EBK MATHEMATICS ALL AROUND
6th Edition
ISBN: 8220103632027
Author: Pirnot
Publisher: Pearson Education (US)
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Chapter 3.3, Problem 12E
To determine
To find:
The truth table for
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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 3 Solutions
EBK MATHEMATICS ALL AROUND
Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...Ch. 3.1 - Sharpening Your Skills In Exercise 110, determine...
Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - In Exercise 1120, identify each statement as...Ch. 3.1 - Consider the following statements: g: Global...Ch. 3.1 - Consider the following statements: g: Global...Ch. 3.1 - Consider the following statements: g: Global...Ch. 3.1 - Consider the following statements: g: Global...Ch. 3.1 - Consider the following statements: t: The radial...Ch. 3.1 - Prob. 26ECh. 3.1 - Consider the following statements: t: The radial...Ch. 3.1 - Consider the following statements: t: The radial...Ch. 3.1 - In Exercises 2934, negate each quantified...Ch. 3.1 - In Exercises 2934, negate each quantified...Ch. 3.1 - In Exercises 2934, negate each quantified...Ch. 3.1 - In Exercises 2934, negate each quantified...Ch. 3.1 - In Exercises 2934, negate each quantified...Ch. 3.1 - In Exercises 2934, negate each quantified...Ch. 3.1 - Applying What Youve Learned Use the following...Ch. 3.1 - Applying What Youve Learned Use the following...Ch. 3.1 - Applying What Youve Learned Use the following...Ch. 3.1 - Applying What Youve Learned Use the following...Ch. 3.1 - Applying What Youve Learned Use the following...Ch. 3.1 - Applying What Youve Learned Use the following...Ch. 3.1 - Consider the happy and sad faces below. Determine...Ch. 3.1 - Consider the happy and sad faces below. Determine...Ch. 3.1 - Consider the happy and sad faces below. Determine...Ch. 3.1 - Consider the happy and sad faces below. Determine...Ch. 3.1 - In Exercises 4548, examine each statement to...Ch. 3.1 - In Exercises 4548, examine each statement to...Ch. 3.1 - In Exercises 4548, examine each statement to...Ch. 3.1 - In Exercises 4548, examine each statement to...Ch. 3.1 - Because the English language is so complex, it is...Ch. 3.1 - Because the English language is so complex, it is...Ch. 3.1 - Because the English language is so complex, it is...Ch. 3.1 - Prob. 52ECh. 3.1 - Prob. 53ECh. 3.1 - Prob. 54ECh. 3.1 - Prob. 55ECh. 3.1 - In 1937, Claude Shannon showed that computer...Ch. 3.1 - Prob. 57ECh. 3.1 - In 1937, Claude Shannon showed that computer...Ch. 3.1 - In Exercises 5962, determine if the following...Ch. 3.1 - In Exercises 5962, determine if the following...Ch. 3.1 - Prob. 61ECh. 3.1 - Prob. 62ECh. 3.1 - Prob. 63ECh. 3.1 - Prob. 64ECh. 3.1 - Prob. 65ECh. 3.1 - Prob. 66ECh. 3.1 - 6772. In symbolic logic, the form of statements is...Ch. 3.1 - Prob. 68ECh. 3.1 - Prob. 69ECh. 3.1 - Prob. 70ECh. 3.1 - Prob. 71ECh. 3.1 - 6772. In symbolic logic, the form of statements is...Ch. 3.1 - Think of real-life situation that you might want...Ch. 3.1 - Provide arguments for or against the view that...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - Prob. 8ECh. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - In Exercise 1-10, assume that p is true, q is...Ch. 3.2 - State whether the numbers given in Exercise 11-14...Ch. 3.2 - State whether the numbers given in Exercise 11-14...Ch. 3.2 - State whether the numbers given in Exercise 11-14...Ch. 3.2 - State whether the numbers given in Exercise 11-14...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 15-24, construct a truth table for...Ch. 3.2 - In Exercise 25-28, determine whether we are using...Ch. 3.2 - In Exercise 25-28, determine whether we are using...Ch. 3.2 - In Exercise 25-28, determine whether we are using...Ch. 3.2 - In Exercise 25-28, determine whether we are using...Ch. 3.2 - In Exercise 29-34, use DeMorgans laws to rewrite...Ch. 3.2 - In Exercise 29-34, use DeMorgans laws to rewrite...Ch. 3.2 - In Exercise 29-34, use DeMorgans laws to rewrite...Ch. 3.2 - In Exercise 29-34, use DeMorgans laws to rewrite...Ch. 3.2 - In Exercise 29-34, use DeMorgans laws to rewrite...Ch. 3.2 - In Exercise 29-34, use DeMorgans laws to rewrite...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - In Exercise 35-42, determine whether the pair of...Ch. 3.2 - Exercise 43-48, deal with three-valued logic....Ch. 3.2 - Exercise 43-48, deal with three-valued logic....Ch. 3.2 - Exercise 43-48, deal with three-valued logic....Ch. 3.2 - Exercise 43-48, deal with three-valued logic....Ch. 3.2 - Exercise 43-48, deal with three-valued logic....Ch. 3.2 - Exercise 43-48, deal with three-valued logic....Ch. 3.2 - Applying What Youve Learned Use the following...Ch. 3.2 - Applying What Youve Learned Use the following...Ch. 3.2 - Prob. 51ECh. 3.2 - Prob. 52ECh. 3.2 - Prob. 53ECh. 3.2 - Prob. 54ECh. 3.2 - Prob. 55ECh. 3.2 - Prob. 56ECh. 3.2 - Prob. 57ECh. 3.2 - Prob. 58ECh. 3.2 - Prob. 59ECh. 3.2 - Prob. 60ECh. 3.2 - Prob. 61ECh. 3.2 - Prob. 62ECh. 3.2 - Use this graph based on data from the National Pet...Ch. 3.2 - Prob. 64ECh. 3.2 - In Section 3.1 page 94, we showed how to represent...Ch. 3.2 - Prob. 66ECh. 3.2 - Prob. 67ECh. 3.2 - Prob. 68ECh. 3.2 - Prob. 69ECh. 3.2 - What advantage do you see in using truth tables to...Ch. 3.2 - Prob. 71ECh. 3.2 - The and connective is necessary in the sense that...Ch. 3.2 - The stroke connective has the following truth...Ch. 3.2 - The stroke connective has the following truth...Ch. 3.2 - The stroke connective has the following truth...Ch. 3.2 - The stroke connective has the following truth...Ch. 3.3 - Prob. 1ECh. 3.3 - Prob. 2ECh. 3.3 - Prob. 3ECh. 3.3 - Prob. 4ECh. 3.3 - Prob. 5ECh. 3.3 - Prob. 6ECh. 3.3 - Prob. 7ECh. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Prob. 10ECh. 3.3 - Prob. 11ECh. 3.3 - Prob. 12ECh. 3.3 - Prob. 13ECh. 3.3 - Prob. 14ECh. 3.3 - Prob. 15ECh. 3.3 - Prob. 16ECh. 3.3 - Prob. 17ECh. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - Prob. 20ECh. 3.3 - Prob. 21ECh. 3.3 - Prob. 22ECh. 3.3 - Prob. 23ECh. 3.3 - Prob. 24ECh. 3.3 - Prob. 25ECh. 3.3 - Prob. 26ECh. 3.3 - Prob. 27ECh. 3.3 - Prob. 28ECh. 3.3 - Assume that you begin with a statement of the form...Ch. 3.3 - Prob. 30ECh. 3.3 - Assume that you begin with a statement of the form...Ch. 3.3 - Assume that you begin with a statement of the form...Ch. 3.3 - In Exercises 3336, write the indicated statement...Ch. 3.3 - In Exercises 3336, write the indicated statement...Ch. 3.3 - Prob. 35ECh. 3.3 - Prob. 36ECh. 3.3 - In Exercises 3740, determine which pairs of...Ch. 3.3 - In Exercises 3740, determine which pairs of...Ch. 3.3 - Prob. 39ECh. 3.3 - Prob. 40ECh. 3.3 - In Exercises 4148, rewrite each statement using...Ch. 3.3 - Prob. 42ECh. 3.3 - In Exercises 4148, rewrite each statement using...Ch. 3.3 - Prob. 44ECh. 3.3 - Prob. 45ECh. 3.3 - Prob. 46ECh. 3.3 - In Exercises 4148, rewrite each statement using...Ch. 3.3 - Prob. 48ECh. 3.3 - Find the truth value for each statement in...Ch. 3.3 - Find the truth value for each statement in...Ch. 3.3 - Find the truth value for each statement in...Ch. 3.3 - Find the truth value for each statement in...Ch. 3.3 - Prob. 53ECh. 3.3 - Prob. 54ECh. 3.3 - Prob. 55ECh. 3.3 - Prob. 56ECh. 3.3 - According to an Accountemps survey appearing in...Ch. 3.3 - According to an Accountemps survey appearing in...Ch. 3.3 - According to an Accountemps survey appearing in...Ch. 3.3 - According to an Accountemps survey appearing in...Ch. 3.3 - Perhaps you have heard the term helicopter...Ch. 3.3 - Perhaps you have heard the term helicopter...Ch. 3.3 - Perhaps you have heard the term helicopter...Ch. 3.3 - Perhaps you have heard the term helicopter...Ch. 3.3 - In Exercises 6568, write the converse, inverse, or...Ch. 3.3 - In Exercises 6568, write the converse, inverse, or...Ch. 3.3 - In Exercises 6568, write the converse, inverse, or...Ch. 3.3 - In Exercises 6568, write the converse, inverse, or...Ch. 3.3 - Prob. 69ECh. 3.3 - Prob. 70ECh. 3.3 - Communicating Mathematics Give an example of a...Ch. 3.3 - Communicating Mathematics Is it possible to have a...Ch. 3.3 - Communicating Mathematics Explain why it is...Ch. 3.3 - Communicating Mathematics Why is it reasonable to...Ch. 3.3 - In Exercises 75 and 76, assume that a credit card...Ch. 3.3 - In Exercises 75 and 76, assume that a credit card...Ch. 3.3 - Challenge Yourself In Exercises 79 and 80, use...Ch. 3.3 - Challenge Yourself In Exercises 79 and 80, use...Ch. 3.3 - Prob. 81ECh. 3.3 - Prob. 82ECh. 3.3 - Prob. 83ECh. 3.3 - Prob. 84ECh. 3.3 - Exercises 85 and 86 are based on the exercise sets...Ch. 3.3 - Exercises 85 and 86 are based on the exercise sets...Ch. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - Prob. 3ECh. 3.4 - Prob. 4ECh. 3.4 - Prob. 5ECh. 3.4 - Prob. 6ECh. 3.4 - Prob. 7ECh. 3.4 - Prob. 8ECh. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.4 - Prob. 13ECh. 3.4 - Prob. 14ECh. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.4 - Prob. 17ECh. 3.4 - Prob. 18ECh. 3.4 - Prob. 19ECh. 3.4 - Prob. 20ECh. 3.4 - Prob. 21ECh. 3.4 - Prob. 22ECh. 3.4 - Prob. 23ECh. 3.4 - Prob. 24ECh. 3.4 - Prob. 25ECh. 3.4 - Prob. 26ECh. 3.4 - Prob. 27ECh. 3.4 - Prob. 28ECh. 3.4 - Prob. 29ECh. 3.4 - Prob. 30ECh. 3.4 - Prob. 31ECh. 3.4 - Prob. 32ECh. 3.4 - Prob. 33ECh. 3.4 - Prob. 34ECh. 3.4 - Prob. 35ECh. 3.4 - Prob. 36ECh. 3.4 - Prob. 37ECh. 3.4 - Prob. 38ECh. 3.4 - Prob. 39ECh. 3.4 - Prob. 40ECh. 3.4 - Prob. 41ECh. 3.4 - Prob. 42ECh. 3.4 - We have emphasized that the form of a logical...Ch. 3.4 - We have emphasized that the form of a logical...Ch. 3.4 - We have emphasized that the form of a logical...Ch. 3.4 - We have emphasized that the form of a logical...Ch. 3.4 - Challenge Yourself Exercises 49-52 are puzzles...Ch. 3.4 - Challenge Yourself Exercises 49-52 are puzzles...Ch. 3.4 - Challenge Yourself Exercises 49-52 are puzzles...Ch. 3.4 - In a complicated argument with many variables, it...Ch. 3.4 - In a complicated argument with many variables, it...Ch. 3.4 - In addition to the argument forms that you studies...Ch. 3.4 - In addition to the argument forms that you studies...Ch. 3.4 - In addition to the argument forms that you studies...Ch. 3.4 - In addition to the argument forms that you studies...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 116, determine whether each syllogism...Ch. 3.5 - In Exercise 1724, complete each syllogism so that...Ch. 3.5 - Prob. 18ECh. 3.5 - In Exercise 1724, complete each syllogism so that...Ch. 3.5 - Prob. 20ECh. 3.5 - In Exercise 1724, complete each syllogism so that...Ch. 3.5 - Prob. 22ECh. 3.5 - In Exercise 1724, complete each syllogism so that...Ch. 3.5 - Prob. 24ECh. 3.5 - In Exercises 25 28, write two syllogisms that can...Ch. 3.5 - In Exercises 25 28, write two syllogisms that can...Ch. 3.5 - In Exercises 25 28, write two syllogisms that can...Ch. 3.5 - In Exercises 25 28, write two syllogisms that can...Ch. 3.5 - Give an example of a valid syllogism that has a...Ch. 3.5 - Give an example of a invalid syllogism that has a...Ch. 3.5 - Draw an Euler diagram for the statements All As...Ch. 3.5 - Draw an Euler diagram for the statements Some As...Ch. 3.5 - Draw an Euler diagram for the statements No As are...Ch. 3.5 - In each of your drawings for Exercises 31 33,...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In Exercises 1-8, assign a truth value between 0...Ch. 3.6 - In a Exercises 9-12, calculate the truth value of...Ch. 3.6 - Prob. 10ECh. 3.6 - Prob. 11ECh. 3.6 - In a Exercises 9-12, calculate the truth value of...Ch. 3.6 - In Exercises 13-16, consider the following fuzzy...Ch. 3.6 - In Exercises 13-16, consider the following fuzzy...Ch. 3.6 - In Exercise 13-16, consider the following fuzzy...Ch. 3.6 - In Exercise 13-16, consider the following fuzzy...Ch. 3.6 - In Exercises 17-24, assume that p has a truth...Ch. 3.6 - In Exercises 17-24, assume that p has a truth...Ch. 3.6 - Prob. 19ECh. 3.6 - In Exercises 17-24, assume that p has a truth...Ch. 3.6 - Prob. 21ECh. 3.6 - Prob. 22ECh. 3.6 - In Exercises 17-24, assume that p has a truth...Ch. 3.6 - Prob. 24ECh. 3.6 - In exercises 25-28, use the method described in...Ch. 3.6 - Prob. 26ECh. 3.6 - In exercises 25-28, use the method described in...Ch. 3.6 - In exercises 25-28, use the method described in...Ch. 3.6 - How are the rules for computing the truth tables...Ch. 3.6 - Discuss some situations in which using fuzzy logic...Ch. 3.6 - Choose a situation you will face in which you must...Ch. 3.6 - Do you have any criticisms of the decision-making...Ch. 3.CR - Prob. 1CRCh. 3.CR - Let v represent the statement I will buy a new...Ch. 3.CR - Let f represent Antonio is fluent in Spanish and...Ch. 3.CR - Negate each quantified statement and then rewrite...Ch. 3.CR - Let p represent some true statement, q represent...Ch. 3.CR - How many rows will be in the table for each...Ch. 3.CR - Construct a truth table for each statement. a....Ch. 3.CR - Negate each statement and then rewrite the...Ch. 3.CR - Which pairs of statements are logically...Ch. 3.CR - Assume we are dealing with three- valued logic and...Ch. 3.CR - Assume that p represent a true statement, q a...Ch. 3.CR - Construct a truth table for each statement. a. pq...Ch. 3.CR - Prob. 13CRCh. 3.CR - Rewrite each statement using the words if then. a....Ch. 3.CR - Section 3.4 15. Identify the form of each...Ch. 3.CR - Determine whether the form represents a valid...Ch. 3.CR - Use a truth table to determine whether the...Ch. 3.CR - In Exercises 18 and 19, use Euler diagrams to...Ch. 3.CR - In Exercises 18 and 19, use Euler diagrams to...Ch. 3.CR - Assume that p and q are fuzzy statements having...Ch. 3.CT - Which of the following are statements? a. New York...Ch. 3.CT - Negate each quantified statement and then rewrite...Ch. 3.CT - Let p represent the statement I will pass my...Ch. 3.CT - Let t represent The Tigers will win the series and...Ch. 3.CT - Prob. 5CTCh. 3.CT - If p is false and q is true and r is false, what...Ch. 3.CT - Prob. 7CTCh. 3.CT - Construct a truth table for each statement. a....Ch. 3.CT - Prob. 9CTCh. 3.CT - Negate each statement and then rewrite the...Ch. 3.CT - Determine whether the following pairs of...Ch. 3.CT - Write in words the converse, inverse, and...Ch. 3.CT - If p is true, q is false, and r is true, what is...Ch. 3.CT - Assume we are dealing with three-valued logic and...Ch. 3.CT - Prob. 15CTCh. 3.CT - Determine whether the form represents a valid...Ch. 3.CT - Identify the form of each argument. If it aint...Ch. 3.CT - In fuzzy logic, we replaced the conditional pq by...Ch. 3.CT - Use a truth table to determine if the argument is...Ch. 3.CT - Use an Euler diagram to determine whether the...
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Similar questions
- 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
- 13) Consider the checkerboard arrangement shown below. Assume that the red checker can move diagonally upward, one square at a time, on the white squares. It may not enter a square if occupied by another checker, but may jump over it. How many routes are there for the red checker to the top of the board?arrow_forwardFill in the blanks to describe squares. The square of a number is that number Question Blank 1 of 4 . The square of negative 12 is written as Question Blank 2 of 4 , but the opposite of the square of 12 is written as Question Blank 3 of 4 . 2 • 2 = 4. Another number that can be multiplied by itself to equal 4 is Question Blank 4 of 4 .arrow_forward12) The prime factors of 1365 are 3, 5, 7 and 13. Determine the total number of divisors of 1365.arrow_forward
- 11) What is the sum of numbers in row #8 of Pascal's Triangle?arrow_forward14) Seven students and three teachers wish to join a committee. Four of them will be selected by the school administration. What is the probability that three students and one teacher will be selected?arrow_forward(1) Write the following quadratic equation in terms of the vertex coordinates.arrow_forward
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