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Question 14 5 pts Consider the parametrically defined curve a. x = 6sin(3t), y=t, z =...
12. Let a curve be defined parametrically by x(1) = 3cost, y(t) = 3 sint, z(1)-...
12. Let a curve be defined parametrically by x(1) = 3cost, y(t) = 3 sint, z(1)- 21. a) Find the equation of the tangent line to the curve att b) Find the curvature of the curve att
Question 6 14 pts Consider the curve C defined by the parametric equations: x f(t) y=...
Question 6 14 pts Consider the curve C defined by the parametric equations: x f(t) y= g(t) = sint -t costt (d) Which picture shows the curve C? Recall the curve C is defined by : x= f(t) cos t g(t) = sint - t y 20 20 10 10F 0 -10 -10 -20 -20 -20 10 -20 10 C 20 -10 0 10 (i) (ii) X 20 20 10 10 0 0 10 -10 -20 -20h -20 10 -20...
Consider the surface z = f(x, y) = sin(x) + cos(y) and the curve C in...
Consider the surface z = f(x, y) = sin(x) + cos(y) and the curve C in the xy plane defined parametrically as x(t) = 2cos(t), y(t) = sin(t) a. Find z'(t). Imagine you are walking directly above the curve C in the direction of increasing t. Find the values of t for which you are walking uphill. Hint:Graph z'(t). Graph f(x, y) for -7 < x < 7 and -7 < y < 7. (You will have to find software...
1. (25 pts] Let F(x, y, z) = (2xy+ + 25)i + (4x²y3 + 2yz3)j +...
1. (25 pts] Let F(x, y, z) = (2xy+ + 25)i + (4x²y3 + 2yz3)j + (5x24 + 3y2z2)k and let C be the curve given parametrically by r(t) = (3t+1)i + tºj + 5tk for 0 <t<1. Evaluate the line integral Sa Spa
[25 pts] Let F(x, y, z) = (2xy4 + 25)i + (4x´y3 + 2yz3)j + (5x24...
[25 pts] Let F(x, y, z) = (2xy4 + 25)i + (4x´y3 + 2yz3)j + (5x24 + 3y2-2)k and let C be the curve given parametrically by r(t) = (3t+1)i + t?j + 5tk for 0 <t< 1. Evaluate the line integral F. dr
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x,...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
3 4. (4 pts) Consider the surface z = z = x²y + y3. (a) Find...
3 4. (4 pts) Consider the surface z = z = x²y + y3. (a) Find the normal direction of the tangent plane to the surface through (1,1,2). (b) Find the equation of the tangent plane in (a). (e) Determine the value a so that the vector 7= -7+27 +ak is parallel to the tangent plane in (a). (d) Find the equation of the tangent line to the level curve of the surface through (1,1).
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t)...
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t) =t+4 12: x(u) = 5 - 3u, y(u) = u +1, (u) = u +2 1. Determine whether li and ly are parallel, skew or intersect. If the lines intersect, find the point of intersection of li and 12. 2. If the lines intersect or are parallel, give an equation for the plane which contains both lines. If the lines are skew, find a...
Question 4 Consider the lines L, D=1+2, y = 2 – 3t, z = 2+t and...
Question 4 Consider the lines L, D=1+2, y = 2 – 3t, z = 2+t and L2 X = 3 - 4s, y=1+ 48, z = -3 + 48. We will use these lines for the questions 4 and 5. Are these lines parallel? Explain your answer below. B IV A - A - Ix E - C o o x G You HTML 11 x Ⓡ 5E T To 12pt Pan Question 5 9 pts Determine where these lines...
1. (25 pts] Let F(x, y, z) = (2xy4 +25)i + (4.x²y3 + 2yz3)j + (5x24...
1. (25 pts] Let F(x, y, z) = (2xy4 +25)i + (4.x²y3 + 2yz3)j + (5x24 + 3y2 -2)k and let C be the curve given parametrically by r(t) = (3t+1)i + t?j + 5tk for 0 <t<1. Evaluate the line integral [F F. dr
12. Let a curve be defined parametrically by x(1) = 3cost, y(t) = 3 sint, z(1)- 21. a) Find the equation of the tangent line to the curve att b) Find the curvature of the curve att
Question 6 14 pts Consider the curve C defined by the parametric equations: x f(t) y= g(t) = sint -t costt (d) Which picture shows the curve C? Recall the curve C is defined by : x= f(t) cos t g(t) = sint - t y 20 20 10 10F 0 -10 -10 -20 -20 -20 10 -20 10 C 20 -10 0 10 (i) (ii) X 20 20 10 10 0 0 10 -10 -20 -20h -20 10 -20...
Consider the surface z = f(x, y) = sin(x) + cos(y) and the curve C in the xy plane defined parametrically as x(t) = 2cos(t), y(t) = sin(t) a. Find z'(t). Imagine you are walking directly above the curve C in the direction of increasing t. Find the values of t for which you are walking uphill. Hint:Graph z'(t). Graph f(x, y) for -7 < x < 7 and -7 < y < 7. (You will have to find software...
1. (25 pts] Let F(x, y, z) = (2xy+ + 25)i + (4x²y3 + 2yz3)j + (5x24 + 3y2z2)k and let C be the curve given parametrically by r(t) = (3t+1)i + tºj + 5tk for 0 <t<1. Evaluate the line integral Sa Spa
[25 pts] Let F(x, y, z) = (2xy4 + 25)i + (4x´y3 + 2yz3)j + (5x24 + 3y2-2)k and let C be the curve given parametrically by r(t) = (3t+1)i + t?j + 5tk for 0 <t< 1. Evaluate the line integral F. dr
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
3 4. (4 pts) Consider the surface z = z = x²y + y3. (a) Find the normal direction of the tangent plane to the surface through (1,1,2). (b) Find the equation of the tangent plane in (a). (e) Determine the value a so that the vector 7= -7+27 +ak is parallel to the tangent plane in (a). (d) Find the equation of the tangent line to the level curve of the surface through (1,1).
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t) =t+4 12: x(u) = 5 - 3u, y(u) = u +1, (u) = u +2 1. Determine whether li and ly are parallel, skew or intersect. If the lines intersect, find the point of intersection of li and 12. 2. If the lines intersect or are parallel, give an equation for the plane which contains both lines. If the lines are skew, find a...
Question 4 Consider the lines L, D=1+2, y = 2 – 3t, z = 2+t and L2 X = 3 - 4s, y=1+ 48, z = -3 + 48. We will use these lines for the questions 4 and 5. Are these lines parallel? Explain your answer below. B IV A - A - Ix E - C o o x G You HTML 11 x Ⓡ 5E T To 12pt Pan Question 5 9 pts Determine where these lines...
1. (25 pts] Let F(x, y, z) = (2xy4 +25)i + (4.x²y3 + 2yz3)j + (5x24 + 3y2 -2)k and let C be the curve given parametrically by r(t) = (3t+1)i + t?j + 5tk for 0 <t<1. Evaluate the line integral [F F. dr
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