 , shift the graph of y= x 3...Ch. 10.4 - Find the vertical asymptotes, if any, and the...Ch. 10.4 - A(n) _______ is the collection of points in a...Ch. 10.4 - For a hyperbola, the foci lie on a line called the...Ch. 10.4 - Answer Problems 9-11 using the figure to the...Ch. 10.4 - Answer Problems 9-11 using the figure to the...Ch. 10.4 - Answer Problems 9-11 using the figure to the...Ch. 10.4 - In a hyperbola, if a=3 and c=5 , then b= ________....Ch. 10.4 - Prob. 13CVCh. 10.4 - Prob. 14CVCh. 10.4 - Prob. 15SBCh. 10.4 - In Problems 15-18, the graph of a hyperbola is...Ch. 10.4 - Prob. 17SBCh. 10.4 - Prob. 18SBCh. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - In Problems 19-28, find an equation for the...Ch. 10.4 - Prob. 29SBCh. 10.4 - In Problems 29-36, find the center, transverse...Ch. 10.4 - In Problems 29-36, find the center, transverse...Ch. 10.4 - Prob. 32SBCh. 10.4 - Prob. 33SBCh. 10.4 - Prob. 34SBCh. 10.4 - Prob. 35SBCh. 10.4 - Prob. 36SBCh. 10.4 - Prob. 37SBCh. 10.4 - In Problems 37-40, write an equation for each...Ch. 10.4 - Prob. 39SBCh. 10.4 - In Problems 37-40, write an equation for each...Ch. 10.4 - Prob. 41SBCh. 10.4 - Prob. 42SBCh. 10.4 - In Problems 41-48, find an equation for the...Ch. 10.4 - Prob. 44SBCh. 10.4 - Prob. 45SBCh. 10.4 - Prob. 46SBCh. 10.4 - Prob. 47SBCh. 10.4 - Prob. 48SBCh. 10.4 - Prob. 49SBCh. 10.4 - In Problems 49-62, find the center, transverse...Ch. 10.4 - Prob. 51SBCh. 10.4 - Prob. 52SBCh. 10.4 - Prob. 53SBCh. 10.4 - In Problems 49-62, find the center, transverse...Ch. 10.4 - Prob. 55SBCh. 10.4 - Prob. 56SBCh. 10.4 - Prob. 57SBCh. 10.4 - Prob. 58SBCh. 10.4 - Prob. 59SBCh. 10.4 - In Problems 49-62, find the center, transverse...Ch. 10.4 - Prob. 61SBCh. 10.4 - In Problems 49-62, find the center, transverse...Ch. 10.4 - Prob. 63SBCh. 10.4 - Prob. 64SBCh. 10.4 - Prob. 65SBCh. 10.4 - Prob. 66SBCh. 10.4 - Prob. 67MPCh. 10.4 - In Problems 67-74, analyze each equation. ( y+2 )...Ch. 10.4 - Prob. 69MPCh. 10.4 - Prob. 70MPCh. 10.4 - In Problems 67-74, analyze each equation. 25 x 2...Ch. 10.4 - In Problems 67-74, analyze each equation. x 2 +36...Ch. 10.4 - In Problems 67-74, analyze each equation. x 2...Ch. 10.4 - In Problems 67-74, analyze each equation. 9 x 2 y...Ch. 10.4 - Fireworks Display Suppose that two people standing...Ch. 10.4 - Lightning Strikes Suppose that two people standing...Ch. 10.4 - Nuclear Power Plaut Some nuclear power plants...Ch. 10.4 - An Explosion Two recording devices are set 2400...Ch. 10.4 - Rutherford’s Experiment In May 1911, Ernest...Ch. 10.4 - Hyperbolic Mirrors Hyperbolas have interesting...Ch. 10.4 - The eccentricity e of a hyperbola is defined as...Ch. 10.4 - A hyperbola for which a=b is called an equilateral...Ch. 10.4 - Two hyperbolas that have the same set of...Ch. 10.4 - Prove that the hyperbola y 2 a 2 x 2 b 2 =1 has...Ch. 10.4 - Show that the graph of an equation of the form A x...Ch. 10.4 - Show that the graph of an equation of the form A x...Ch. 10.4 - Prob. 87RYKCh. 10.4 - Prob. 88RYKCh. 10.4 - Prob. 89RYKCh. 10.4 - Prob. 90RYKCh. 10.5 - The sum formula for the sine function is sin( A+B...Ch. 10.5 - Prob. 2AYPCh. 10.5 - Prob. 3AYPCh. 10.5 - Prob. 4AYPCh. 10.5 - Prob. 5CVCh. 10.5 - Prob. 6CVCh. 10.5 - Prob. 7CVCh. 10.5 - Prob. 8CVCh. 10.5 - Prob. 9CVCh. 10.5 - Prob. 10CVCh. 10.5 - Prob. 11SBCh. 10.5 - Prob. 12SBCh. 10.5 - Prob. 13SBCh. 10.5 - Prob. 14SBCh. 10.5 - Prob. 15SBCh. 10.5 - Prob. 16SBCh. 10.5 - Prob. 17SBCh. 10.5 - Prob. 18SBCh. 10.5 - Prob. 19SBCh. 10.5 - Prob. 20SBCh. 10.5 - Prob. 21SBCh. 10.5 - Prob. 22SBCh. 10.5 - Prob. 23SBCh. 10.5 - Prob. 24SBCh. 10.5 - Prob. 25SBCh. 10.5 - Prob. 26SBCh. 10.5 - Prob. 27SBCh. 10.5 - Prob. 28SBCh. 10.5 - Prob. 29SBCh. 10.5 - Prob. 30SBCh. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - Prob. 35SBCh. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 31-42, rotate the axes so that the new...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - In Problems 43-52, identify the graph of each...Ch. 10.5 - Prob. 53AECh. 10.5 - Prob. 54AECh. 10.5 - Prob. 55AECh. 10.5 - Prob. 56AECh. 10.5 - Use the rotation formulas ( 5 ) to show that...Ch. 10.5 - Show that the graph of the equation x 1/2 + y 1/2...Ch. 10.5 - Formulate a strategy for analyzing and graphing an...Ch. 10.5 - Explain how your strategy presented in Problem 59...Ch. 10.5 - Problems 61-64 are based on material learned...Ch. 10.5 - Problems 61-64 are based on material learned...Ch. 10.5 - Problems 61-64 are based on material learned...Ch. 10.5 - Problems 61-64 are based on material learned...Ch. 10.6 - Prob. 1AYPCh. 10.6 - Transform the equation r=6cos from polar...Ch. 10.6 - A _______ is the set of points P in a plane such...Ch. 10.6 - The eccentricity e of a parabola is ________, of...Ch. 10.6 - True or False If ( r, ) are polar coordinates, the...Ch. 10.6 - Prob. 6CVCh. 10.6 - Prob. 7SBCh. 10.6 - Prob. 8SBCh. 10.6 - Prob. 9SBCh. 10.6 - Prob. 10SBCh. 10.6 - Prob. 11SBCh. 10.6 - Prob. 12SBCh. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - In Problems 13-24, analyze each equation and graph...Ch. 10.6 - Prob. 23SBCh. 10.6 - Prob. 24SBCh. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - In Problems 25-36, convert each polar equation to...Ch. 10.6 - Prob. 37SBCh. 10.6 - Prob. 38SBCh. 10.6 - Prob. 39SBCh. 10.6 - Prob. 40SBCh. 10.6 - Prob. 41SBCh. 10.6 - Prob. 42SBCh. 10.6 - Prob. 43AECh. 10.6 - Prob. 44AECh. 10.6 - Derive equation (d) in Table 5: r= ep 1esinCh. 10.6 - Orbit of Mercury The planet Mercury travels around...Ch. 10.6 - Prob. 47RYKCh. 10.6 - Problems 47-50 are based on material learned...Ch. 10.6 - Problems 47-50 are based on material learned...Ch. 10.6 - Problems 47-50 are based on material learned...Ch. 10.7 - The function f( x )=3sin( 4x ) has amplitude...Ch. 10.7 - Let x=f( t ) and y=g( t ) , where f and g are two...Ch. 10.7 - The parametric equations x=2sint , y=3cost define...Ch. 10.7 - Prob. 4CVCh. 10.7 - True or False Parametric equations defining a...Ch. 10.7 - True or False Curves defined using parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - In Problems 7-26, graph the curve whose parametric...Ch. 10.7 - y=4x1Ch. 10.7 - y=8x+3Ch. 10.7 - y= x 2 +1Ch. 10.7 - y= x 3Ch. 10.7 - y= x 3Ch. 10.7 - y= x 4 +1Ch. 10.7 - x= y 3/2Ch. 10.7 - x= yCh. 10.7 - In Problems 35-38, find parametric equations that...Ch. 10.7 - In Problems 35-38, find parametric equations that...Ch. 10.7 - In Problems 35-38, find parametric equations that...Ch. 10.7 - In Problems 35-38, find parametric equations that...Ch. 10.7 - In Problems 39-42, find parametric equations for...Ch. 10.7 - In Problems 39-42, find parametric equations for...Ch. 10.7 - In Problems 39-42, find parametric equations for...Ch. 10.7 - In Problems 39-42, find parametric equations for...Ch. 10.7 - In Problems 43 and 44, the parametric equations of...Ch. 10.7 - In Problems 43 and 44, the parametric equations of...Ch. 10.7 - In Problems 45-48, use a graphing utility to graph...Ch. 10.7 - In Problems 45-48, use a graphing utility to graph...Ch. 10.7 - In Problems 45-48, use a graphing utility to graph...Ch. 10.7 - In Problems 45-48, use a graphing utility to graph...Ch. 10.7 - Projectile Motion Bob throws a ball straight up...Ch. 10.7 - Projectile Motion Alice throws a ball straight up...Ch. 10.7 - Catching a Train Bill’s train leaves at 8:06 AM...Ch. 10.7 - Catching a Bus Jodi’s bus leaves at 5:30 PM and...Ch. 10.7 - Projectile Motion Ichiro throws a baseball with an...Ch. 10.7 - Projectile Motion Mark Texeira hit a baseball with...Ch. 10.7 - Projectile Motion Suppose that Adam hits a golf...Ch. 10.7 - Projectile Motion Suppose that Karla hits a golf...Ch. 10.7 - Uniform Motion AToyota Camry (traveling east at 40...Ch. 10.7 - Uniform Motion A Cessna (heading south at 120 mph)...Ch. 10.7 - The Green Monster The left field wall at Fenway...Ch. 10.7 - Projectile Motion The position of a projectile...Ch. 10.7 - Show that the parametric equations for a line...Ch. 10.7 - Hypocycloid The hypocycloid is a curve defined by...Ch. 10.7 - In Problem 62, we graphed the hypocycloid. Now...Ch. 10.7 - Problems 65-68 are based on material learned...Ch. 10.7 - Problems 65-68 are based on material learned...Ch. 10.7 - Problems 65-68 are based on material learned...Ch. 10.7 - Problems 65-68 are based on material learned...Ch. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - Prob. 4RECh. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - Prob. 9RECh. 10.R - In Problems 1-10, identify each equation. If it is...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 11-18, find an equation of the conic...Ch. 10.R - In Problems 19-23, identify each conic without...Ch. 10.R - Prob. 20RECh. 10.R - Prob. 21RECh. 10.R - Prob. 22RECh. 10.R - Prob. 23RECh. 10.R - In Problems 24-26, rotate the axes so that the new...Ch. 10.R - In Problems 24-26, rotate the axes so that the new...Ch. 10.R - In Problems 24-26, rotate the axes so that the new...Ch. 10.R - Prob. 27RECh. 10.R - Prob. 28RECh. 10.R - Prob. 29RECh. 10.R - Prob. 30RECh. 10.R - Prob. 31RECh. 10.R - In Problems 32-34, graph the curve whose...Ch. 10.R - Prob. 33RECh. 10.R - In Problems 32-34, graph the curve whose...Ch. 10.R - Prob. 35RECh. 10.R - Prob. 36RECh. 10.R - Prob. 37RECh. 10.R - Prob. 38RECh. 10.R - Prob. 39RECh. 10.R - Prob. 40RECh. 10.R - Prob. 41RECh. 10.R - Prob. 42RECh. 10.R - Prob. 43RECh. 10.R - Prob. 44RE
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- 4. 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_forward3. 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_forward
- The final answer is 8/π(sinx) + 8/3π(sin 3x)+ 8/5π(sin5x)....arrow_forwardKeity 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_forward
- 2. (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_forward1. (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_forward
- 3. (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_forward1. (i) (ii) which are not. What does it mean to say that a set ECR2 is closed? [1 Mark] Identify which of the following subsets of R2 are closed and (a) A = [-1, 1] × (1, 3) (b) B = [-1, 1] x {1,3} (c) C = {(1/n², 1/n2) ER2 | n EN} Provide a sketch and a brief explanation to each of your answers. [6 Marks] (iii) Give an example of a closed set which does not have interior points. [3 Marks]arrow_forwardA company specializing in lubrication products for vintage motors produce two blended oils, Smaza and Nefkov. They make a profit of K5,000.00 per litre of Smaza and K4,000.00 per litre of Nefkov. A litre of Smaza requires 0.4 litres of heavy oil and 0.6 litres of light oil. A litre of Nefkov requires 0.8 litres of heavy oil and 0.2 litres of light oil. The company has 100 litres of heavy oil and 80 litres of light oil. How many litres of each product should they make to maximize profits and what level of profit will they obtain? Show all your workings.arrow_forward
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