MML PRECALCULUS ENHANCED
7th Edition
ISBN: 9780134119250
Author: Sullivan
Publisher: INTER PEAR
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Textbook Question
Chapter 6.1, Problem 43SB
In Problems 35—46, convert each angle in radians to degrees.
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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]
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]
Chapter 6 Solutions
MML PRECALCULUS ENHANCED
Ch. 6.1 - What is the formula for the circumference C of a...Ch. 6.1 - If an object has a speed of r feet per second and...Ch. 6.1 - An angle is in _____ _____ if its vertex is at...Ch. 6.1 - A _____ _____ is a positive angle whose vertex is...Ch. 6.1 - If the radius of a circle is r and the length of...Ch. 6.1 - On a circle of radius r , a central angle of ...Ch. 6.1 - 180 = _____ radians a. 2 b. c. 3 2 d. 2Ch. 6.1 - An object travels on a circle of radius r with...Ch. 6.1 - True or False The angular speed of an object...Ch. 6.1 - True or False For circular motion on a circle of...
Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 11-22, draw each angle in standard...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 23-34, convert each angle in degrees...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 35—46, convert each angle in radians...Ch. 6.1 - In Problems 47-52, convert each angle in degrees...Ch. 6.1 - In Problems 47-52, convert each angle in degrees...Ch. 6.1 - In Problems 47-52, convert each angle in degrees...Ch. 6.1 - In Problems 47-52, convert each angle in degrees...Ch. 6.1 - In Problems 47-52, convert each angle in degrees...Ch. 6.1 - In Problems 47-52, convert each angle in degrees...Ch. 6.1 - In Problems 53-58, convert each angle in radians...Ch. 6.1 - In Problems 53-58, convert each angle in radians...Ch. 6.1 - In Problems 53-58, convert each angle in radians...Ch. 6.1 - In Problems 53-58, convert each angle in radians...Ch. 6.1 - In Problems 53-58, convert each angle in radians...Ch. 6.1 - In Problems 53-58, convert each angle in radians...Ch. 6.1 - In Problems 59-64, convert each angle to a decimal...Ch. 6.1 - In Problems 59-64, convert each angle to a decimal...Ch. 6.1 - Prob. 61SBCh. 6.1 - Prob. 62SBCh. 6.1 - In Problems 59-64, convert each angle to a decimal...Ch. 6.1 - Prob. 64SBCh. 6.1 - In Problems 65-70, convert each angle to DMS form....Ch. 6.1 - In Problems 65-70, convert each angle to DMS form....Ch. 6.1 - In Problems 65-70, convert each angle to DMS form....Ch. 6.1 - In Problems 65-70, convert each angle to DMS form....Ch. 6.1 - In Problems 65-70, convert each angle to DMS form....Ch. 6.1 - In Problems 65-70, convert each angle to DMS form....Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 71-78, s denotes the length of the are...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - In Problems 79-86, A denotes the area of the...Ch. 6.1 - Prob. 87SBCh. 6.1 - Prob. 88SBCh. 6.1 - Prob. 89SBCh. 6.1 - Prob. 90SBCh. 6.1 - Movement of a Minute Hand The minute hand of a...Ch. 6.1 - Prob. 92AECh. 6.1 - Prob. 93AECh. 6.1 - Prob. 94AECh. 6.1 - Prob. 95AECh. 6.1 - Prob. 96AECh. 6.1 - Prob. 97AECh. 6.1 - Prob. 98AECh. 6.1 - Prob. 99AECh. 6.1 - Prob. 100AECh. 6.1 - Prob. 101AECh. 6.1 - Prob. 102AECh. 6.1 - Prob. 103AECh. 6.1 - Prob. 104AECh. 6.1 - Prob. 105AECh. 6.1 - Car Wheels The radius of each wheel of a car is 15...Ch. 6.1 - In Problems 107-110, the latitude of a location L...Ch. 6.1 - Prob. 108AECh. 6.1 - In Problems 107-110, the latitude of a location L...Ch. 6.1 - In Problems 107-110, the latitude of a location L...Ch. 6.1 - Speed of the Moon The mean distance of the moon...Ch. 6.1 - Speed of Earth The mean distance of Earth from the...Ch. 6.1 - Pulleys Two pulleys, one with radius 2 inches and...Ch. 6.1 - Ferris Wheels A neighborhood carnival has a Ferris...Ch. 6.1 - Computing the Speed of a River Current To...Ch. 6.1 - Spin Balancing Tires A spin balancer rotates the...Ch. 6.1 - The Cable Cars of San Francisco At the Cable Car...Ch. 6.1 - Difference in Time of Sunrise Naples, Florida, is...Ch. 6.1 - Let the Dog Roam A dog is attached to a 9-foot...Ch. 6.1 - Area of a Region The measure of are BE is 2 ....Ch. 6.1 - Keeping Up with the Sun How fast would you have to...Ch. 6.1 - Nautical Miles A nautical mile equals the length...Ch. 6.1 - Approximating the Circumference of Earth...Ch. 6.1 - Prob. 124AECh. 6.1 - Pulleys Two pulleys, one with radius r 1 and the...Ch. 6.1 - Do you prefer to measure angles using degrees or...Ch. 6.1 - What is 1 radian? What is 1 degree?Ch. 6.1 - Which angle has the larger measure: 1 degree or 1...Ch. 6.1 - Explain the difference between linear speed and...Ch. 6.1 - For a circle of radius r , a central angle of ...Ch. 6.1 - Discuss why ships and airplanes use nautical miles...Ch. 6.1 - Investigate the way that speed bicycles work. In...Ch. 6.1 - In Example 6, we found that the distance between...Ch. 6.1 - Problems 134-137 are based on material learned...Ch. 6.1 - Problems 134-137 are based on material learned...Ch. 6.1 - Problems 134-137 are based on material learned...Ch. 6.1 - Problems 134-137 are based on material learned...Ch. 6.2 - In a right triangle, with legs a and b and...Ch. 6.2 - The value of the function f( x )=3x7 at 5 is...Ch. 6.2 - True or False For a function y=f( x ) , for each x...Ch. 6.2 - If two triangles are similar, then corresponding...Ch. 6.2 - What point is symmetric with respect to the y-axis...Ch. 6.2 - Prob. 6AYPCh. 6.2 - Which function takes as input a real number t that...Ch. 6.2 - The point on the unit circle that corresponds to =...Ch. 6.2 - The point on the unit circle that corresponds to =...Ch. 6.2 - The point on the unit circle that corresponds to =...Ch. 6.2 - For any angle in standard position, let P=( x,y )...Ch. 6.2 - True or False Exact values can be found for the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 13-20, P=( x,y ) is the point on the...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 21-30, find the exact value. Do not...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 31-46, find the exact value of each...Ch. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - Prob. 52SBCh. 6.2 - Prob. 53SBCh. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - Prob. 55SBCh. 6.2 - Prob. 56SBCh. 6.2 - Prob. 57SBCh. 6.2 - Prob. 58SBCh. 6.2 - Prob. 59SBCh. 6.2 - Prob. 60SBCh. 6.2 - Prob. 61SBCh. 6.2 - Prob. 62SBCh. 6.2 - Prob. 63SBCh. 6.2 - Prob. 64SBCh. 6.2 - Prob. 65SBCh. 6.2 - Prob. 66SBCh. 6.2 - In Problems 47-64, find the exact values of the...Ch. 6.2 - Prob. 68SBCh. 6.2 - Prob. 69SBCh. 6.2 - Prob. 70SBCh. 6.2 - Prob. 71SBCh. 6.2 - Prob. 72SBCh. 6.2 - Prob. 73SBCh. 6.2 - Prob. 74SBCh. 6.2 - Prob. 75SBCh. 6.2 - Prob. 76SBCh. 6.2 - Prob. 77SBCh. 6.2 - Prob. 78SBCh. 6.2 - Prob. 79SBCh. 6.2 - Prob. 80SBCh. 6.2 - Prob. 81SBCh. 6.2 - Prob. 82SBCh. 6.2 - Prob. 83SBCh. 6.2 - Prob. 84SBCh. 6.2 - Prob. 85SBCh. 6.2 - Prob. 86SBCh. 6.2 - Find the exact value of: sin 40 +sin 130 +sin...Ch. 6.2 - Prob. 88SBCh. 6.2 - Prob. 89SBCh. 6.2 - Prob. 90SBCh. 6.2 - Prob. 91SBCh. 6.2 - Prob. 92SBCh. 6.2 - Prob. 93SBCh. 6.2 - Prob. 94SBCh. 6.2 - Prob. 95SBCh. 6.2 - Prob. 96SBCh. 6.2 - Prob. 97SBCh. 6.2 - Prob. 98SBCh. 6.2 - Prob. 99SBCh. 6.2 - Prob. 100SBCh. 6.2 - Prob. 101SBCh. 6.2 - Prob. 102SBCh. 6.2 - Prob. 103SBCh. 6.2 - Prob. 104SBCh. 6.2 - Prob. 105SBCh. 6.2 - Prob. 106SBCh. 6.2 - Prob. 107MPCh. 6.2 - Prob. 108MPCh. 6.2 - Prob. 109MPCh. 6.2 - Prob. 110MPCh. 6.2 - Prob. 111MPCh. 6.2 - Prob. 112MPCh. 6.2 - In Problems 107-116, f( x )=sinx , g( x )=cosx ,...Ch. 6.2 - Prob. 114MPCh. 6.2 - Prob. 115MPCh. 6.2 - Prob. 116MPCh. 6.2 - Prob. 117AECh. 6.2 - Prob. 118AECh. 6.2 - Use a calculator in radian mode to complete the...Ch. 6.2 - Use a calculator in radian mode to complete the...Ch. 6.2 - Prob. 121AECh. 6.2 - For Problems 121-124, use the following...Ch. 6.2 - Prob. 123AECh. 6.2 - Prob. 124AECh. 6.2 - Prob. 125AECh. 6.2 - Prob. 126AECh. 6.2 - Calculating the Time of a Trip Two oceanfront...Ch. 6.2 - Designing Fine Decorative Pieces A designer of...Ch. 6.2 - Use the following to answer Problems 129-132. The...Ch. 6.2 - Use the following to answer Problems 129-132. The...Ch. 6.2 - Use the following to answer Problems 129-132. The...Ch. 6.2 - Use the following to answer Problems 129-132. The...Ch. 6.2 - Let be the measure of an angle, in radians, in...Ch. 6.2 - Let be the measure of an angle, in radians, in...Ch. 6.2 - Projectile Motion An object is propelled upward at...Ch. 6.2 - If , 0 is the angle between the positive x-axis...Ch. 6.2 - In Problems 137 and 138, use the figure to...Ch. 6.2 - In Problems 137 and 138, use the figure to...Ch. 6.2 - Prob. 139DWCh. 6.2 - Prob. 140DWCh. 6.2 - How would you explain the meaning of the sine...Ch. 6.2 - Prob. 142DWCh. 6.2 - Problems 143-146 are based on material learned...Ch. 6.2 - Problems 143-146 are based on material learned...Ch. 6.2 - Problems 143-146 are based on material learned...Ch. 6.2 - Problems 143-146 are based on material learned...Ch. 6.3 - The domain of the function f(x)= x+1 2x+1 is _____...Ch. 6.3 - A function for which f(x)=f(x) for all x in the...Ch. 6.3 - True or False The function f(x)= x is even....Ch. 6.3 - True or False The equation x 2 +2x= (x+1) 2 1 is...Ch. 6.3 - The sine, cosine, cosecant, and secant functions...Ch. 6.3 - The domain of the tangent function is _____ .Ch. 6.3 - Which of the following is not in the range of the...Ch. 6.3 - Which of the following functions is even? a....Ch. 6.3 - sin 2 + cos 2 = _____ .Ch. 6.3 - True or False sec= 1 sinCh. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - ln Problems 11-26, use the fact that the...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - In Problems 27—34, name the quadrant in which...Ch. 6.3 - Prob. 34SBCh. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In problems 35-42, sin and cos are given. Find the...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 43-58, find the exact value of each of...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 59-76, use the even-odd properties to...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - In Problems 77-88, use properties of the...Ch. 6.3 - If sin=0.3 , find the value of: sin+sin( +2 )+sin(...Ch. 6.3 - If cos=0.2 , find the value of: cos+cos( +2 )+cos(...Ch. 6.3 - If tan=3 , find the value of: tan+tan( + )+tan( +2...Ch. 6.3 - If cot=2 , find the value of: cot+cot( - )+cot( -2...Ch. 6.3 - Find the exact value of: sin 1 + sin2 + sin3 ++...Ch. 6.3 - Find the exact value of: cos 1 + cos2 + cos3 ++...Ch. 6.3 - What is the domain of the sine function?Ch. 6.3 - What is the domain of the cosine function?Ch. 6.3 - For what numbers is f( )=tan not defined?Ch. 6.3 - For what numbers is f( )=cot not defined?Ch. 6.3 - For what numbers is f( )=sec not defined?Ch. 6.3 - For what numbers is f( )=csc not defined?Ch. 6.3 - What is the range of the sine function?Ch. 6.3 - What is the range of the cosine function?Ch. 6.3 - What is the range of the tangent function?Ch. 6.3 - What is the range of the cotangent function?Ch. 6.3 - What is the range of the secant function?Ch. 6.3 - What is the range of the cosecant function?Ch. 6.3 - Is the sine function even, odd, or neither? Is its...Ch. 6.3 - Is the cosine function even, odd, or neither? Is...Ch. 6.3 - Is the tangent function even, odd, or neither? Is...Ch. 6.3 - Is the cotangent function even, odd, or neither?...Ch. 6.3 - Is the cotangent function even, odd, or neither?...Ch. 6.3 - Is the cotangent function even, odd, or neither?...Ch. 6.3 - In Problems 113-118, use the periodic and even-odd...Ch. 6.3 - In Problems 113-118, use the periodic and even-odd...Ch. 6.3 - In Problems 113-118, use the periodic and even-odd...Ch. 6.3 - In Problems 113-118, use the periodic and even-odd...Ch. 6.3 - In Problems 113-118, use the periodic and even-odd...Ch. 6.3 - In Problems 113-118, use the periodic and even-odd...Ch. 6.3 - Calculating the Time of a Trip From a parking lot,...Ch. 6.3 - Calculating the Time of a Trip Two oceanfront...Ch. 6.3 - Show that the range of the tangent function is the...Ch. 6.3 - Show that the range of the cotangent function is...Ch. 6.3 - Show that the period of f( )=sin is 2 . [Hint:...Ch. 6.3 - show that the period of f( )=cos is 2 .Ch. 6.3 - show that the period of f( )=sec is 2 .Ch. 6.3 - show that the period of f( )=csc is 2 .Ch. 6.3 - show that the period of f( )=tan is .Ch. 6.3 - show that the period of f( )=cot is .Ch. 6.3 - Prove the reciprocal identities given in formula...Ch. 6.3 - Prove the quotient identities given in formula...Ch. 6.3 - Establish the identity: (sincos) 2 + (sinsin) 2 +...Ch. 6.3 - Write down five properties of the tangent...Ch. 6.3 - Describe your understanding of the meaning of a...Ch. 6.3 - Explain how to find the value of sin 390 using...Ch. 6.3 - Explain how to find the value of cos( 45 ) using...Ch. 6.3 - Explain how to find the value of sin 390 and cos(...Ch. 6.3 - Problems 137-140 are based on material learned...Ch. 6.3 - Problems 137-140 are based on material learned...Ch. 6.3 - Problems 137-140 are based on material learned...Ch. 6.3 - Problems 137-140 are based on material learned...Ch. 6.4 - Use transformations to graph y=3 x 2 . (pp....Ch. 6.4 - Use transformations to graph y= 2x . (pp. 106-114)Ch. 6.4 - The maximum value of y=sinx , 0x2 , is ____ and...Ch. 6.4 - The function y=Asin( x ) , A0 ,has amplitude 3 and...Ch. 6.4 - The function y=3cos( 6x ) has amplitude ____ and...Ch. 6.4 - True or False The graphs of y=sinx and y=cosx are...Ch. 6.4 - True or false For y=2sin( x ) , the amplitude is 2...Ch. 6.4 - True or False The graph of the sine function has...Ch. 6.4 - One period of the graph of y=sin( x ) or y=cos( x...Ch. 6.4 - To graph y=3sin( 2x ) using key points, the...Ch. 6.4 - f( x )=sinx (a) What is the y-intercept of the...Ch. 6.4 - g( x )=cosx (a) What is the y-intercept of the...Ch. 6.4 - In Problems 13-22, determine the amplitude and...Ch. 6.4 - In Problems 13-22, determine the amplitude and...Ch. 6.4 - In Problems 13-22, determine the amplitude and...Ch. 6.4 - Prob. 16SBCh. 6.4 - Prob. 17SBCh. 6.4 - In Problems 13-22, determine the amplitude and...Ch. 6.4 - In Problems 13-22, determine the amplitude and...Ch. 6.4 - Prob. 20SBCh. 6.4 - Prob. 21SBCh. 6.4 - Prob. 22SBCh. 6.4 - Prob. 23SBCh. 6.4 - In Problems 23-32, match the given function to one...Ch. 6.4 - Prob. 25SBCh. 6.4 - Prob. 26SBCh. 6.4 - Prob. 27SBCh. 6.4 - Prob. 28SBCh. 6.4 - In Problems 23-32, match the given function to one...Ch. 6.4 - In Problems 23-32, match the given function to one...Ch. 6.4 - In Problems 23-32, match the given function to one...Ch. 6.4 - In Problems 23-32, match the given function to one...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - In Problems 33-56, graph each function using...Ch. 6.4 - Prob. 57SBCh. 6.4 - Prob. 58SBCh. 6.4 - In Problems 57-60, write the equation of a sine...Ch. 6.4 - In Problems 57-60, write the equation of a sine...Ch. 6.4 - Prob. 61SBCh. 6.4 - Prob. 62SBCh. 6.4 - Prob. 63SBCh. 6.4 - Prob. 64SBCh. 6.4 - Prob. 65SBCh. 6.4 - Prob. 66SBCh. 6.4 - Prob. 67SBCh. 6.4 - Prob. 68SBCh. 6.4 - Prob. 69SBCh. 6.4 - Prob. 70SBCh. 6.4 - Prob. 71SBCh. 6.4 - Prob. 72SBCh. 6.4 - Prob. 73SBCh. 6.4 - Prob. 74SBCh. 6.4 - Prob. 75MPCh. 6.4 - Prob. 76MPCh. 6.4 - Prob. 77MPCh. 6.4 - Prob. 78MPCh. 6.4 - In Problems 79-82, find ( fg ) (x) and (gf) ( x )...Ch. 6.4 - Prob. 80MPCh. 6.4 - In Problems 79-82, find ( fg ) (x) and (gf) ( x )...Ch. 6.4 - In Problems 79-82, find ( fg ) (x) and (gf) ( x )...Ch. 6.4 - In Problems 83 and 84, graph each function. f( x...Ch. 6.4 - In Problems 83 and 84, graph each function. g( x...Ch. 6.4 - Alternating Current (ac) Circuits The current I ,...Ch. 6.4 - Prob. 86AECh. 6.4 - Alternating Current (ac) Generators The voltage V...Ch. 6.4 - Alternating Current (ac) Generators The voltage V...Ch. 6.4 - Alternating Current (ac) Generators The voltage V...Ch. 6.4 - Bridge Clearance A one-lane highway runs through a...Ch. 6.4 - Blood Pressure Blood pressure is a way of...Ch. 6.4 - Ferris Wheel The function h( t )=100cos( 15 t...Ch. 6.4 - Hours of Daylight For a certain town in Alaska,...Ch. 6.4 - Holding Pattern The function d( t )=50cos( 10 t...Ch. 6.4 - Biorhythms In the theory of biorhythms, a sine...Ch. 6.4 - Graph y=| cosx |,2x2 .Ch. 6.4 - Graph y=| sinx |,2x2 .Ch. 6.4 - Prob. 98AECh. 6.4 - In Problems 98-101, the graphs of the given pairs...Ch. 6.4 - In Problems 98-101, the graphs of the given pairs...Ch. 6.4 - In Problems 98-101, the graphs of the given pairs...Ch. 6.4 - Prob. 102DWCh. 6.4 - Explain the term amplitude as it relates to the...Ch. 6.4 - Explain the term period as it relates to the graph...Ch. 6.4 - Explain how the amplitude and period of a...Ch. 6.4 - Find an application in your major field that leads...Ch. 6.4 - Problems 107-110 are based on material learned...Ch. 6.4 - Problems 107-110 are based on material learned...Ch. 6.4 - Problems 107-110 are based on material learned...Ch. 6.4 - Prob. 110RYKCh. 6.5 - The graph of y= 3x6 x4 has a vertical asymptote....Ch. 6.5 - True or False If x=3 is a vertical asymptote of a...Ch. 6.5 - The graph of y=tanx is symmetric with respect to...Ch. 6.5 - The graph of y=secx is symmetric with respect to...Ch. 6.5 - It is easiest to graph y=secx by first sketching...Ch. 6.5 - True or False The graphs of...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 7-16, if necessary, refer to the...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 17-40, graph each function. Be sure to...Ch. 6.5 - In Problems 41-44, find the average rale of change...Ch. 6.5 - In Problems 41-44, find the average rale of change...Ch. 6.5 - In Problems 41-44, find the average rale of change...Ch. 6.5 - In Problems 41-44, find the average rale of change...Ch. 6.5 - In Problems 45-48, find ( fg )( x )and( gf )( x )...Ch. 6.5 - In Problems 45-48, find ( fg )( x )and( gf )( x )...Ch. 6.5 - In Problems 45-48, find ( fg )( x )and( gf )( x )...Ch. 6.5 - In Problems 45-48, find ( fg )( x )and( gf )( x )...Ch. 6.5 - In Problems 49 and 50, graph each function. f( x...Ch. 6.5 - In Problems 49 and 50, graph each function. g( x...Ch. 6.5 - Carrying a Ladder around a Corner Two hallways,...Ch. 6.5 - A Rotating Beacon Suppose that a fire truck is...Ch. 6.5 - Exploration Graph y=tanxandy=cot( x+ 2 ) Do you...Ch. 6.5 - Problems 54-57 are based on material learned...Ch. 6.5 - Problems 54-57 are based on material learned...Ch. 6.5 - Problems 54-57 are based on material learned...Ch. 6.5 - Problems 54-57 are based on material learned...Ch. 6.6 - For the graph of y=Asin( x ) , the number is...Ch. 6.6 - True or False A graphing utility requires only two...Ch. 6.6 - Prob. 3SBCh. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - Prob. 8SBCh. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 3-14, find the amplitude, period, and...Ch. 6.6 - In Problems 15-18, write the equation of a sine...Ch. 6.6 - In Problems 15-18, write the equation of a sine...Ch. 6.6 - In Problems 15-18, write the equation of a sine...Ch. 6.6 - In Problems 15-18, write the equation of a sine...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - In Problems 19-26, apply the methods of this and...Ch. 6.6 - Alternating Current (ac) Circuits The current I ,...Ch. 6.6 - Alternating Current (ac) Circuits The current I ,...Ch. 6.6 - Hurricanes Hurricanes are categorized using the...Ch. 6.6 - Monthly Temperature The data below represent the...Ch. 6.6 - Monthly Temperature The given data represent the...Ch. 6.6 - Monthly Temperature The following data represent...Ch. 6.6 - Tides The length of time between consecutive high...Ch. 6.6 - Tides The length of time between consecutive high...Ch. 6.6 - Hours of Daylight According to the Old Farmer’s...Ch. 6.6 - Hours of Daylight According to the Old Farmer's...Ch. 6.6 - Hours of Daylight According to the Old Farmer's...Ch. 6.6 - Hours of Daylight According to the Old Fanner's...Ch. 6.6 - Prob. 39DWCh. 6.6 - Find an application in your major field that leads...Ch. 6.6 - Prob. 41RYKCh. 6.6 - Prob. 42RYKCh. 6.6 - Problems 41-44 are based on material learned...Ch. 6.6 - Problems 41-44 are based on material learned...
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- 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]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_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_forwardThe 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_forward
- 2. (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_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_forward
- 2. 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_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_forward
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