EBK CALCULUS & ITS APPLICATIONS
14th Edition
ISBN: 8220103679527
Author: Asmar
Publisher: YUZU
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Chapter 6.5, Problem 37E
To determine
The value of
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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 6 Solutions
EBK CALCULUS & ITS APPLICATIONS
Ch. 6.1 - Determine the following: a. t7/2dt b....Ch. 6.1 - Find a function f(t) that satisfies f(t)=3t+5 and...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Determine the following: 4x3dxCh. 6.1 - Determine the following: 13xdx
Ch. 6.1 - Determine the following: 7dxCh. 6.1 - Determine the following: k2dx ((kisaconstant).Ch. 6.1 - Determine the following: xcdx(cisaconstant0)...Ch. 6.1 - Determine the following: xx2dx.Ch. 6.1 - Determine the following: (2x+x2)dx.Ch. 6.1 - Determine the following: 17xdx.Ch. 6.1 - Determine the following: xxdx.Ch. 6.1 - Determine the following: (2x+2x)dx.Ch. 6.1 - Determine the following: (x2x2+13x)dx.Ch. 6.1 - Determine the following: (72x3x3)dx.Ch. 6.1 - Determine the following: 3e2xdx.Ch. 6.1 - Determine the following: exdx.Ch. 6.1 - Determine the following: edx.Ch. 6.1 - Determine the following: 72e2xdx.Ch. 6.1 - Determine the following: 2(e2x+1)dx.Ch. 6.1 - Determine the following: (3ex+2xe0.5x2)dx.Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Figure 4 shows the graphs of several functions...Ch. 6.1 - Figure 5 shows the graphs of several functions...Ch. 6.1 - Which of the following is lnxdx ? a.1x+C b.xlnxx+C...Ch. 6.1 - Which of the following is xx+1dx?...Ch. 6.1 - Figure 6 contains the graph of a function F(x). On...Ch. 6.1 - Figure 7 contains an antiderivative of the...Ch. 6.1 - The function g(x) in Fig. 8, resulted from...Ch. 6.1 - The function g(x) in Fig.9 resulted from shifting...Ch. 6.1 - Height of a Ball A ball is thrown upward from a...Ch. 6.1 - Free Fall A rock is dropped from the top of a...Ch. 6.1 - Rate of Production Let P(t) be the total output of...Ch. 6.1 - Rate of Production After t hours of operation, a...Ch. 6.1 - Heat DiffusionA package of frozen strawberries is...Ch. 6.1 - Epidemic A flu epidemic hits a town. Let P(t) be...Ch. 6.1 - Profit A small tie shop finds that at a sales...Ch. 6.1 - Prob. 62ECh. 6.1 - Prob. 63ECh. 6.1 - U.S. Natural Gas Production Since 1987, the rate...Ch. 6.1 - Prob. 65ECh. 6.1 - Prob. 66ECh. 6.1 - Prob. 67ECh. 6.1 - Prob. 68ECh. 6.2 - Evaluate 01e2x1exdx.Ch. 6.2 - If f(t)=1t, find f(2)f(0).Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - Given 01f(x)dx=3.5 and 14f(x)dx=5, find 04f(x)dx.Ch. 6.2 - Given 11f(x)dx=0 and 110f(x)dx=4, find 110f(x)dx.Ch. 6.2 - Given 13f(x)dx=3 and 13g(x)dx=1, find...Ch. 6.2 - Given 0.53f(x)dx=0 and 0.53(2g(x)+f(x))dx=4, find...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - Refer to Fig. 4 and evaluate 02f(x)dx. Figure 4Ch. 6.2 - Refer to Fig. 5 and evaluate 03f(x)dx. Figure 5Ch. 6.2 - Refer to Fig. 6 and evaluate 11f(t)dt. Figure 6Ch. 6.2 - Refer to Fig. 7 and evaluate 12f(t)dt. Figure 7Ch. 6.2 - Net Change in Position A rock is dropped from the...Ch. 6.2 - Net change in Position The velocity at time t...Ch. 6.2 - Net Change in Position The velocity at time t...Ch. 6.2 - Velocity of a Skydiver The velocity of a skydiver...Ch. 6.2 - Net Change in Cost A companys marginal cost...Ch. 6.2 - Prob. 36ECh. 6.2 - Net Increase of an Investment An investment grew...Ch. 6.2 - Depreciation of Real Estate A property with an...Ch. 6.2 - Population Model with Emigration The rate of...Ch. 6.2 - Paying Down a Mortgage You took a 200,000 home...Ch. 6.2 - Mortgage Using the data from the previous...Ch. 6.2 - Radioactive Decay A sample of radioactive material...Ch. 6.2 - Prob. 43ECh. 6.2 - Level of Water in a Tank A conical-shaped tank is...Ch. 6.3 - Repeat Example 6 using midpoints of the...Ch. 6.3 - Repeat Example 6 using left endpoints of the...Ch. 6.3 - In exercises 16, compute the area of the shaded...Ch. 6.3 - In exercises 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - Prob. 12ECh. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - In Exercises 1618, draw the region whose area is...Ch. 6.3 - In Exercises 1618, draw the region whose area is...Ch. 6.3 - In Exercises 1618, draw the region whose area is...Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Prob. 25ECh. 6.3 - Find the real number b0 so that the area under the...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3740, use a Riemann sum to...Ch. 6.3 - In Exercises 3740, use a Riemann sum to...Ch. 6.3 - In Exercises 3740, use a Riemann sum to...Ch. 6.3 - Prob. 40ECh. 6.3 - Use a Riemann sum with n=4 and left endpoints to...Ch. 6.3 - Prob. 42ECh. 6.3 - The graph of the function f(x)=1x2 on the interval...Ch. 6.3 - Use a Riemann sum with n=5 and midpoints to...Ch. 6.3 - Estimate the area (in square feet) of the...Ch. 6.3 - Prob. 46ECh. 6.3 - Prob. 47ECh. 6.3 - Prob. 48ECh. 6.3 - Technology Exercises. The area under the graph of...Ch. 6.3 - Prob. 50ECh. 6.3 - Prob. 51ECh. 6.3 - Prob. 52ECh. 6.4 - Find the area between the curves y=x+3 and...Ch. 6.4 - A company plans to increase its production from 10...Ch. 6.4 - Write a definite integral or sum of definite...Ch. 6.4 - Write a definite integral or sum of definite...Ch. 6.4 - Shade the portion of Fig. 23 whose area is given...Ch. 6.4 - Shade the portion ofFig. 24 whose area is given by...Ch. 6.4 - Let f(x) be the function pictured in Fig. 25....Ch. 6.4 - Let g(x) be the function pictured in Fig. 26....Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region between y=x23x and the...Ch. 6.4 - Find the area of the region between y=x2 and...Ch. 6.4 - Find the area in Fig. 27 of the region bounded by...Ch. 6.4 - Find the area of the region bounded by y=1/x,y=4x...Ch. 6.4 - Height of a Helicopter A helicopter is rising...Ch. 6.4 - Assembly line productionAfter t hour of operation,...Ch. 6.4 - Cost Suppose that the marginal cost function for a...Ch. 6.4 - ProfitSuppose that the marginal profit function...Ch. 6.4 - Marginal Profit Let M(x) be a companys marginal...Ch. 6.4 - Marginal Profit Let M(x) be a companys marginal...Ch. 6.4 - Prob. 37ECh. 6.4 - VelocitySuppose that the velocity of a car at time...Ch. 6.4 - Deforestation and Fuel wood Deforestation is one...Ch. 6.4 - Refer to Exercise 39. The rate of new tree growth...Ch. 6.4 - After an advertising campaign, a companys marginal...Ch. 6.4 - Profit and Area The marginal profit for a certain...Ch. 6.4 - Velocity and Distance Two rockets are fired...Ch. 6.4 - Distance TraveledCars A and B start at the same...Ch. 6.4 - Displacement versus Distance Traveled The velocity...Ch. 6.4 - Displacement versus Distance Traveled The velocity...Ch. 6.4 - Prob. 47ECh. 6.4 - Prob. 48ECh. 6.4 - Prob. 49ECh. 6.4 - Prob. 50ECh. 6.5 - A rock dropped from a bridge has a velocity of 32t...Ch. 6.5 - An Investment yields a continuous income stream of...Ch. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Prob. 4ECh. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Prob. 6ECh. 6.5 - Average Temperature During a certain 12-hour...Ch. 6.5 - Average PopulationAssuming that a countrys...Ch. 6.5 - Average Amount of Radium. One hundred grams of...Ch. 6.5 - Average Amount of Money. One hundred dollars is...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Prob. 15ECh. 6.5 - Prob. 16ECh. 6.5 - Prob. 17ECh. 6.5 - Prob. 18ECh. 6.5 - Prob. 19ECh. 6.5 - Prob. 20ECh. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.5 - Prob. 23ECh. 6.5 - Future Value Suppose that money is deposited...Ch. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - Volume of Solids of Revolution Find the volume of...Ch. 6.5 - Volume of Solids of Revolution Find the volume of...Ch. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - Prob. 32ECh. 6.5 - Prob. 33ECh. 6.5 - Prob. 34ECh. 6.5 - Prob. 35ECh. 6.5 - Prob. 36ECh. 6.5 - Prob. 37ECh. 6.5 - For the Riemann sum...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - Prob. 42ECh. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - Prob. 45ECh. 6.5 - Prob. 46ECh. 6 - What does it mean to antidifferentiate a function?Ch. 6 - Prob. 2CCECh. 6 - Prob. 3CCECh. 6 - Prob. 4CCECh. 6 - Prob. 5CCECh. 6 - Prob. 6CCECh. 6 - Prob. 7CCECh. 6 - Prob. 8CCECh. 6 - Prob. 9CCECh. 6 - Prob. 10CCECh. 6 - Prob. 11CCECh. 6 - Calculate the following integrals. 32dxCh. 6 - Prob. 2RECh. 6 - Calculate the following integrals. x+1dxCh. 6 - Calculate the following integrals. 2x+4dxCh. 6 - Calculate the following integrals. 2(x3+3x21)dxCh. 6 - Calculate the following integrals. x+35dxCh. 6 - Calculate the following integrals. ex/2dxCh. 6 - Calculate the following integrals. 5x7dxCh. 6 - Calculate the following integrals. (3x44x3)dxCh. 6 - Calculate the following integrals. (2x+3)7dxCh. 6 - Calculate the following integrals. 4xdxCh. 6 - Calculate the following integrals. (5xx5)dxCh. 6 - Calculate the following integrals. 11(x+1)2dxCh. 6 - Calculate the following integrals. 01/8x3dxCh. 6 - Calculate the following integrals. 122x+4dxCh. 6 - Calculate the following integrals. 201(2x+11x+4)dxCh. 6 - Calculate the following integrals. 124x5dxCh. 6 - Calculate the following integrals. 2308x+1dxCh. 6 - Calculate the following integrals. 141x2dxCh. 6 - Calculate the following integrals. 36e2(x/3)dxCh. 6 - Calculate the following integrals. 05(5+3x)1dxCh. 6 - Calculate the following integrals. 2232e3xdxCh. 6 - Calculate the following integrals. 0ln2(exex)dxCh. 6 - Calculate the following integrals. ln2ln3(ex+ex)dxCh. 6 - Calculate the following integrals. 0ln3ex+exe2xdxCh. 6 - Calculate the following integrals. 013+e2xexdxCh. 6 - Find the area under the curve y=(3x2)3 from x=1 to...Ch. 6 - Find the area under the curve y=1+x from x=1 to...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - Find the area of the region bounded by the curves...Ch. 6 - Find the area of the region between the curves...Ch. 6 - Find the function f(x) for which...Ch. 6 - Find the function f(x) for which f(x)=e5x,f(0)=1.Ch. 6 - Describe all solutions of the following...Ch. 6 - Let k be a constant, and let y=f(t) be a function...Ch. 6 - Prob. 43RECh. 6 - Prob. 44RECh. 6 - A drug is injected into a patient at the rate of...Ch. 6 - A rock thrown straight up into the air has a...Ch. 6 - Prob. 47RECh. 6 - Prob. 48RECh. 6 - Prob. 49RECh. 6 - Prob. 50RECh. 6 - Find the consumers surplus for the demand curve...Ch. 6 - Three thousand dollars is deposited in the bank at...Ch. 6 - Find the average value of f(x)=1/x3 from x=13 to...Ch. 6 - Prob. 54RECh. 6 - In Fig. 2, three regions are labelled with their...Ch. 6 - Prob. 56RECh. 6 - Prob. 57RECh. 6 - Prob. 58RECh. 6 - Prob. 59RECh. 6 - Prob. 60RECh. 6 - Prob. 61RECh. 6 - Prob. 62RECh. 6 - Prob. 63RECh. 6 - Prob. 64RECh. 6 - Prob. 65RECh. 6 - Prob. 66RECh. 6 - Prob. 67RECh. 6 - Prob. 68RECh. 6 - Prob. 69RECh. 6 - Prob. 70RECh. 6 - Prob. 71RECh. 6 - Prob. 72RECh. 6 - Prob. 73RECh. 6 - Prob. 74RE
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- 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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