CALCULUS+ITS APPLICATIONS
12th Edition
ISBN: 9780135164884
Author: BITTINGER
Publisher: PEARSON
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Chapter CR, Problem 70CR
To determine
To evaluate: the integral
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The final answer is 8/π(sinx) + 8/3π(sin 3x)+ 8/5π(sin5x)....
Keity
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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 Marks
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]
Chapter CR Solutions
CALCULUS+ITS APPLICATIONS
Ch. CR - Prob. 1CRCh. CR - Prob. 2CRCh. CR - Prob. 3CRCh. CR - Prob. 4CRCh. CR - Prob. 5CRCh. CR - Prob. 6CRCh. CR - Prob. 7CRCh. CR - Prob. 8CRCh. CR - Prob. 9CRCh. CR - Prob. 10CR
Ch. CR - Prob. 11CRCh. CR - Prob. 12CRCh. CR - Prob. 13CRCh. CR - Prob. 14CRCh. CR - Prob. 15CRCh. CR - Prob. 16CRCh. CR - Prob. 17CRCh. CR - Prob. 18CRCh. CR - Prob. 19CRCh. CR - Prob. 20CRCh. CR - Prob. 21CRCh. CR - Prob. 22CRCh. CR - Prob. 23CRCh. CR - Prob. 24CRCh. CR - Prob. 25CRCh. CR - Prob. 26CRCh. CR - Prob. 27CRCh. CR - Prob. 28CRCh. CR - Prob. 29CRCh. CR - Prob. 30CRCh. CR - Prob. 31CRCh. CR - Prob. 32CRCh. CR - Prob. 33CRCh. CR - Prob. 34CRCh. CR - Prob. 35CRCh. CR - Prob. 36CRCh. CR - Prob. 37CRCh. CR - Prob. 40CRCh. CR - Prob. 41CRCh. CR - Prob. 42CRCh. CR - Prob. 43CRCh. CR - Prob. 44CRCh. CR - Prob. 45CRCh. CR - Prob. 46CRCh. CR - Prob. 47CRCh. CR - Prob. 48CRCh. CR - Prob. 49CRCh. CR - Prob. 50CRCh. CR - Prob. 52CRCh. CR - Prob. 54CRCh. CR - Prob. 55CRCh. CR - Prob. 56CRCh. CR - Prob. 57CRCh. CR - Prob. 58CRCh. CR - Prob. 59CRCh. CR - Prob. 61CRCh. CR - Prob. 62CRCh. CR - Prob. 63CRCh. CR - Suppose the rate of change of y with respect to x...Ch. CR - Prob. 65CRCh. CR - Prob. 66CRCh. CR - Prob. 67CRCh. CR - Prob. 68CRCh. CR - Prob. 69CRCh. CR - Prob. 70CRCh. CR - Prob. 71CRCh. CR - Prob. 72CR
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- 2. (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_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_forward
- 3. (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_forwardA company specializing in lubrication products for vintage motors produce two blended oils, Smaza and Nefkov. They make a profit of K5,000.00 per litre of Smaza and K4,000.00 per litre of Nefkov. A litre of Smaza requires 0.4 litres of heavy oil and 0.6 litres of light oil. A litre of Nefkov requires 0.8 litres of heavy oil and 0.2 litres of light oil. The company has 100 litres of heavy oil and 80 litres of light oil. How many litres of each product should they make to maximize profits and what level of profit will they obtain? Show all your workings.arrow_forward
- 1. Show that the vector field F(x, y, z) = (2x sin ye³)ix² cos yj + (3xe³ +5)k satisfies the necessary conditions for a conservative vector field, and find a potential function for F.arrow_forward1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude of the gravitational force between two objects with masses m and M is |F| mMG |r|2 where r is the distance between the objects, and G is the gravitational constant. Assume that the object with mass M is located at the origin in R³. Then, the gravitational force field acting on the object at the point r = (x, y, z) is given by F(x, y, z) = mMG r3 r. mMG mMG Show that the scalar vector field f(x, y, z) = = is a potential function for r √√x² + y² . Fi.e. show that F = Vf. Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).arrow_forward2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.arrow_forward
- write it down for better understanding pleasearrow_forward1. Suppose F(t) gives the temperature in degrees Fahrenheit t minutes after 1pm. With a complete sentence, interpret the equation F(10) 68. (Remember this means explaining the meaning of the equation without using any mathy vocabulary!) Include units. (3 points) =arrow_forward2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below. a. Evaluate f(-3). If you have multiple steps, be sure to connect your expressions with EQUALS SIGNS. (3 points)arrow_forward
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