EBK CALCULUS & ITS APPLICATIONS
14th Edition
ISBN: 8220103679527
Author: Asmar
Publisher: YUZU
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Textbook Question
Chapter 2.4, Problem 13E
Sketch the graphs of the following functions.
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Students have asked these similar questions
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 2 Solutions
EBK CALCULUS & ITS APPLICATIONS
Ch. 2.1 - Does the slope of the curve in Fig. 17 increases...Ch. 2.1 - At which labelled point on the graph in Fig. 18 is...Ch. 2.1 - Exercises 1-4 refer to graphs (a)-(f) in Fig.19...Ch. 2.1 - Exercises 1-4 refer to graphs (a)-(f) in Fig.19...Ch. 2.1 - Exercises 1-4 refer to graphs (a)-(f) in Fig.19...Ch. 2.1 - Exercises 1-4 refer to graphs (a)-(f) in Fig.19...Ch. 2.1 - Describe each of the following graphs. Your...Ch. 2.1 - Describe each of the following graphs. Your...Ch. 2.1 - Describe each of the following graphs. Your...Ch. 2.1 - Describe each of the following graphs. Your...
Ch. 2.1 - Describe each of the following graphs. Your...Ch. 2.1 - Prob. 10ECh. 2.1 - Describe each of the following graphs. Your...Ch. 2.1 - Prob. 12ECh. 2.1 - Describe the way the slope changes as you move...Ch. 2.1 - Prob. 14ECh. 2.1 - Describe the way the slope changes on the graph in...Ch. 2.1 - Prob. 16ECh. 2.1 - Exercise 17 and 18 refer to the graph in Fig 20....Ch. 2.1 - Exercise 17 and 18 refer to the graph in Fig 20....Ch. 2.1 - Prob. 19ECh. 2.1 - In Exercises 19-22, draw the graph of a function...Ch. 2.1 - In Exercises 19-22, draw the graph of a function...Ch. 2.1 - Prob. 22ECh. 2.1 - Annual World Consumption of Oil The annual world...Ch. 2.1 - Prob. 24ECh. 2.1 - A Patients Temperature At noon, a childs...Ch. 2.1 - Prob. 26ECh. 2.1 - Blood Flow through the Brain One method of...Ch. 2.1 - Pollution Suppose that some organic waste products...Ch. 2.1 - Number of U.S. Farms Figure 22 gives to number of...Ch. 2.1 - Prob. 30ECh. 2.1 - Prob. 31ECh. 2.1 - Let P(t) be the population of a bacteria culture...Ch. 2.1 - In Exercises 3336, sketch the graph of a function...Ch. 2.1 - In Exercises 3336, sketch the graph of a function...Ch. 2.1 - In Exercises 3336, sketch the graph of a function...Ch. 2.1 - In Exercises 3336, sketch the graph of a function...Ch. 2.1 - Consider a smooth curve with no undefined points....Ch. 2.1 - If the function f(x) has a relative minimum at x=a...Ch. 2.1 - Technology Exercises Graph the function...Ch. 2.1 - Prob. 40ECh. 2.1 - Technology Exercises Simultaneously graph the...Ch. 2.2 - Make a good sketch of the function f(x) near the...Ch. 2.2 - The graph of f(x)=x3 is shown in Fig. 15. Is the...Ch. 2.2 - The graph of y=f(x) is shown in Fig. 16. Explain...Ch. 2.2 - Exercises 14 refer to the functions whose graphs...Ch. 2.2 - Exercises 14 refer to the functions whose graphs...Ch. 2.2 - Exercises 14 refer to the functions whose graphs...Ch. 2.2 - Exercises 14 refer to the functions whose graphs...Ch. 2.2 - Which one of the graph in Fig. 18 could represent...Ch. 2.2 - Which one of the graphs in Fig. 18 could represent...Ch. 2.2 - In Exercises 712, sketch the graph of a function...Ch. 2.2 - In Exercises 712, sketch the graph of a function...Ch. 2.2 - In Exercises 712, sketch the graph of a function...Ch. 2.2 - In Exercises 712, sketch the graph of a function...Ch. 2.2 - In Exercises 712, sketch the graph of a function...Ch. 2.2 - In Exercises 712, sketch the graph of a function...Ch. 2.2 - In Exercises 1318, use the given information to...Ch. 2.2 - In Exercises 1318, use the given information to...Ch. 2.2 - In Exercises 1318, use the given information to...Ch. 2.2 - In Exercises 1318, use the given information to...Ch. 2.2 - In Exercises 1318, use the given information to...Ch. 2.2 - In Exercises 1318, use the given information to...Ch. 2.2 - Refer to the graph in Fig. 19. 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Changes in Temperature T(t) is the temperature...Ch. 2.2 - Prob. 39ECh. 2.2 - Prob. 40ECh. 2.2 - Prob. 41ECh. 2.2 - 42. Match each observation (a)(e) with a...Ch. 2.2 - Prob. 43ECh. 2.2 - Drug Diffusion in the Bloodstream After a drug is...Ch. 2.2 - Prob. 45ECh. 2.2 - Prob. 46ECh. 2.3 - Which of the curves in Fig.15 could possibly be...Ch. 2.3 - Which of the curves in Fig.16 could be the graph...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Prob. 14ECh. 2.3 - Prob. 15ECh. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Each of the graphs of the functions in Exercises...Ch. 2.3 - Prob. 24ECh. 2.3 - Sketch the following curves, indicating all...Ch. 2.3 - Sketch the following curves, indicating all...Ch. 2.3 - Sketch the following curves, indicating all...Ch. 2.3 - Sketch the following curves, indicating all...Ch. 2.3 - Sketch the following curves, indicating all...Ch. 2.3 - Sketch the following curves, indicating all...Ch. 2.3 - Prob. 31ECh. 2.3 - Prob. 32ECh. 2.3 - Prob. 33ECh. 2.3 - Let a,b,c,d be fixed numbers with a0, and let...Ch. 2.3 - The graph of each function in Exercises 35 40 has...Ch. 2.3 - Prob. 36ECh. 2.3 - The graph of each function in Exercises 35 40 has...Ch. 2.3 - The graph of each function in Exercises 35 40 has...Ch. 2.3 - The graph of each function in Exercises 35 40 has...Ch. 2.3 - The graph of each function in Exercises 35 40 has...Ch. 2.3 - In Exercises 41 and 42, determine which function...Ch. 2.3 - In Exercises 41 and 42, determine which function...Ch. 2.3 - Consider the graph of g(x) in Fig. 17. a. If g(x)...Ch. 2.3 - U. S. Population The population (in millions) of...Ch. 2.3 - Index-Fund Fees When a mutual fund company charges...Ch. 2.3 - Prob. 46ECh. 2.3 - Technology Exercises Draw the graph of...Ch. 2.3 - Technology Exercises Draw the graph of...Ch. 2.3 - Technology Exercises Draw the graph of...Ch. 2.3 - Technology Exercises Draw the graph of...Ch. 2.4 - Determine whether each of the following functions...Ch. 2.4 - Prob. 2CYUCh. 2.4 - Prob. 3CYUCh. 2.4 - Find the x intercepts of the given function....Ch. 2.4 - Prob. 2ECh. 2.4 - Find the x intercepts of the given function....Ch. 2.4 - Prob. 4ECh. 2.4 - Find the x intercepts of the given function....Ch. 2.4 - Find the x intercepts of the given function....Ch. 2.4 - Show that the function f(x)=13x32x2+5x has no...Ch. 2.4 - Prob. 8ECh. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Prob. 14ECh. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Prob. 16ECh. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Prob. 18ECh. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Sketch the graphs of the following functions....Ch. 2.4 - Prob. 21ECh. 2.4 - Prob. 22ECh. 2.4 - Sketch the graphs of the following functions for...Ch. 2.4 - Prob. 24ECh. 2.4 - Sketch the graphs of the following functions for...Ch. 2.4 - Prob. 26ECh. 2.4 - Sketch the graphs of the following functions for...Ch. 2.4 - Prob. 28ECh. 2.4 - Prob. 29ECh. 2.4 - Prob. 30ECh. 2.4 - Prob. 31ECh. 2.4 - Prob. 32ECh. 2.4 - Find the quadratic function f(x)=ax2+bx+c that...Ch. 2.4 - Prob. 34ECh. 2.4 - Prob. 35ECh. 2.4 - Prob. 36ECh. 2.4 - Prob. 37ECh. 2.4 - Technology Exercises Height of Tropical Grass The...Ch. 2.5 - Volume A canvas wind shelter for the beach has a...Ch. 2.5 - Prob. 2CYUCh. 2.5 - For what x does the function g(x)=10+40xx2 have...Ch. 2.5 - Find the maximum value of the function f(x)=12xx2,...Ch. 2.5 - Find the minimum value of f(t)=t36t2+40, t0 and...Ch. 2.5 - For what t does the function f(t)=t2-24t have its...Ch. 2.5 - Optimization with Constraint Find the maximum of...Ch. 2.5 - Optimization with Constraint Find two positive...Ch. 2.5 - Optimization with Constraint Find the minimum of...Ch. 2.5 - In Exercise 7, can there be a maximum for Q=x2+y2...Ch. 2.5 - Minimizing a Sum Find the positive values of x and...Ch. 2.5 - Maximizing a Product Find the positive values of...Ch. 2.5 - Area There are 320 available to fence in a...Ch. 2.5 - Volume Figure 12 (b) shows an open rectangular box...Ch. 2.5 - Volume Postal requirements specify that parcels...Ch. 2.5 - Perimeter Consider the problem of finding the...Ch. 2.5 - Cost A rectangular garden of area 75 square feet...Ch. 2.5 - Cost A closed rectangular box with a square base...Ch. 2.5 - Surface Area Find the dimensions of the closed...Ch. 2.5 - Volume A canvas wind shelter for the beach has a...Ch. 2.5 - Area A farmer has 1500 available to build an...Ch. 2.5 - Area Find the dimensions of the rectangular garden...Ch. 2.5 - Maximizing a Product Find two positive numbers,...Ch. 2.5 - Minimizing a Sum Find two positive numbers, xandy,...Ch. 2.5 - Area Figure 140 (a) shows a Norman window, which...Ch. 2.5 - Surface Area A large soup can is to be designed so...Ch. 2.5 - In Example 3 we can solve the constraint equation...Ch. 2.5 - Cost A ship uses 5x2 dollars of fuel per hour when...Ch. 2.5 - Cost A cable is to be installed from one corner,...Ch. 2.5 - Area A rectangular page is to contain 50 square...Ch. 2.5 - Distance Find the point on the graph of y=x that...Ch. 2.5 - Prob. 30ECh. 2.5 - Distance Find the point on the line y=2x+5 that is...Ch. 2.5 - Technology Exercise Inscribed Rectangle of Maximum...Ch. 2.6 - In the inventory problem of Example 2, suppose...Ch. 2.6 - In the inventory problem Example 2, Suppose that...Ch. 2.6 - Inventory Problem Figure 6 shows the inventory...Ch. 2.6 - Refer to Fig. 6. Suppose that The ordering cost...Ch. 2.6 - Inventory Control A pharmacist wants to establish...Ch. 2.6 - Inventory Control A furniture store expects to...Ch. 2.6 - Inventory Control A California distributor of...Ch. 2.6 - Economic Lot Size The Great American Tire Co....Ch. 2.6 - Prob. 7ECh. 2.6 - Prob. 8ECh. 2.6 - Prob. 9ECh. 2.6 - Prob. 10ECh. 2.6 - Area Starting with a 100-foot-long stone wall, a...Ch. 2.6 - Prob. 12ECh. 2.6 - Length A rectangular corral of 54 square meters is...Ch. 2.6 - Refer to Exercise 13. If the cost of the fencing...Ch. 2.6 - Revenue Shakespeares Pizza sells 1000 large vegi...Ch. 2.6 - Prob. 16ECh. 2.6 - Cost A storage shed is to be built in the shape of...Ch. 2.6 - Cost A supermarket is to be designed as a...Ch. 2.6 - Volume A certain airline requires that rectangular...Ch. 2.6 - Area An athletic field [Fig.8] consists of a...Ch. 2.6 - Volume An open rectangular box is to be...Ch. 2.6 - Volume A closed rectangular box is to be...Ch. 2.6 - Amount of Oxygen in a Lake Let f(t) be the amount...Ch. 2.6 - Prob. 24ECh. 2.6 - Area Consider a parabolic arch whose shape may be...Ch. 2.6 - Prob. 26ECh. 2.6 - Surface Area An open rectangular box of volume 400...Ch. 2.6 - If f(x) is defined on the interval 0x5 and f(x) is...Ch. 2.6 - Technology Exercises Volume A pizza box is formed...Ch. 2.6 - Technology Exercises Consumption of Coffee in the...Ch. 2.7 - Prob. 1CYUCh. 2.7 - Rework Example 4 under the condition that the...Ch. 2.7 - On a certain route, a regional airline carries...Ch. 2.7 - Minimizing Marginal Cost Given the cost function...Ch. 2.7 - Minimizing Marginal Cost If a total cost function...Ch. 2.7 - Maximizing Revenue Cost The revenue function for a...Ch. 2.7 - Maximizing Revenue The revenue function for a...Ch. 2.7 - Cost and Profit A one-product firm estimates that...Ch. 2.7 - Maximizing Profit A small tie shop sells ties for...Ch. 2.7 - Demand and Revenue The demand equation for a...Ch. 2.7 - Maximizing Revenue The demand equation for a...Ch. 2.7 - Profit Some years ago, it was estimated that the...Ch. 2.7 - Maximizing Area Consider a rectangle in the xy-...Ch. 2.7 - Demand, Revenue, and Profit Until recently...Ch. 2.7 - Demand and Revenue The average ticket price for a...Ch. 2.7 - Demand and Revenue An artist is planning to sell...Ch. 2.7 - Demand and Revenue A swimming club offers...Ch. 2.7 - Prob. 15ECh. 2.7 - Prob. 16ECh. 2.7 - Price Setting The monthly demand equation for an...Ch. 2.7 - Taxes, Profit, and Revenue The demand equation for...Ch. 2.7 - Interest Rate A savings and loan association...Ch. 2.7 - Prob. 20ECh. 2.7 - Revenue The revenue for a manufacturer is R(x)...Ch. 2.7 - Prob. 22ECh. 2 - State as many terms used to describe graphs of...Ch. 2 - What is the difference between having a relative...Ch. 2 - Give three characterizations of what it means for...Ch. 2 - What does it mean to say that the graph of f(x)...Ch. 2 - Prob. 5CCECh. 2 - Prob. 6CCECh. 2 - Prob. 7CCECh. 2 - Prob. 8CCECh. 2 - Prob. 9CCECh. 2 - Prob. 10CCECh. 2 - Prob. 11CCECh. 2 - Prob. 12CCECh. 2 - Prob. 13CCECh. 2 - Prob. 14CCECh. 2 - Outline the procedure for solving an optimization...Ch. 2 - Prob. 16CCECh. 2 - Figure (1) contains the graph of f(x), the...Ch. 2 - Figure (2) shows the graph of function f(x) and...Ch. 2 - In Exercise 36, draw the graph of a function f(x)...Ch. 2 - In Exercise 36, draw the graph of a function f(x)...Ch. 2 - In Exercise 36, draw the graph of a function f(x)...Ch. 2 - In Exercise 36, draw the graph of a function f(x)...Ch. 2 - Exercise 712, refer to the graph in Fig. 3. List...Ch. 2 - Exercise 712, refer to the graph in Fig. 3. List...Ch. 2 - Exercise 712, refer to the graph in Fig. 3. List...Ch. 2 - Exercise 712, refer to the graph in Fig. 3. List...Ch. 2 - Exercise 712, refer to the graph in Fig. 3. List...Ch. 2 - Exercise 712, refer to the graph in Fig. 3. List...Ch. 2 - Properties of various functions are described...Ch. 2 - Properties of various functions are described...Ch. 2 - Properties of various functions are described...Ch. 2 - Properties of various functions are described...Ch. 2 - Properties of various functions are described...Ch. 2 - Properties of various functions are described...Ch. 2 - Properties of various functions are described...Ch. 2 - Prob. 20RECh. 2 - In Fig. 4 (a) and 4 (b), the t axis represents...Ch. 2 - U.S. Electric Energy United States electrical...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following parabolas. Include there x...Ch. 2 - Sketch the following curves. y=2x3+3x2+1Ch. 2 - Sketch the following curves. y=x332x26xCh. 2 - Sketch the following curves. y=x33x2+3x2Ch. 2 - Sketch the following curves. y=100+36x6x2x3Ch. 2 - Sketch the following curves. y=113+3xx213x3Ch. 2 - Sketch the following curves. y=x33x29x+7Ch. 2 - Sketch the following curves. y=13x32x25xCh. 2 - Sketch the following curves. y=x36x215x+50Ch. 2 - Sketch the following curves. y=x42x2Ch. 2 - Sketch the following curves. y=x44x3Ch. 2 - Sketch the following curves. y=x5+20x+3(x0)Ch. 2 - Sketch the following curves. y=12x+2x+1(x0)Ch. 2 - Let f(x)=(x2+2)3/2. Show that the graph of f(x)...Ch. 2 - Show that the function f(x)=(2x2+3)3/2 is...Ch. 2 - Let f(x) be a function whose derivative is...Ch. 2 - Let f(x) be a function whose derivative is...Ch. 2 - Position Velocity and Acceleration A car traveling...Ch. 2 - The water level in a reservoir varies during the...Ch. 2 - Population near New York City Let f(x) be the...Ch. 2 - For what x does the function f(x)=14x2x+2,0x8,...Ch. 2 - Find the maximum value of the function...Ch. 2 - Find the minimum value of the function...Ch. 2 - Surface Area An open rectangular box is to be 4...Ch. 2 - Volume A closed rectangular box with a square base...Ch. 2 - Volume A long rectangular sheet of metal 30 inches...Ch. 2 - Maximizing the Total Yield A small orchard yields...Ch. 2 - Inventory Control A publishing company sells...Ch. 2 - Profit if the demand equation for a monopolist is...Ch. 2 - Minimizing time Jane wants to drive her tractor...Ch. 2 - Maximizing Revenue A travel agency offers a boat...
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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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