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This is about linear algebra. Please use LU method(perhaps) to solve it.

 

Calculate elementary matrices that transform A into upper triangular form  

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1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

 

1.)  Some punch is made by mixing grape juice and bubbly water in a ratio of 4 to 3. Give at least five different pairs of quantities of grape juice and bubbly water that you could use to make the punch mixture in that ratio. Include at least two examples that involve fractions or mixed numberquantities. Show or explain why all the mixtures are in that same ratio of 5 to 2 using a method your choice (i.e. double-sided number line, strip diagram, batch table).

 

 

2.)  Using a ratio of 3 to 8;

 

a.)  Give two different ways of using this ratio with different ‘part-to-part’ perspectives.

 

 

b.)  Give two different ways of using this ratio with a ‘part-to-whole’ perspective.

 

 

3.)  A drink mixture is made by mixing cola and lemon-lime soda in a ratio of 5 to 3. How much cola and how much lemon-lime soda will you need to make 240 cups of the drink mixture?  Complete either (a) OR (b).

 

(a) Show how to solve the problem from the multiple-batches perspective by reasoning about multiplication and division with quantities. You can support your work with a table or double-sided number line.

 

(b) Show how to solve the problem from the variable-parts perspective by reasoning about multiplication and division with quantities. You can support your work with a strip diagram.

 

 

4.)  Traveling at a constant speed, a race car goes 15 miles every 6 minutes.  Use the unit rate method to solve the following:  (a) How far does the race car go in 15 minutes?  (b) How long does it take the race car to go 20 miles?

 

 

5.)  A type of fertilizer is made by mixing nitrogen and phosphate in an 8 to 3 ratio. For 35 kilograms of nitrogen, how many kilograms of phosphate are needed to make that type of fertilizer?

 

 

6.)  To make jewelry, jewelers often mix gold and copper in a 7 to 5 ratio. How much gold should a jeweler -mix with 42 grams of copper?​

 

 

7.)  You need 3 cups of grape juice for a recipe that makes 4 servings. How many cups of grape juice will you need if you want to make 10 servings of the same recipe?

8.)  Consider all ramps that could be built for which the ratio of height to length is 2 cm to 7 cm. Let x be the unspecified length of a ramp in feet, which can vary, and let y be the corresponding height of the ramp in feet. Derive and explain an equation of the form y = c•x for a suitable constant c (which you should find).

 

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