Loose-Leaf Version for Calculus: Early Transcendentals Single Variable
4th Edition
ISBN: 9781319312886
Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert
Publisher: W. H. Freeman
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Question
Chapter 3.5, Problem 42E
To determine
The intervals on which the acceleration is positive, negative and zero from the given graph.
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Students have asked these similar questions
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
Loose-Leaf Version for Calculus: Early Transcendentals Single Variable
Ch. 3.1 - Prob. 1PQCh. 3.1 - Prob. 2PQCh. 3.1 - Prob. 3PQCh. 3.1 - Prob. 4PQCh. 3.1 - Prob. 5PQCh. 3.1 - Prob. 6PQCh. 3.1 - Prob. 7PQCh. 3.1 - Prob. 1ECh. 3.1 - Prob. 2ECh. 3.1 - Prob. 3E
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Prob. 24ECh. 3.8 - Prob. 25ECh. 3.8 - Prob. 26ECh. 3.8 - Prob. 27ECh. 3.8 - Prob. 28ECh. 3.8 - Prob. 29ECh. 3.8 - Prob. 30ECh. 3.8 - Prob. 31ECh. 3.8 - Prob. 32ECh. 3.8 - Prob. 33ECh. 3.8 - Prob. 34ECh. 3.8 - Prob. 35ECh. 3.8 - Prob. 36ECh. 3.8 - Prob. 37ECh. 3.8 - Prob. 38ECh. 3.8 - Prob. 39ECh. 3.8 - Prob. 40ECh. 3.8 - Prob. 41ECh. 3.8 - Prob. 42ECh. 3.8 - Prob. 43ECh. 3.8 - Prob. 44ECh. 3.8 - Prob. 45ECh. 3.8 - Prob. 46ECh. 3.8 - Prob. 47ECh. 3.8 - Prob. 48ECh. 3.8 - Prob. 49ECh. 3.8 - Prob. 50ECh. 3.8 - Prob. 51ECh. 3.8 - Prob. 52ECh. 3.8 - Prob. 53ECh. 3.8 - Prob. 54ECh. 3.8 - Prob. 55ECh. 3.8 - Prob. 56ECh. 3.8 - Prob. 57ECh. 3.8 - Prob. 58ECh. 3.8 - Prob. 59ECh. 3.8 - Prob. 60ECh. 3.8 - Prob. 61ECh. 3.8 - Prob. 62ECh. 3.8 - Prob. 63ECh. 3.8 - Prob. 64ECh. 3.8 - Prob. 65ECh. 3.8 - Prob. 66ECh. 3.8 - Prob. 67ECh. 3.8 - Prob. 68ECh. 3.8 - Prob. 69ECh. 3.8 - Prob. 70ECh. 3.8 - Prob. 71ECh. 3.8 - Prob. 72ECh. 3.8 - Prob. 73ECh. 3.8 - Prob. 74ECh. 3.8 - Prob. 75ECh. 3.8 - Prob. 76ECh. 3.8 - Prob. 77ECh. 3.8 - Prob. 78ECh. 3.8 - Prob. 79ECh. 3.8 - Prob. 80ECh. 3.8 - Prob. 81ECh. 3.8 - Prob. 82ECh. 3.8 - Prob. 83ECh. 3.8 - Prob. 84ECh. 3.8 - Prob. 85ECh. 3.8 - Prob. 86ECh. 3.8 - Prob. 87ECh. 3.8 - Prob. 88ECh. 3.8 - Prob. 89ECh. 3.8 - Prob. 90ECh. 3.8 - Prob. 91ECh. 3.8 - Prob. 92ECh. 3.8 - Prob. 93ECh. 3.9 - Prob. 1PQCh. 3.9 - Prob. 2PQCh. 3.9 - Prob. 3PQCh. 3.9 - Prob. 4PQCh. 3.9 - Prob. 5PQCh. 3.9 - Prob. 1ECh. 3.9 - Prob. 2ECh. 3.9 - Prob. 3ECh. 3.9 - Prob. 4ECh. 3.9 - Prob. 5ECh. 3.9 - Prob. 6ECh. 3.9 - Prob. 7ECh. 3.9 - Prob. 8ECh. 3.9 - Prob. 9ECh. 3.9 - Prob. 10ECh. 3.9 - Prob. 11ECh. 3.9 - Prob. 12ECh. 3.9 - Prob. 13ECh. 3.9 - Prob. 14ECh. 3.9 - Prob. 15ECh. 3.9 - Prob. 16ECh. 3.9 - Prob. 17ECh. 3.9 - Prob. 18ECh. 3.9 - Prob. 19ECh. 3.9 - Prob. 20ECh. 3.9 - Prob. 21ECh. 3.9 - Prob. 22ECh. 3.9 - Prob. 23ECh. 3.9 - Prob. 24ECh. 3.9 - Prob. 25ECh. 3.9 - Prob. 26ECh. 3.9 - Prob. 27ECh. 3.9 - Prob. 28ECh. 3.9 - Prob. 29ECh. 3.9 - Prob. 30ECh. 3.9 - Prob. 31ECh. 3.9 - Prob. 32ECh. 3.9 - Prob. 33ECh. 3.9 - Prob. 34ECh. 3.9 - Prob. 35ECh. 3.9 - Prob. 36ECh. 3.9 - Prob. 37ECh. 3.9 - Prob. 38ECh. 3.9 - Prob. 39ECh. 3.9 - Prob. 40ECh. 3.9 - Prob. 41ECh. 3.9 - Prob. 42ECh. 3.9 - Prob. 43ECh. 3.9 - Prob. 44ECh. 3.9 - Prob. 45ECh. 3.9 - Prob. 46ECh. 3.9 - Prob. 47ECh. 3.9 - Prob. 48ECh. 3.9 - Prob. 49ECh. 3.9 - Prob. 50ECh. 3.9 - Prob. 51ECh. 3.9 - Prob. 52ECh. 3.9 - Prob. 53ECh. 3.9 - Prob. 54ECh. 3.9 - Prob. 55ECh. 3.9 - Prob. 56ECh. 3.9 - Prob. 57ECh. 3.9 - Prob. 58ECh. 3.9 - Prob. 59ECh. 3.9 - Prob. 60ECh. 3.9 - Prob. 61ECh. 3.9 - Prob. 62ECh. 3.9 - Prob. 63ECh. 3.9 - Prob. 64ECh. 3.9 - Prob. 65ECh. 3.9 - Prob. 66ECh. 3.9 - Prob. 67ECh. 3.9 - Prob. 68ECh. 3.9 - Prob. 69ECh. 3.9 - Prob. 70ECh. 3.9 - Prob. 71ECh. 3.9 - Prob. 72ECh. 3.9 - Prob. 73ECh. 3.9 - Prob. 74ECh. 3.9 - Prob. 75ECh. 3.9 - Prob. 76ECh. 3.9 - Prob. 77ECh. 3.9 - Prob. 78ECh. 3.9 - Prob. 79ECh. 3.9 - Prob. 80ECh. 3.9 - Prob. 81ECh. 3.9 - Prob. 82ECh. 3.9 - Prob. 83ECh. 3.9 - Prob. 84ECh. 3.9 - Prob. 85ECh. 3.9 - Prob. 86ECh. 3.9 - Prob. 87ECh. 3.9 - Prob. 88ECh. 3.10 - Prob. 1PQCh. 3.10 - Prob. 2PQCh. 3.10 - Prob. 3PQCh. 3.10 - Prob. 4PQCh. 3.10 - Prob. 5PQCh. 3.10 - Prob. 6PQCh. 3.10 - Prob. 1ECh. 3.10 - Prob. 2ECh. 3.10 - Prob. 3ECh. 3.10 - Prob. 4ECh. 3.10 - Prob. 5ECh. 3.10 - Prob. 6ECh. 3.10 - Prob. 7ECh. 3.10 - Prob. 8ECh. 3.10 - Prob. 9ECh. 3.10 - Prob. 10ECh. 3.10 - Prob. 11ECh. 3.10 - Prob. 12ECh. 3.10 - Prob. 13ECh. 3.10 - Prob. 14ECh. 3.10 - Prob. 15ECh. 3.10 - Prob. 16ECh. 3.10 - Prob. 17ECh. 3.10 - Prob. 18ECh. 3.10 - Prob. 19ECh. 3.10 - Prob. 20ECh. 3.10 - Prob. 21ECh. 3.10 - Prob. 22ECh. 3.10 - Prob. 23ECh. 3.10 - Prob. 24ECh. 3.10 - Prob. 25ECh. 3.10 - Prob. 26ECh. 3.10 - Prob. 27ECh. 3.10 - Prob. 28ECh. 3.10 - Prob. 29ECh. 3.10 - Prob. 30ECh. 3.10 - Prob. 31ECh. 3.10 - Prob. 32ECh. 3.10 - Prob. 33ECh. 3.10 - Prob. 34ECh. 3.10 - Prob. 35ECh. 3.10 - Prob. 36ECh. 3.10 - Prob. 37ECh. 3.10 - Prob. 38ECh. 3.10 - Prob. 39ECh. 3.10 - Prob. 40ECh. 3.10 - Prob. 41ECh. 3.10 - Prob. 42ECh. 3.10 - Prob. 43ECh. 3.10 - Prob. 44ECh. 3.10 - Prob. 45ECh. 3 - Prob. 1CRECh. 3 - Prob. 2CRECh. 3 - Prob. 3CRECh. 3 - Prob. 4CRECh. 3 - Prob. 5CRECh. 3 - Prob. 6CRECh. 3 - Prob. 7CRECh. 3 - Prob. 8CRECh. 3 - Prob. 9CRECh. 3 - Prob. 10CRECh. 3 - Prob. 11CRECh. 3 - Prob. 12CRECh. 3 - Prob. 13CRECh. 3 - Prob. 14CRECh. 3 - Prob. 15CRECh. 3 - Prob. 16CRECh. 3 - Prob. 17CRECh. 3 - Prob. 18CRECh. 3 - Prob. 19CRECh. 3 - Prob. 20CRECh. 3 - Prob. 21CRECh. 3 - Prob. 22CRECh. 3 - Prob. 23CRECh. 3 - Prob. 24CRECh. 3 - Prob. 25CRECh. 3 - Prob. 26CRECh. 3 - Prob. 27CRECh. 3 - Prob. 28CRECh. 3 - Prob. 29CRECh. 3 - Prob. 30CRECh. 3 - Prob. 31CRECh. 3 - Prob. 32CRECh. 3 - Prob. 33CRECh. 3 - Prob. 34CRECh. 3 - Prob. 35CRECh. 3 - Prob. 36CRECh. 3 - Prob. 37CRECh. 3 - Prob. 38CRECh. 3 - Prob. 39CRECh. 3 - Prob. 40CRECh. 3 - Prob. 41CRECh. 3 - Prob. 42CRECh. 3 - Prob. 43CRECh. 3 - Prob. 44CRECh. 3 - Prob. 45CRECh. 3 - Prob. 46CRECh. 3 - Prob. 47CRECh. 3 - Prob. 48CRECh. 3 - Prob. 49CRECh. 3 - Prob. 50CRECh. 3 - Prob. 51CRECh. 3 - Prob. 52CRECh. 3 - Prob. 53CRECh. 3 - Prob. 54CRECh. 3 - Prob. 55CRECh. 3 - Prob. 56CRECh. 3 - Prob. 57CRECh. 3 - Prob. 58CRECh. 3 - Prob. 59CRECh. 3 - Prob. 60CRECh. 3 - Prob. 61CRECh. 3 - Prob. 62CRECh. 3 - Prob. 63CRECh. 3 - Prob. 64CRECh. 3 - Prob. 65CRECh. 3 - Prob. 66CRECh. 3 - Prob. 67CRECh. 3 - Prob. 68CRECh. 3 - Prob. 69CRECh. 3 - Prob. 70CRECh. 3 - Prob. 71CRECh. 3 - Prob. 72CRECh. 3 - Prob. 73CRECh. 3 - Prob. 74CRECh. 3 - Prob. 75CRECh. 3 - Prob. 76CRECh. 3 - Prob. 77CRECh. 3 - Prob. 78CRECh. 3 - Prob. 79CRECh. 3 - Prob. 80CRECh. 3 - Prob. 81CRECh. 3 - Prob. 82CRECh. 3 - Prob. 83CRECh. 3 - Prob. 84CRECh. 3 - Prob. 85CRECh. 3 - Prob. 86CRECh. 3 - Prob. 87CRECh. 3 - Prob. 88CRECh. 3 - Prob. 89CRECh. 3 - Prob. 90CRECh. 3 - Prob. 91CRECh. 3 - Prob. 92CRECh. 3 - Prob. 93CRECh. 3 - Prob. 94CRECh. 3 - Prob. 95CRECh. 3 - Prob. 96CRECh. 3 - Prob. 97CRECh. 3 - Prob. 98CRECh. 3 - Prob. 99CRECh. 3 - Prob. 100CRECh. 3 - Prob. 101CRECh. 3 - Prob. 102CRECh. 3 - Prob. 103CRECh. 3 - Prob. 104CRECh. 3 - Prob. 105CRECh. 3 - Prob. 106CRECh. 3 - Prob. 107CRECh. 3 - Prob. 108CRECh. 3 - Prob. 109CRECh. 3 - Prob. 110CRECh. 3 - Prob. 111CRECh. 3 - Prob. 112CRECh. 3 - Prob. 113CRECh. 3 - Prob. 114CRECh. 3 - Prob. 115CRECh. 3 - Prob. 116CRECh. 3 - Prob. 117CRECh. 3 - Prob. 118CRECh. 3 - Prob. 119CRECh. 3 - Prob. 120CRE
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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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