CALCULUS AND ITS APPLICATIONS BRIEF
12th Edition
ISBN: 9780135998229
Author: BITTINGER
Publisher: PEARSON
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Textbook Question
Chapter 2.6, Problem 61E
A population
- Find the tripling time of this population.
- Find the percent increase of the population after two tripling periods.
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Students have asked these similar questions
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 2 Solutions
CALCULUS AND ITS APPLICATIONS BRIEF
Ch. 2.1 - Prob. 2ECh. 2.1 - Prob. 3ECh. 2.1 - Prob. 4ECh. 2.1 - Graph each function. Then identify the domain,...Ch. 2.1 - Prob. 6ECh. 2.1 - Prob. 7ECh. 2.1 - Prob. 9ECh. 2.1 - Prob. 10ECh. 2.1 - For Exercises 9-16, an initial investment amount...Ch. 2.1 - Prob. 12E
Ch. 2.1 - Prob. 13ECh. 2.1 - Prob. 14ECh. 2.1 - For Exercises 9-16, an initial investment amount...Ch. 2.1 - Prob. 16ECh. 2.1 - Prob. 17ECh. 2.1 - Prob. 18ECh. 2.1 - Prob. 19ECh. 2.1 - Prob. 20ECh. 2.1 - For Exercises 17-26, use a calculator to find each...Ch. 2.1 - Prob. 22ECh. 2.1 - For Exercises 17-26, use a calculator to find each...Ch. 2.1 - Prob. 24ECh. 2.1 - For Exercises 17-26, use a calculator to find each...Ch. 2.1 - Prob. 26ECh. 2.1 - Given ln4=1.3863 and ln5=1.6094, use properties of...Ch. 2.1 - Prob. 28ECh. 2.1 - Given ln4=1.3863 and ln5=1.6094, use properties of...Ch. 2.1 - Prob. 30ECh. 2.1 - Given ln4=1.3863 and ln5=1.6094, use properties of...Ch. 2.1 - Prob. 32ECh. 2.1 - Given ln4=1.3863 and ln5=1.6094, use properties of...Ch. 2.1 - Prob. 34ECh. 2.1 - Given ln4=1.3863 and ln5=1.6094, use properties of...Ch. 2.1 - Prob. 36ECh. 2.1 - Given ln4=1.3863 and ln5=1.6094, use properties of...Ch. 2.1 - Prob. 38ECh. 2.1 - Prob. 39ECh. 2.1 - Prob. 40ECh. 2.1 - Prob. 41ECh. 2.1 - Prob. 42ECh. 2.1 - Prob. 43ECh. 2.1 - Prob. 44ECh. 2.1 - Solve for t. Round the answer to three decimal...Ch. 2.1 - Prob. 46ECh. 2.1 - Prob. 47ECh. 2.1 - Prob. 48ECh. 2.1 - Solve for t. Round the answer to three decimal...Ch. 2.1 - Prob. 50ECh. 2.1 - Prob. 51ECh. 2.1 - Prob. 52ECh. 2.1 - Find the domain of each logarithmic function and...Ch. 2.1 - Prob. 54ECh. 2.1 - Find the domain of each logarithmic function and...Ch. 2.1 - Prob. 56ECh. 2.1 - Find the domain of each logarithmic function and...Ch. 2.1 - Prob. 58ECh. 2.1 - Prob. 59ECh. 2.1 - Prob. 60ECh. 2.1 - Prob. 61ECh. 2.1 - Prob. 62ECh. 2.1 - Solve each logarithmic equation. Round the answer...Ch. 2.1 - Prob. 64ECh. 2.1 - Prob. 65ECh. 2.1 - Prob. 66ECh. 2.1 - Solve each logarithmic equation. Round the answer...Ch. 2.1 - Prob. 68ECh. 2.1 - U.S. travel exports. U.S. travel exports (goods...Ch. 2.1 - Prob. 70ECh. 2.1 - Compound interest: future value. Dennis deposits...Ch. 2.1 - Prob. 72ECh. 2.1 - Prob. 73ECh. 2.1 - Prob. 74ECh. 2.1 - Prob. 75ECh. 2.1 - Prob. 76ECh. 2.1 - Prob. 77ECh. 2.1 - Prob. 78ECh. 2.1 - Prob. 79ECh. 2.1 - Prob. 80ECh. 2.1 - Cooling liquid. A cup of hot coffee is placed on a...Ch. 2.1 - Prob. 82ECh. 2.1 - Prob. 83ECh. 2.1 - Prob. 84ECh. 2.1 - In Exercises 8594, solve for x. 85. e2x5ex+4=0....Ch. 2.1 - Prob. 86ECh. 2.1 - Prob. 87ECh. 2.1 - Prob. 88ECh. 2.1 - In Exercises 8594, solve for x. 89. e2xex12=0Ch. 2.1 - Prob. 90ECh. 2.1 - Prob. 91ECh. 2.1 - Prob. 92ECh. 2.1 - Prob. 93ECh. 2.1 - Prob. 94ECh. 2.1 - Prob. 95ECh. 2.1 - Prob. 96ECh. 2.1 - Prob. 97ECh. 2.1 - Prob. 99ECh. 2.1 - Prob. 100ECh. 2.1 - Prob. 101ECh. 2.1 - Prob. 102ECh. 2.1 - Prob. 103ECh. 2.2 - Differentiate. 1. g(x)=e2xCh. 2.2 - Prob. 2ECh. 2.2 - Differentiate. 3. g(x)=3e5xCh. 2.2 - Prob. 4ECh. 2.2 - Differentiate. 5. G(x)=x35e2xCh. 2.2 - Prob. 6ECh. 2.2 - Prob. 7ECh. 2.2 - Prob. 8ECh. 2.2 - Prob. 9ECh. 2.2 - Prob. 10ECh. 2.2 - Differentiate. 11. f(x)=x22x+2exCh. 2.2 - Prob. 12ECh. 2.2 - Differentiate. 13. f(x)=ex2+8xCh. 2.2 - Prob. 14ECh. 2.2 - Differentiate. 15. y=ex1Ch. 2.2 - Prob. 16ECh. 2.2 - Differentiate. 17. y=ex+x3xexCh. 2.2 - Prob. 18ECh. 2.2 - Differentiate. 19. g(x)=4x2+3xex27xCh. 2.2 - Prob. 20ECh. 2.2 - Prob. 21ECh. 2.2 - Prob. 22ECh. 2.2 - Differentiate. 23. r(t)=t2+2tet2Ch. 2.2 - Differentiate. 24. f(t)=t35te4t3Ch. 2.2 - Prob. 25ECh. 2.2 - Prob. 26ECh. 2.2 - Prob. 27ECh. 2.2 - Prob. 28ECh. 2.2 - Prob. 29ECh. 2.2 - Prob. 30ECh. 2.2 - Find the second derivative. 31. d(x)=e2x+1Ch. 2.2 - Prob. 32ECh. 2.2 - Prob. 33ECh. 2.2 - Prob. 34ECh. 2.2 - Find the second derivative. 35. w(x)=xexCh. 2.2 - Prob. 36ECh. 2.2 - Find the second derivative. 37. f(t)=(2t+3)e3tCh. 2.2 - Prob. 38ECh. 2.2 - Find the second derivative. 39. z(x)=e2x+12Ch. 2.2 - Prob. 40ECh. 2.2 - Find the second derivative. 41. w(t)=t2+2t+3e5tCh. 2.2 - Prob. 42ECh. 2.2 - Prob. 43ECh. 2.2 - Prob. 44ECh. 2.2 - Find the second derivative. 45. f(t)=e3t1Ch. 2.2 - Prob. 46ECh. 2.2 - Marginal cost. The total cost, in millions of...Ch. 2.2 - Prob. 48ECh. 2.2 - Growth of a retirement fund. Maria invests $20,000...Ch. 2.2 - Prob. 50ECh. 2.2 - Depreciation. Perriots Restaurant purchased...Ch. 2.2 - Prob. 52ECh. 2.2 - Prob. 53ECh. 2.2 - Prob. 54ECh. 2.2 - Prob. 59ECh. 2.2 - Prob. 60ECh. 2.2 - Prob. 61ECh. 2.2 - Prob. 62ECh. 2.2 - Prob. 63ECh. 2.2 - Prob. 64ECh. 2.2 - Prob. 65ECh. 2.2 - Prob. 66ECh. 2.2 - Prob. 67ECh. 2.2 - Prob. 68ECh. 2.2 - Prob. 69ECh. 2.2 - Prob. 70ECh. 2.2 - Prob. 71ECh. 2.2 - Prob. 72ECh. 2.2 - For each of the functions in Exercises 70-73,...Ch. 2.2 - Prob. 75ECh. 2.3 - Differentiate. 1. y=9lnxCh. 2.3 - Prob. 2ECh. 2.3 - Prob. 3ECh. 2.3 - Prob. 4ECh. 2.3 - Differentiate. 5. fx=ln10xCh. 2.3 - Prob. 6ECh. 2.3 - Differentiate. 7. y=x6lnxCh. 2.3 - Prob. 8ECh. 2.3 - Differentiate. 9. y=lnxx5Ch. 2.3 - Prob. 10ECh. 2.3 - Differentiate. 11. y=lnx24Hint:lnAB=lnAlnBCh. 2.3 - Prob. 12ECh. 2.3 - Differentiate. 13. y=ln3x2+2x1Ch. 2.3 - Differentiate. 14. y=ln7x2+5x+2Ch. 2.3 - Differentiate. 15. f(x)=lnx2+5xCh. 2.3 - Differentiate. 16. f(x)=lnx27xCh. 2.3 - Differentiate. 17. g(x)=(lnx)4 (Hint: Use the...Ch. 2.3 - Differentiate. 18. g(x)=(lnx)3Ch. 2.3 - Differentiate. 19. h(x)=lnx2x3+1e2xCh. 2.3 - Differentiate. 20. h(x)=ln2x4e3xx2+x+15Ch. 2.3 - Find the equation of the line tangent to the graph...Ch. 2.3 - Find the equation of the line tangent to the graph...Ch. 2.3 - Find the equation of the line tangent to the graph...Ch. 2.3 - Find the equation of the line tangent to the graph...Ch. 2.3 - Prob. 25ECh. 2.3 - Advertising. A model for consumers' response to...Ch. 2.3 - Prob. 27ECh. 2.3 - Prob. 28ECh. 2.3 - Forgetting. As part of a study, students in a...Ch. 2.3 - Walking speed. Bornstein and Bornstein found in a...Ch. 2.3 - Prob. 31ECh. 2.3 - Prob. 32ECh. 2.3 - Prob. 33ECh. 2.3 - Prob. 34ECh. 2.3 - Prob. 35ECh. 2.3 - Prob. 36ECh. 2.3 - Prob. 37ECh. 2.3 - Prob. 38ECh. 2.3 - Prob. 39ECh. 2.3 - Prob. 40ECh. 2.3 - Prob. 41ECh. 2.3 - In Exercise 34, the time t, in weeks, needed for...Ch. 2.3 - Prob. 43ECh. 2.3 - Prob. 44ECh. 2.3 - Let y1=ax and y2=lnx. Find a such that the graph...Ch. 2.3 - Prob. 46ECh. 2.4 - Find f if f(x)=4f(x).Ch. 2.4 - Find g if g(x)=6g(x).Ch. 2.4 - Find the function that satisfies dA/dt=9A.Ch. 2.4 - Find the function that satisfies dP/dt=3P(t).Ch. 2.4 - Find the function that satisfies dQ/dt=kQ.Ch. 2.4 - Find the function that satisfies dR/dt=kR.Ch. 2.4 - U.S. patents. Between 2006 and 2016, the number of...Ch. 2.4 - Franchise expansion. Pete Zah's, Inc., is selling...Ch. 2.4 - Compound interest. If an amount P0 is invested in...Ch. 2.4 - Compound interest. If an amount P0 is deposited in...Ch. 2.4 - Bottled water sales. The volume of bottled water...Ch. 2.4 - Apps downloads. Since June 2014, the number of...Ch. 2.4 - Art masterpieces. In 2004, a collector paid...Ch. 2.4 - Prob. 14ECh. 2.4 - Federal receipts. In 2013, U.S. federal receipts...Ch. 2.4 - Prob. 16ECh. 2.4 - Prob. 17ECh. 2.4 - Prob. 18ECh. 2.4 - Value of Manhattan Island. Peter Minuit of the...Ch. 2.4 - Total revenue. Intel, a computer chip...Ch. 2.4 - The U.S. Forever Stamp. The U.S. Postal Service...Ch. 2.4 - Prob. 22ECh. 2.4 - Effect of advertising. Suppose that SpryBorg Inc....Ch. 2.4 - Prob. 24ECh. 2.4 - Prob. 25ECh. 2.4 - Prob. 26ECh. 2.4 - Prob. 27ECh. 2.4 - Prob. 28ECh. 2.4 - Limited population growth: Human Population....Ch. 2.4 - Prob. 30ECh. 2.4 - 44. Limited population growth. A lake is stocked...Ch. 2.4 - Prob. 32ECh. 2.4 - Hullian learning model. The Hullian learning model...Ch. 2.4 - Spread of infection. Spread by skin-to-skin...Ch. 2.4 - Prob. 35ECh. 2.4 - Prob. 36ECh. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - We have now studied models for linear, quadratic,...Ch. 2.4 - Prob. 48ECh. 2.4 - Prob. 49ECh. 2.4 - Prob. 50ECh. 2.4 - Prob. 51ECh. 2.5 - Prob. 1ECh. 2.5 - Prob. 2ECh. 2.5 - Prob. 3ECh. 2.5 - Prob. 4ECh. 2.5 - Prob. 5ECh. 2.5 - Prob. 6ECh. 2.5 - Prob. 7ECh. 2.5 - Prob. 8ECh. 2.5 - Prob. 9ECh. 2.5 - Prob. 10ECh. 2.5 - Prob. 11ECh. 2.5 - Prob. 12ECh. 2.5 - Population decay. The population of Cortez Breaks...Ch. 2.5 - Prob. 14ECh. 2.5 - Prob. 15ECh. 2.5 - Prob. 16ECh. 2.5 - Prob. 17ECh. 2.5 - Prob. 18ECh. 2.5 - Prob. 19ECh. 2.5 - Prob. 20ECh. 2.5 - Prob. 21ECh. 2.5 - Prob. 22ECh. 2.5 - Prob. 23ECh. 2.5 - Prob. 24ECh. 2.5 - Prob. 25ECh. 2.5 - Prob. 26ECh. 2.5 - Radioactive decay. For Exercises 23-26, complete...Ch. 2.5 - Radioactive decay. For Exercises 23-26, complete...Ch. 2.5 - Carbon dating. How old is an ivory tusk that has...Ch. 2.5 - Carbon dating. How old is a piece of wood that has...Ch. 2.5 - 21. Cancer Treatment. Iodine-125 is often used to...Ch. 2.5 - Carbon dating. How old is a Chinese artifact that...Ch. 2.5 - Prob. 33ECh. 2.5 - Present value. Following the birth of a child, a...Ch. 2.5 - Present value. Following the birth of their child,...Ch. 2.5 - Present value. Desmond wants to have $15,000...Ch. 2.5 - 27. Sports salaries. An athlete signs a contract...Ch. 2.5 - 28. Actor’s salaries. An actor signs a film...Ch. 2.5 - Prob. 39ECh. 2.5 - Prob. 40ECh. 2.5 - Salvage value. Lucas Mining estimates that the...Ch. 2.5 - Prob. 42ECh. 2.5 - Prob. 43ECh. 2.5 - Prob. 47ECh. 2.5 - Prob. 48ECh. 2.5 - Prob. 49ECh. 2.5 - 37. Decline in beef consumption. Annual...Ch. 2.5 - Prob. 51ECh. 2.5 - Prob. 52ECh. 2.5 - 40. Cooling. After warming the water in a hot tub...Ch. 2.5 - 41. Cooling. The temperature in a whirlpool bath...Ch. 2.5 - Forensics. A coroner arrives at a murder scene at...Ch. 2.5 - 43. Forensics. A coroner arrives at 11 p.m. She...Ch. 2.5 - Prisoner-of-war protest. The initial weight of a...Ch. 2.5 - 45. Political Protest. A monk weighing 170 lb...Ch. 2.5 - 46. Atmospheric Pressure. Atmospheric pressure P...Ch. 2.5 - 47. Satellite power. The power supply of a...Ch. 2.5 - Prob. 61ECh. 2.5 - For each of the scatterplots in Exercise 49-58,...Ch. 2.5 - Prob. 63ECh. 2.5 - Prob. 64ECh. 2.5 - Prob. 65ECh. 2.5 - Prob. 66ECh. 2.5 - For each of the scatterplots in Exercise 49-58,...Ch. 2.5 - Prob. 68ECh. 2.5 - Prob. 69ECh. 2.5 - For each of the scatterplots in Exercise 49-58,...Ch. 2.5 - Prob. 71ECh. 2.5 - A sample of an element lost 25% of its mass in 5...Ch. 2.5 - 60. A vehicle lost 15% of its value in 2 yr....Ch. 2.5 - The Beer-Lambert Law. A beam of light enters a...Ch. 2.5 - The Beer-Lambert Law. A beam of light enters a...Ch. 2.5 - Prob. 76ECh. 2.5 - An interest rate decreases from 8% to 7.2%....Ch. 2.5 - Prob. 78ECh. 2.6 - Prob. 1ECh. 2.6 - Prob. 2ECh. 2.6 - Prob. 3ECh. 2.6 - Prob. 4ECh. 2.6 - Prob. 5ECh. 2.6 - Prob. 6ECh. 2.6 - Prob. 7ECh. 2.6 - Prob. 8ECh. 2.6 - In Exercises 1-12, find an exponential function of...Ch. 2.6 - Prob. 10ECh. 2.6 - Prob. 11ECh. 2.6 - Prob. 12ECh. 2.6 - Differentiate.
1.
Ch. 2.6 - Differentiate. y=7xCh. 2.6 - Prob. 15ECh. 2.6 - Prob. 16ECh. 2.6 - Prob. 17ECh. 2.6 - Prob. 18ECh. 2.6 - Prob. 19ECh. 2.6 - Prob. 20ECh. 2.6 - Differentiate. y=7x4+2Ch. 2.6 - Differentiate.
8.
Ch. 2.6 - Differentiate. 23. f(t)=100(0.52)tCh. 2.6 - Prob. 24ECh. 2.6 - Prob. 25ECh. 2.6 - Prob. 26ECh. 2.6 - Prob. 27ECh. 2.6 - Prob. 28ECh. 2.6 - Prob. 29ECh. 2.6 - Prob. 30ECh. 2.6 - Differentiate. 31. y=5log6x2+xCh. 2.6 - Prob. 32ECh. 2.6 - Prob. 33ECh. 2.6 - Prob. 34ECh. 2.6 - Prob. 35ECh. 2.6 - Prob. 36ECh. 2.6 - Prob. 37ECh. 2.6 - Prob. 38ECh. 2.6 - Prob. 39ECh. 2.6 - Prob. 40ECh. 2.6 - Prob. 41ECh. 2.6 - Prob. 42ECh. 2.6 - Prob. 43ECh. 2.6 - Prob. 44ECh. 2.6 - Prob. 45ECh. 2.6 - Prob. 46ECh. 2.6 - Prob. 47ECh. 2.6 - Prob. 48ECh. 2.6 - Prob. 49ECh. 2.6 - Prob. 50ECh. 2.6 - Recycling glass. In 2012,34.1 of all glass...Ch. 2.6 - Prob. 52ECh. 2.6 - Prob. 53ECh. 2.6 - Prob. 54ECh. 2.6 - Prob. 55ECh. 2.6 - Prob. 56ECh. 2.6 - Prob. 57ECh. 2.6 - Prob. 58ECh. 2.6 - Prob. 59ECh. 2.6 - Prob. 60ECh. 2.6 - A population P0 doubles every 5yr. Find the...Ch. 2.6 - Prob. 62ECh. 2.6 - Prob. 63ECh. 2.6 - Prob. 64ECh. 2.6 - Prob. 65ECh. 2.6 - Prob. 66ECh. 2.6 - Prob. 67ECh. 2.6 - Prob. 68ECh. 2 - Prob. 1RECh. 2 - In Exercises 1-6, match each equation in column A...Ch. 2 - In Exercises 1-6, match each equation in column A...Ch. 2 - Prob. 4RECh. 2 - In Exercises 1-6, match each equation in column A...Ch. 2 - Prob. 6RECh. 2 - Prob. 7RECh. 2 - Prob. 8RECh. 2 - Prob. 9RECh. 2 - Prob. 10RECh. 2 - Prob. 11RECh. 2 - Prob. 12RECh. 2 - Prob. 13RECh. 2 - Prob. 14RECh. 2 - Prob. 16RECh. 2 - Prob. 17RECh. 2 - Prob. 18RECh. 2 - Prob. 19RECh. 2 - Prob. 20RECh. 2 - Prob. 21RECh. 2 - Prob. 22RECh. 2 - Prob. 23RECh. 2 - Prob. 24RECh. 2 - Prob. 25RECh. 2 - Prob. 26RECh. 2 - Prob. 27RECh. 2 - Prob. 28RECh. 2 - Prob. 29RECh. 2 - Prob. 30RECh. 2 - Prob. 31RECh. 2 - Prob. 32RECh. 2 - Prob. 33RECh. 2 - Prob. 34RECh. 2 - Prob. 35RECh. 2 - Prob. 36RECh. 2 - Prob. 37RECh. 2 - Prob. 38RECh. 2 - Prob. 39RECh. 2 - Business: price of a prime-rib dinner. Suppose the...Ch. 2 - Prob. 42RECh. 2 - Prob. 43RECh. 2 - Prob. 44RECh. 2 - Prob. 45RECh. 2 - Prob. 46RECh. 2 - Prob. 47RECh. 2 - Prob. 48RECh. 2 - Prob. 49RECh. 2 - Prob. 50RECh. 2 - Prob. 52RECh. 2 - Prob. 53RECh. 2 - Prob. 54RECh. 2 - Prob. 55RECh. 2 - Prob. 56RECh. 2 - Prob. 57RECh. 2 - Prob. 58RECh. 2 - Differentiate. y=2e3xCh. 2 - Differentiate. y=(lnx)4Ch. 2 - Differentiate.
3.
Ch. 2 - Differentiate. f(x)=lnx7Ch. 2 - Differentiate.
5.
Ch. 2 - Differentiate. f(x)=3exlnxCh. 2 - Differentiate.
7.
Ch. 2 - Prob. 8TCh. 2 - Prob. 9TCh. 2 - Prob. 10TCh. 2 - Prob. 11TCh. 2 - Prob. 12TCh. 2 - Prob. 13TCh. 2 - Prob. 15TCh. 2 - Prob. 16TCh. 2 - Prob. 17TCh. 2 - 18. Life Science: decay rate. The decay rate of...Ch. 2 - Prob. 19TCh. 2 - Business: effect of advertising. Twin City...Ch. 2 - Prob. 21TCh. 2 - Prob. 22TCh. 2 - Differentiate: y=x(lnx)22xlnx+2x.Ch. 2 - Prob. 24TCh. 2 - Prob. 25TCh. 2 - Prob. 26TCh. 2 - Prob. 1ETECh. 2 - Use the exponential function to predict gross...Ch. 2 - Prob. 3ETECh. 2 - Prob. 4ETECh. 2 - Prob. 5ETECh. 2 - Prob. 7ETECh. 2 - Prob. 8ETECh. 2 - Prob. 9ETECh. 2 - Prob. 10ETECh. 2 - Prob. 11ETECh. 2 - Prob. 12ETE
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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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