Calculus: Early Transcendental Functions (MindTap Course List)
6th Edition
ISBN: 9781285774770
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 9.2, Problem 43E
To determine
To calculate: The convergence or divergence of series
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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 9 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
Ch. 9.1 - Writing the Terms of a Sequence In exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In exercises 5-10,...Ch. 9.1 - Prob. 6ECh. 9.1 - Writing the Terms of a Sequence In Exercises 11...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 11...Ch. 9.1 - Prob. 9ECh. 9.1 - Matching In Exercises 13-16, match the sequence...
Ch. 9.1 - Matching In Exercises 13-16, match the sequence...Ch. 9.1 - Matching In Exercises 13-16, match the sequence...Ch. 9.1 - Prob. 13ECh. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.1 - Prob. 17ECh. 9.1 - Prob. 18ECh. 9.1 - Prob. 19ECh. 9.1 - Prob. 20ECh. 9.1 - Prob. 21ECh. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Prob. 24ECh. 9.1 - Prob. 25ECh. 9.1 - Prob. 26ECh. 9.1 - Prob. 27ECh. 9.1 - Prob. 28ECh. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Prob. 30ECh. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Prob. 32ECh. 9.1 - Prob. 33ECh. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Prob. 36ECh. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Prob. 44ECh. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Prob. 60ECh. 9.1 - Using a Theorem In Exercises 61-64, (a) use...Ch. 9.1 - Prob. 62ECh. 9.1 - Prob. 63ECh. 9.1 - Prob. 64ECh. 9.1 - Increasing Sequence Let {an} be an increasing...Ch. 9.1 - Prob. 67ECh. 9.1 - Compound Interest A deposit of $100 is made in an...Ch. 9.1 - Inflation When the rate of inflation is 412 per...Ch. 9.1 - Prob. 66ECh. 9.1 -
69. Sequence Is it possible for a sequence to...Ch. 9.1 - Prob. 70ECh. 9.1 - Prob. 71ECh. 9.1 - HOW DO YOU SEE IT? The graphs of two sequences are...Ch. 9.1 - Prob. 73ECh. 9.1 - Prob. 75ECh. 9.1 - Prob. 76ECh. 9.1 - Prob. 77ECh. 9.1 - Modeling DataThe amounts of the federal debt an...Ch. 9.1 - Prob. 79ECh. 9.1 - Prob. 80ECh. 9.1 - Prob. 81ECh. 9.1 - True or False? In Exercises 77-80, determine...Ch. 9.1 - Fibonacci Sequence In a study of the progeny of...Ch. 9.1 - Using a Theorem Show that the converse of Theorem...Ch. 9.1 - Prob. 85ECh. 9.1 - Prob. 86ECh. 9.1 - Prob. 87ECh. 9.1 - Prob. 88ECh. 9.1 - Prob. 89ECh. 9.1 - Using a Sequence Find a divergent sequence an such...Ch. 9.1 - Prob. 91ECh. 9.1 - Prob. 92ECh. 9.1 - T0=2,T1=3,T2=6, and for n3...Ch. 9.2 - Sequence and Series Describe the difference...Ch. 9.2 - Geometric Series Define a geometric series, state...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Prob. 4ECh. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Prob. 7ECh. 9.2 - Verifying Divergence In Exercises 11-18, verify...Ch. 9.2 - Verifying Divergence In Exercises 11-18, verify...Ch. 9.2 - Verifying Divergence In Exercises 11-18, verify...Ch. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Prob. 18ECh. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Prob. 20ECh. 9.2 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.2 - Prob. 22ECh. 9.2 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.2 - Prob. 24ECh. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Using a Geometric Series In Exercises 39-44, (a)...Ch. 9.2 - Prob. 36ECh. 9.2 - Prob. 37ECh. 9.2 - Using a Geometric Series In Exercises 39-44, (a)...Ch. 9.2 - Prob. 39ECh. 9.2 - Prob. 40ECh. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Prob. 42ECh. 9.2 - Prob. 43ECh. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Prob. 48ECh. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Prob. 53ECh. 9.2 - Prob. 54ECh. 9.2 - Using a Series (a) You delete a finite number of...Ch. 9.2 - Prob. 55ECh. 9.2 - Prob. 58ECh. 9.2 - Prob. 59ECh. 9.2 - Prob. 61ECh. 9.2 - Prob. 62ECh. 9.2 - Prob. 63ECh. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Prob. 65ECh. 9.2 - Prob. 66ECh. 9.2 - Prob. 67ECh. 9.2 - Prob. 68ECh. 9.2 - Prob. 69ECh. 9.2 - Prob. 70ECh. 9.2 - Prob. 71ECh. 9.2 - Depreciation A company buys a machine for $475,000...Ch. 9.2 - Prob. 73ECh. 9.2 - Prob. 74ECh. 9.2 - Prob. 75ECh. 9.2 - Prob. 76ECh. 9.2 - Prob. 77ECh. 9.2 - Prob. 78ECh. 9.2 - Probability A fair coin is tossed repeatedly. The...Ch. 9.2 - Prob. 80ECh. 9.2 - Area The sides of a square are 16 inches in...Ch. 9.2 - Prob. 82ECh. 9.2 - Prob. 83ECh. 9.2 - Prob. 84ECh. 9.2 - Prob. 85ECh. 9.2 - Sphereflake The sphereflakeshown below is a...Ch. 9.2 - Prob. 87ECh. 9.2 - Prob. 88ECh. 9.2 - Prob. 89ECh. 9.2 - Prob. 90ECh. 9.2 - Prob. 91ECh. 9.2 - Prob. 92ECh. 9.2 - Prob. 93ECh. 9.2 - Prob. 94ECh. 9.2 - Prob. 95ECh. 9.2 - Prob. 96ECh. 9.2 - Prob. 97ECh. 9.2 - Prob. 98ECh. 9.2 - Prob. 99ECh. 9.2 - Remainder Let an be a convergent series, and let...Ch. 9.2 - Prob. 101ECh. 9.2 - Prob. 102ECh. 9.2 - Prob. 103ECh. 9.2 - Prob. 104ECh. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 122, confirm...Ch. 9.3 - Prob. 11ECh. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Prob. 10ECh. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Prob. 16ECh. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Prob. 18ECh. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Prob. 20ECh. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Prob. 26ECh. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Prob. 29ECh. 9.3 - Prob. 30ECh. 9.3 - Prob. 31ECh. 9.3 - Prob. 32ECh. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Prob. 37ECh. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Numerical and Graphical Analysis Use a graphing...Ch. 9.3 - Prob. 40ECh. 9.3 - Prob. 41ECh. 9.3 - Prob. 42ECh. 9.3 - EXPLORING CONCEPTS Think About It Without...Ch. 9.3 - EXPLORING CONCEPTS Using a Function Let f be a...Ch. 9.3 - Prob. 45ECh. 9.3 - Prob. 46ECh. 9.3 - Prob. 47ECh. 9.3 - Prob. 48ECh. 9.3 - Prob. 49ECh. 9.3 - Prob. 50ECh. 9.3 - Finding Values In Exercises 45-50, find the...Ch. 9.3 - Prob. 52ECh. 9.3 - Proof Let f be a positive, continuous, and...Ch. 9.3 - Prob. 54ECh. 9.3 - Prob. 55ECh. 9.3 - Prob. 56ECh. 9.3 - Prob. 57ECh. 9.3 - Prob. 58ECh. 9.3 - Prob. 59ECh. 9.3 - Prob. 60ECh. 9.3 - Finding a Value In Exercises 59-62, use the result...Ch. 9.3 - Prob. 62ECh. 9.3 - Finding a Value In Exercises 59-62, use the result...Ch. 9.3 - Prob. 64ECh. 9.3 - Prob. 65ECh. 9.3 - Prob. 66ECh. 9.3 - Prob. 67ECh. 9.3 - Prob. 68ECh. 9.3 - Prob. 69ECh. 9.3 - Prob. 70ECh. 9.3 - Prob. 71ECh. 9.3 - Prob. 72ECh. 9.3 - Prob. 73ECh. 9.3 - Prob. 74ECh. 9.3 - Prob. 75ECh. 9.3 - Prob. 76ECh. 9.3 - Prob. 77ECh. 9.3 - Prob. 78ECh. 9.3 - Prob. 79ECh. 9.3 - Prob. 80ECh. 9.3 - Prob. 81ECh. 9.3 - Prob. 82ECh. 9.4 - Prob. 1ECh. 9.4 - Prob. 2ECh. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exersices5-16,...Ch. 9.4 - Using the Direct Comparison Test In Exersices5-16,...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Prob. 12ECh. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Prob. 16ECh. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Prob. 18ECh. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Prob. 31ECh. 9.4 - Prob. 32ECh. 9.4 - Prob. 33ECh. 9.4 - Prob. 34ECh. 9.4 - Prob. 35ECh. 9.4 - Prob. 36ECh. 9.4 - Prob. 37ECh. 9.4 - Prob. 38ECh. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Prob. 40ECh. 9.4 - Prob. 41ECh. 9.4 - Prob. 42ECh. 9.4 - Prob. 43ECh. 9.4 - Comparing Series It appears that the series...Ch. 9.4 - Prob. 45ECh. 9.4 - Prob. 44ECh. 9.4 - Prob. 47ECh. 9.4 - Prob. 48ECh. 9.4 - Prob. 49ECh. 9.4 - Prob. 50ECh. 9.4 - Prob. 51ECh. 9.4 - Prob. 52ECh. 9.4 - Prob. 53ECh. 9.4 - Prob. 54ECh. 9.4 - Prob. 55ECh. 9.4 - Prob. 56ECh. 9.4 - Prob. 57ECh. 9.4 - Prob. 58ECh. 9.4 - Prob. 59ECh. 9.4 - Prob. 60ECh. 9.4 - Prob. 61ECh. 9.4 - Prob. 62ECh. 9.4 - Prob. 63ECh. 9.4 - Prob. 64ECh. 9.4 - Prob. 65ECh. 9.4 - Prob. 66ECh. 9.4 - Prob. 67ECh. 9.5 - Absolute and Conditional Convergence In your own...Ch. 9.5 - Numerical and Graphical Analysis In Exercises 5-8,...Ch. 9.5 - Prob. 2ECh. 9.5 - Prob. 3ECh. 9.5 - Numerical and Graphical Analysis In Exercises 5-8,...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Prob. 12ECh. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Prob. 18ECh. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Prob. 20ECh. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Prob. 26ECh. 9.5 - Prob. 27ECh. 9.5 - Prob. 28ECh. 9.5 - Approximating the Sum of an Alternating Series In...Ch. 9.5 - Prob. 30ECh. 9.5 - Prob. 31ECh. 9.5 - Prob. 32ECh. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Prob. 34ECh. 9.5 - Prob. 35ECh. 9.5 - Prob. 36ECh. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Prob. 40ECh. 9.5 - Prob. 41ECh. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Prob. 44ECh. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Prob. 48ECh. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Prob. 52ECh. 9.5 - Prob. 53ECh. 9.5 - Prob. 54ECh. 9.5 - Prob. 55ECh. 9.5 - Prob. 56ECh. 9.5 - Prob. 57ECh. 9.5 - Prob. 59ECh. 9.5 - Prob. 60ECh. 9.5 - Prob. 61ECh. 9.5 - Prob. 62ECh. 9.5 - Finding Values In Exercises 63 and 64, find the...Ch. 9.5 - Finding Values In Exercises 63 and 64, find the...Ch. 9.5 - Prob. 65ECh. 9.5 - Prob. 66ECh. 9.5 - Prob. 67ECh. 9.5 - Prob. 68ECh. 9.5 - Prob. 69ECh. 9.5 - Prob. 70ECh. 9.5 - Prob. 71ECh. 9.5 - Prob. 72ECh. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Prob. 74ECh. 9.5 - Prob. 75ECh. 9.5 - Prob. 76ECh. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Prob. 81ECh. 9.5 - Prob. 82ECh. 9.6 - Prob. 96ECh. 9.6 - Prob. 1ECh. 9.6 - Prob. 2ECh. 9.6 - Prob. 3ECh. 9.6 - Prob. 4ECh. 9.6 - Prob. 5ECh. 9.6 - Prob. 6ECh. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Prob. 9ECh. 9.6 - Prob. 10ECh. 9.6 - Prob. 11ECh. 9.6 - Prob. 12ECh. 9.6 - Prob. 13ECh. 9.6 - Prob. 14ECh. 9.6 - Prob. 15ECh. 9.6 - Prob. 16ECh. 9.6 - Prob. 17ECh. 9.6 - Prob. 18ECh. 9.6 - Prob. 19ECh. 9.6 - Prob. 20ECh. 9.6 - Prob. 21ECh. 9.6 - Prob. 22ECh. 9.6 - Prob. 23ECh. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Prob. 25ECh. 9.6 - Prob. 26ECh. 9.6 - Prob. 27ECh. 9.6 - Prob. 28ECh. 9.6 - Prob. 29ECh. 9.6 - Prob. 30ECh. 9.6 - Prob. 31ECh. 9.6 - Prob. 32ECh. 9.6 - Prob. 33ECh. 9.6 - Prob. 34ECh. 9.6 - Prob. 35ECh. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Prob. 38ECh. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Prob. 43ECh. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Prob. 48ECh. 9.6 - Prob. 49ECh. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Prob. 52ECh. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Prob. 54ECh. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Prob. 56ECh. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Prob. 58ECh. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Prob. 60ECh. 9.6 - Prob. 61ECh. 9.6 - Prob. 62ECh. 9.6 - Prob. 63ECh. 9.6 - Prob. 64ECh. 9.6 - Prob. 65ECh. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Prob. 67ECh. 9.6 - Prob. 68ECh. 9.6 - Prob. 69ECh. 9.6 - Prob. 70ECh. 9.6 - Prob. 71ECh. 9.6 - Prob. 72ECh. 9.6 - Prob. 73ECh. 9.6 - Prob. 74ECh. 9.6 - Prob. 75ECh. 9.6 - Prob. 76ECh. 9.6 - Prob. 77ECh. 9.6 - Prob. 78ECh. 9.6 - Prob. 79ECh. 9.6 - Prob. 80ECh. 9.6 - Prob. 81ECh. 9.6 - Prob. 82ECh. 9.6 - Prob. 83ECh. 9.6 - Prob. 84ECh. 9.6 - Prob. 85ECh. 9.6 - Prob. 86ECh. 9.6 - Prob. 87ECh. 9.6 - Prob. 88ECh. 9.6 - Prob. 89ECh. 9.6 - Prob. 90ECh. 9.6 - Prob. 91ECh. 9.6 - Prob. 92ECh. 9.6 - Prob. 93ECh. 9.6 - Prob. 94ECh. 9.6 - Think About It You are told that the terms of a...Ch. 9.6 - Prob. 97ECh. 9.6 - Prob. 98ECh. 9.6 - Prob. 99ECh. 9.6 - Prob. 100ECh. 9.6 - Prob. 101ECh. 9.6 - Prob. 102ECh. 9.6 - Prob. 103ECh. 9.6 - Prob. 104ECh. 9.6 - Prob. 105ECh. 9.6 - Prob. 106ECh. 9.6 - Prob. 107ECh. 9.6 - Prob. 108ECh. 9.6 - Prob. 109ECh. 9.6 - Prob. 110ECh. 9.7 - CONCEPT CHECK Polynomial Approximation An...Ch. 9.7 - Prob. 62ECh. 9.7 - Prob. 63ECh. 9.7 - Matching In Exercises 5-8, match the Taylor...Ch. 9.7 - Prob. 2ECh. 9.7 - Prob. 3ECh. 9.7 - Prob. 4ECh. 9.7 - Finding a First-Degree Polynomial Approximation In...Ch. 9.7 - Finding a First-Degree Polynomial Approximation In...Ch. 9.7 - Prob. 7ECh. 9.7 - Finding a First-Degree Polynomial Approximation In...Ch. 9.7 - Graphical and Numerical Analysis In Exercises 13...Ch. 9.7 - Prob. 10ECh. 9.7 - Conjecture Consider the function f(x)=cosx and its...Ch. 9.7 - Prob. 12ECh. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Prob. 15ECh. 9.7 - Prob. 16ECh. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Prob. 21ECh. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Finding a Taylor Polynomial In Exercises 2530,...Ch. 9.7 - Prob. 27ECh. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Prob. 30ECh. 9.7 - Prob. 31ECh. 9.7 - Prob. 32ECh. 9.7 - Numerical and Graphical Approximations (a) Use the...Ch. 9.7 - Prob. 34ECh. 9.7 - Prob. 35ECh. 9.7 - Prob. 36ECh. 9.7 - Identifying Maclaurin Polynomials In Exercises...Ch. 9.7 - Prob. 38ECh. 9.7 - Prob. 39ECh. 9.7 - Prob. 40ECh. 9.7 - Approximating a Function Value In Exercises 39-44,...Ch. 9.7 - Prob. 42ECh. 9.7 - Prob. 43ECh. 9.7 - Prob. 44ECh. 9.7 - Using Taylors Theorem In Exercises 45-50, use...Ch. 9.7 - Prob. 46ECh. 9.7 - Prob. 47ECh. 9.7 - Prob. 48ECh. 9.7 - Finding a Degree In Exercises 51-56, determine the...Ch. 9.7 - Prob. 50ECh. 9.7 - Prob. 51ECh. 9.7 - Prob. 52ECh. 9.7 - Prob. 53ECh. 9.7 - Prob. 54ECh. 9.7 - Prob. 55ECh. 9.7 - Finding Values In Exercises 59-62, determine the...Ch. 9.7 - Prob. 57ECh. 9.7 - Prob. 58ECh. 9.7 - Prob. 60ECh. 9.7 - Prob. 61ECh. 9.7 - Prob. 64ECh. 9.7 - Prob. 65ECh. 9.7 - Prob. 66ECh. 9.7 - Prob. 67ECh. 9.7 - Prob. 68ECh. 9.7 - Prob. 69ECh. 9.7 - Prob. 70ECh. 9.7 - Prob. 71ECh. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Prob. 2ECh. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Prob. 24ECh. 9.8 - Prob. 25ECh. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Prob. 28ECh. 9.8 - Prob. 29ECh. 9.8 - Prob. 30ECh. 9.8 - Prob. 31ECh. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Prob. 34ECh. 9.8 - Prob. 35ECh. 9.8 - Prob. 36ECh. 9.8 - Prob. 37ECh. 9.8 - Prob. 38ECh. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Prob. 40ECh. 9.8 - Writing an Equivalent Series In Exercises 45-48,...Ch. 9.8 - Writing an Equivalent Series In Exercises 45-48,...Ch. 9.8 - Prob. 43ECh. 9.8 - Prob. 44ECh. 9.8 - Finding Intervals of Convergence In Exercises...Ch. 9.8 - Prob. 46ECh. 9.8 - Finding Intervals of Convergence In Exercises...Ch. 9.8 - Prob. 48ECh. 9.8 - Prob. 49ECh. 9.8 - Prob. 50ECh. 9.8 - Prob. 51ECh. 9.8 - Prob. 52ECh. 9.8 - Prob. 53ECh. 9.8 - Prob. 54ECh. 9.8 - Prob. 55ECh. 9.8 - Prob. 56ECh. 9.8 - Prob. 57ECh. 9.8 - Prob. 58ECh. 9.8 - Prob. 59ECh. 9.8 - Prob. 60ECh. 9.8 - Prob. 61ECh. 9.8 - Prob. 62ECh. 9.8 - Prob. 63ECh. 9.8 - Prob. 64ECh. 9.8 - Bessel FunctionThe Bessel function of order 0 is...Ch. 9.8 - Prob. 66ECh. 9.8 - Prob. 67ECh. 9.8 - Prob. 68ECh. 9.8 - Prob. 69ECh. 9.8 - Prob. 70ECh. 9.8 - Prob. 71ECh. 9.8 - Prob. 72ECh. 9.8 - Prob. 73ECh. 9.8 - True or False? In Exercises 73-76, determine...Ch. 9.8 - Prob. 75ECh. 9.8 - Prob. 76ECh. 9.8 - Prob. 77ECh. 9.8 - Prob. 78ECh. 9.8 - Prob. 79ECh. 9.8 - Prob. 80ECh. 9.8 - Prob. 81ECh. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Prob. 5ECh. 9.9 - Prob. 6ECh. 9.9 - Prob. 7ECh. 9.9 - Prob. 8ECh. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Prob. 11ECh. 9.9 - Prob. 12ECh. 9.9 - Prob. 13ECh. 9.9 - Prob. 14ECh. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Prob. 16ECh. 9.9 - Prob. 17ECh. 9.9 - Prob. 18ECh. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Prob. 22ECh. 9.9 - Prob. 23ECh. 9.9 - Prob. 24ECh. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Prob. 26ECh. 9.9 - Prob. 27ECh. 9.9 - Prob. 28ECh. 9.9 - Prob. 29ECh. 9.9 - Approximating a Sum In Exercises 31 and 32, (a)...Ch. 9.9 - Prob. 31ECh. 9.9 - Prob. 32ECh. 9.9 - Prob. 33ECh. 9.9 - Prob. 34ECh. 9.9 - Prob. 35ECh. 9.9 - Prob. 36ECh. 9.9 - Using a Power Series In Exercises 37-40, use the...Ch. 9.9 - Prob. 38ECh. 9.9 - Prob. 39ECh. 9.9 - Prob. 40ECh. 9.9 - Prob. 41ECh. 9.9 - Prob. 42ECh. 9.9 - Prob. 43ECh. 9.9 - Prob. 44ECh. 9.9 - Proof Prove that arctanx+arctany=arctanx+y1xy for...Ch. 9.9 - Prob. 46ECh. 9.9 - Prob. 47ECh. 9.9 - Prob. 48ECh. 9.9 - Prob. 49ECh. 9.9 - Prob. 50ECh. 9.9 - Prob. 51ECh. 9.9 - Prob. 52ECh. 9.9 - Prob. 53ECh. 9.9 - Prob. 54ECh. 9.9 - Prob. 55ECh. 9.9 - Prob. 56ECh. 9.9 - Prob. 57ECh. 9.9 - Prob. 58ECh. 9.9 - Prob. 59ECh. 9.9 - Prob. 60ECh. 9.9 - Prob. 61ECh. 9.9 - Prob. 62ECh. 9.10 - Prob. 1ECh. 9.10 - Prob. 2ECh. 9.10 - Prob. 3ECh. 9.10 - Prob. 4ECh. 9.10 - Prob. 5ECh. 9.10 - Prob. 6ECh. 9.10 - Prob. 7ECh. 9.10 - Prob. 8ECh. 9.10 - Prob. 9ECh. 9.10 - Prob. 10ECh. 9.10 - Prob. 11ECh. 9.10 - Prob. 12ECh. 9.10 - Prob. 13ECh. 9.10 - Proof In Exercises 13-16, prove that the Maclaurin...Ch. 9.10 - Prob. 15ECh. 9.10 - Prob. 16ECh. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Prob. 18ECh. 9.10 - Prob. 20ECh. 9.10 - Prob. 22ECh. 9.10 - Prob. 24ECh. 9.10 - Prob. 26ECh. 9.10 - Prob. 17ECh. 9.10 - Using a Binomial Series In Exercises 1726, use the...Ch. 9.10 - Prob. 23ECh. 9.10 - Prob. 25ECh. 9.10 - Prob. 27ECh. 9.10 - Prob. 28ECh. 9.10 - Prob. 29ECh. 9.10 - Prob. 30ECh. 9.10 - Prob. 31ECh. 9.10 - Prob. 32ECh. 9.10 - Prob. 33ECh. 9.10 - Prob. 34ECh. 9.10 - Prob. 35ECh. 9.10 - Prob. 36ECh. 9.10 - Prob. 37ECh. 9.10 - Prob. 38ECh. 9.10 - Prob. 39ECh. 9.10 - Prob. 40ECh. 9.10 - Prob. 45ECh. 9.10 - Prob. 46ECh. 9.10 - Prob. 41ECh. 9.10 - Prob. 42ECh. 9.10 - Prob. 43ECh. 9.10 - Prob. 44ECh. 9.10 - Prob. 47ECh. 9.10 - Prob. 48ECh. 9.10 - Prob. 49ECh. 9.10 - Prob. 50ECh. 9.10 - Prob. 51ECh. 9.10 - Prob. 52ECh. 9.10 - Prob. 53ECh. 9.10 - Prob. 54ECh. 9.10 - Prob. 55ECh. 9.10 - Prob. 56ECh. 9.10 - Prob. 57ECh. 9.10 - Prob. 58ECh. 9.10 - Prob. 59ECh. 9.10 - Prob. 60ECh. 9.10 - Prob. 61ECh. 9.10 - Prob. 62ECh. 9.10 - Prob. 63ECh. 9.10 - Prob. 64ECh. 9.10 - Prob. 65ECh. 9.10 - Prob. 66ECh. 9.10 - Prob. 67ECh. 9.10 - Prob. 68ECh. 9.10 - Prob. 69ECh. 9.10 - Prob. 70ECh. 9.10 - Prob. 71ECh. 9.10 - Prob. 72ECh. 9.10 - Prob. 73ECh. 9.10 - Prob. 74ECh. 9.10 - Prob. 75ECh. 9.10 - Prob. 76ECh. 9.10 - Prob. 77ECh. 9.10 - Prob. 78ECh. 9.10 - Prob. 79ECh. 9.10 - Prob. 80ECh. 9.10 - Prob. 81ECh. 9.10 - HOW DO YOU SEE IT?Match the polynomial with its...Ch. 9.10 - Prob. 83ECh. 9.10 - Prob. 84ECh. 9.10 - Prob. 85ECh. 9.10 - 86. Investigation
(a) Find the power series...Ch. 9.10 - 87. Proof Prove that for any real x.
Ch. 9.10 - Prob. 88ECh. 9.10 - Prob. 89ECh. 9.10 - Prob. 90ECh. 9.10 - Prob. 91ECh. 9.10 - Prob. 92ECh. 9.10 - Prob. 93ECh. 9.10 - Prob. 94ECh. 9.10 - Prob. 95ECh. 9.10 - Prob. 96ECh. 9 - Prob. 1RECh. 9 - Prob. 2RECh. 9 - Prob. 3RECh. 9 - Prob. 4RECh. 9 - Prob. 5RECh. 9 - Prob. 6RECh. 9 - Prob. 7RECh. 9 - Prob. 8RECh. 9 - Finding the Limit of a Sequence In Exercises s...Ch. 9 - Prob. 10RECh. 9 - Prob. 14RECh. 9 - Determining Convergence or Divergence In Exercises...Ch. 9 - Prob. 11RECh. 9 - Prob. 16RECh. 9 - Prob. 18RECh. 9 - Prob. 12RECh. 9 - Prob. 13RECh. 9 - Prob. 17RECh. 9 - Prob. 19RECh. 9 - Prob. 20RECh. 9 - Prob. 21RECh. 9 - Prob. 22RECh. 9 - Prob. 23RECh. 9 - Prob. 24RECh. 9 - Finding Partial Sums In Exercises 27 and 28, find...Ch. 9 - Prob. 26RECh. 9 - Prob. 27RECh. 9 - Prob. 28RECh. 9 - Prob. 29RECh. 9 - Prob. 30RECh. 9 - Prob. 31RECh. 9 - Prob. 32RECh. 9 - Prob. 33RECh. 9 - Prob. 34RECh. 9 - Prob. 35RECh. 9 - Prob. 36RECh. 9 - Prob. 37RECh. 9 - Prob. 38RECh. 9 - Prob. 40RECh. 9 - Prob. 39RECh. 9 - Prob. 42RECh. 9 - Prob. 41RECh. 9 - Prob. 43RECh. 9 - Prob. 44RECh. 9 - Prob. 45RECh. 9 - Prob. 46RECh. 9 - Prob. 47RECh. 9 - Prob. 48RECh. 9 - Prob. 49RECh. 9 - Prob. 50RECh. 9 - Prob. 51RECh. 9 - Prob. 52RECh. 9 - Prob. 53RECh. 9 - Prob. 54RECh. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Prob. 56RECh. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Prob. 59RECh. 9 - Prob. 58RECh. 9 - Prob. 60RECh. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Prob. 62RECh. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Numerical. Graphical, and Analytic Analysis In...Ch. 9 - Prob. 68RECh. 9 - Review In Exercises 71-76, determine the...Ch. 9 - Prob. 70RECh. 9 - Prob. 71RECh. 9 - Prob. 72RECh. 9 - Finding a Degree In Exercises 81 and 82, determine...Ch. 9 - Prob. 74RECh. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Prob. 76RECh. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Prob. 78RECh. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Prob. 80RECh. 9 - Prob. 81RECh. 9 - Prob. 82RECh. 9 - Prob. 83RECh. 9 - Prob. 84RECh. 9 - Finding a Geometric Power Series In Exercises 93...Ch. 9 - Prob. 86RECh. 9 - Finding a Power Series In Exercises 95 and 96,...Ch. 9 - Finding a Power Series In Exercises 87 and 88,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Prob. 90RECh. 9 - Prob. 91RECh. 9 - Prob. 92RECh. 9 - Prob. 93RECh. 9 - Prob. 94RECh. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Prob. 96RECh. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Prob. 100RECh. 9 - Prob. 101RECh. 9 - Prob. 102RECh. 9 - Prob. 103RECh. 9 - Forming Maclaurin Series Determine the first four...Ch. 9 - Finding a Maclaurin Series In Exercises 113-116,...Ch. 9 - Prob. 106RECh. 9 - Prob. 107RECh. 9 - Prob. 108RECh. 9 - Prob. 109RECh. 9 - Prob. 110RECh. 9 - Cantor Set The Cantor set (Georg Cantor,...Ch. 9 - Prob. 2PSCh. 9 - Prob. 3PSCh. 9 - Prob. 4PSCh. 9 - Prob. 5PSCh. 9 - Prob. 6PSCh. 9 - Prob. 7PSCh. 9 - Prob. 8PSCh. 9 - Prob. 9PSCh. 9 - Prob. 10PSCh. 9 - Prob. 11PSCh. 9 - Prob. 12PSCh. 9 - Prob. 13PSCh. 9 - Prob. 14PSCh. 9 - Prob. 15PSCh. 9 - Prob. 16PS
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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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