![4. (11 marks] If the height above sea level is given by the function H(x,y) = 100+ (x + 1)?y2 and r(t) = 2t i++) is a path th](http://img.homeworklib.com/questions/683ed110-7adc-11eb-aa63-133bb08f6bc2.png?x-oss-process=image/resize,w_560)
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4. (11 marks] If the height above sea level is given by the function H(x,y) =...
Consider the following. w = In(x2 + y), x = 2t, y = 5 - t...
Consider the following. w = In(x2 + y), x = 2t, y = 5 - t (a) Find af by using the appropriate Chain Rule. (b) Find by converting w to a function of t before differentiating. -/1 POINTS LARCALC11 13.R.054. Differentiate implicitly to find oux x2 = 9 x + y -11 POINTS LARCALC11 13.R.069. Find an equation of the tangent plane to the surface at the given point. z = x2 + y2 + 9, (1, 2, 14)
(11) Suppose a cinder cone's height function (in meters) if given by h(x, y) = 900...
(11) Suppose a cinder cone's height function (in meters) if given by h(x, y) = 900 - 5x2 + 4y2, where r and y represent east-west and north-south respectively; (0,0,900) is the top of the cinder cone. Suppose you are on the cinder cone and are 8 meters north of the vertical line through the top of the cinder cone. Find the slope of your path if you hiking in the southwest direction.
17 marks] Consider the functionf(x, y) = (y - 1) (x- 1). (a) Find a unit normal vector to the contour line given by f=0...
17 marks] Consider the functionf(x, y) = (y - 1) (x- 1). (a) Find a unit normal vector to the contour line given by f=0.5 at the point (x,y) (1.5,2). Do not forget to check that this point is on the contour line. (b) Consider the curve L given by r(t) = 2ti+3tj with 0 ts1 and the Cartesian basis vectors i= (1,0) and j= (0,1) of R2. Determine using the chain rule and use this result to determine all...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
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...
Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine...
Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine the equation for the flow line of F passing through the point (1,1) in terms of and y (b) Let u R> R3 be a C3 path parametrised in terms of t. Evaluate and simplify d dt
Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine the equation for the flow line of F passing through the...
Question 3. Consider the function h: R3 → R h(x, y, 2) = (x2 + y2...
Question 3. Consider the function h: R3 → R h(x, y, 2) = (x2 + y2 + 2) +3/(x2 + 2xy + y) (a) What is the maximal domain of h? Describe it in words. (it may help to factor the denominator in the second term) > 0 for any a, (b) It is difficult to immediately find the range of h. Using the fact that a show that h cannot take negative values. Can h be an onto function?...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
Consider the function f(x) = x2 - 8x + 8. The slope of the tangent line...
Consider the function f(x) = x2 - 8x + 8. The slope of the tangent line at x = 5 is 2. Find the equation of this tangent line. (Write your answer in the form y=mx+b with no spaces - as always, if the computer marks you incorrect even if you have the answer correct but just in a different form than I use, email me and I will fix your points.) Given the function f(x)= 4x2 - 3x +...
1. Sketch a few of the level curves of the function f(x, y) = surface z...
1. Sketch a few of the level curves of the function f(x, y) = surface z = y2 and then use these to graph the f (x, y) 2. Evaluate the following limits if they exist. If they don't, explain why not. (a lim (x,y)(0,0) + 4y2 x4-y4 (b lim (x,y)(0,0) x2 + y2 cos 2 y2) - 1 lim (c (z,y)(0,0 2ry (x, y)(0,0) Is the function f(x, y) continuous at (0,0)? 3 = (х, у) — (0,0) 2x2y...
Consider the following. w = In(x2 + y), x = 2t, y = 5 - t (a) Find af by using the appropriate Chain Rule. (b) Find by converting w to a function of t before differentiating. -/1 POINTS LARCALC11 13.R.054. Differentiate implicitly to find oux x2 = 9 x + y -11 POINTS LARCALC11 13.R.069. Find an equation of the tangent plane to the surface at the given point. z = x2 + y2 + 9, (1, 2, 14)
(11) Suppose a cinder cone's height function (in meters) if given by h(x, y) = 900 - 5x2 + 4y2, where r and y represent east-west and north-south respectively; (0,0,900) is the top of the cinder cone. Suppose you are on the cinder cone and are 8 meters north of the vertical line through the top of the cinder cone. Find the slope of your path if you hiking in the southwest direction.
17 marks] Consider the functionf(x, y) = (y - 1) (x- 1). (a) Find a unit normal vector to the contour line given by f=0.5 at the point (x,y) (1.5,2). Do not forget to check that this point is on the contour line. (b) Consider the curve L given by r(t) = 2ti+3tj with 0 ts1 and the Cartesian basis vectors i= (1,0) and j= (0,1) of R2. Determine using the chain rule and use this result to determine all...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
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...
Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine the equation for the flow line of F passing through the point (1,1) in terms of and y (b) Let u R> R3 be a C3 path parametrised in terms of t. Evaluate and simplify d dt
Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine the equation for the flow line of F passing through the...
Question 3. Consider the function h: R3 → R h(x, y, 2) = (x2 + y2 + 2) +3/(x2 + 2xy + y) (a) What is the maximal domain of h? Describe it in words. (it may help to factor the denominator in the second term) > 0 for any a, (b) It is difficult to immediately find the range of h. Using the fact that a show that h cannot take negative values. Can h be an onto function?...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
Consider the function f(x) = x2 - 8x + 8. The slope of the tangent line at x = 5 is 2. Find the equation of this tangent line. (Write your answer in the form y=mx+b with no spaces - as always, if the computer marks you incorrect even if you have the answer correct but just in a different form than I use, email me and I will fix your points.) Given the function f(x)= 4x2 - 3x +...
1. Sketch a few of the level curves of the function f(x, y) = surface z = y2 and then use these to graph the f (x, y) 2. Evaluate the following limits if they exist. If they don't, explain why not. (a lim (x,y)(0,0) + 4y2 x4-y4 (b lim (x,y)(0,0) x2 + y2 cos 2 y2) - 1 lim (c (z,y)(0,0 2ry (x, y)(0,0) Is the function f(x, y) continuous at (0,0)? 3 = (х, у) — (0,0) 2x2y...
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