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Lent Term Week 5 (lectures 29 and 30): In this question you will look at countours of a function, tagent planes, plane intersections, the gradient function, and directional derivatives.

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As with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.

Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.

Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.

", "variables": {"m": {"description": "

This is the x coordinate of our climber on the mountain.

", "definition": "random(100 , 200 , 300 , 400 , 500)", "name": "m", "group": "Variables for part f", "templateType": "anything"}, "u": {"description": "

This is a direction vector in the x,y plane.

", "definition": "vector(3,4)", "name": "u", "group": "Variables for part f", "templateType": "anything"}, "f_u_m_n": {"description": "

This is the drectional derivative of f at the point (m,n,f(m,n)) in the direction u.

", "definition": "dot(nabla_f_m_n , (1/5) * u)", "name": "f_u_m_n", "group": "Variables for part f", "templateType": "anything"}, "mod_nabla_f_m_n": {"description": "

This is the modulus of nabla_f_m_n. I.e. the rate of increase in the direction of greatest increase of f at the point (m,n,f(m,n))

", "definition": "(dot(nabla_f_m_n , nabla_f_m_n))^(1/2)", "name": "mod_nabla_f_m_n", "group": "Variables for part f", "templateType": "anything"}, "c": {"description": "

This is the z value of the point that we want the student to find the tangent plane of. By its definition and the definition of a and b, it is always positive.

", "definition": "a^2 - b^2", "name": "c", "group": "Variables for parts b to d", "templateType": "anything"}, "b": {"description": "

This is the y value of the point that we want the student to find the tangent plane of.

", "definition": "random(1..5)", "name": "b", "group": "Variables for parts b to d", "templateType": "anything"}, "a": {"description": "

This is the x value of the point that we want the student to find the tangent plane of. It is defined so that it is larger than the y value, b. Hence, the z value, c, which is a^2 - b^2 by definition, is positive.

", "definition": "b + random(1..5)", "name": "a", "group": "Variables for parts b to d", "templateType": "anything"}, "nabla_f_m_n": {"description": "

This is the direction of greatest increase of f at the point (m,n,f(m,n))

", "definition": "vector(-0.002*m , -0.008*n)", "name": "nabla_f_m_n", "group": "Variables for part f", "templateType": "anything"}, "f_m_n": {"description": "

This is the z coordinate of our climber on the mountain.

", "definition": "4000 - 0.001*m*m - 0.004*n*n", "name": "f_m_n", "group": "Variables for part f", "templateType": "anything"}, "n": {"description": "

This is the y coordinate of our climber on the mountain.

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What is the definition of the graph of a function $f : \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$.

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The graph of a function $f : \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$ is the subset of $\\mathbb{R}^3$ consisting of all points of the form $\\big( x ,y , f(x,y) \\big)$ such that $x, y \\in \\mathbb{R}$; i.e. all points $\\big( x ,y , z \\big)$ that satisfy $x,y \\in \\mathbb{R}$ and $z=f(x,y)$.

", "

The graph of a function $f : \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$ is the subset of $\\mathbb{R}^3$ consisting of all points of the form $\\big( x , 0 , f(x,0) \\big)$ such that $x \\in \\mathbb{R}$; i.e. all points $\\big( x ,0 , z \\big)$ that satisfy $x \\in \\mathbb{R}$ and $z=f(x,0)$.

", "

The graph of a function $f : \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$ is the subset of $\\mathbb{R}^3$ consisting of all points of the form $\\big( 0 ,y , f(0,y) \\big)$ such that $y \\in \\mathbb{R}$; i.e. all points $\\big( 0 ,y , z \\big)$ that satisfy $y \\in \\mathbb{R}$ and $z=f(0,y)$.

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Consider the function $f: \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$ defined by $f(x,y) = x^2 - y^2$ for all $x,y \\in \\mathbb{R}$.

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On a sheet of paper, draw the countours $f(x,y) = c$ for $c = 0 , 1 , 4,$ and $9$ on a single copy of the $x,y$ plane. In each case for $c$, you can start by considering the points on the contour where $y=0$, and also how the graph behaves when $x$ is large. What are the equations of the assymptotes that each contour share? (Please enter the equation of the assymptote with positive gradient first).

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$y =$ [[0]] and
$y =$ [[1]]

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The cartesian equation of the tangent plane to the graph of $f$ at the general point $\\big( a , b , f(a,b) \\big)$ is given by $z - f(a,b) = f_x (a,b) (x - a) + f_y (a,b) (y - b)$.

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First let us calculate $f_x (x , y)$ and $f_y (x , y)$.

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$f_x (x , y) =$

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$f_y (x , y) =$

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Now we can now calculate $f_x (\\var{{a}} , \\var{{b}} )$ and $f_y (\\var{{a}} , \\var{{b}} )$.

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$f_x (\\var{{a}} , \\var{{b}} ) =$

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$f_y (\\var{{a}} , \\var{{b}} ) =$

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Hence, using the cartesian equation for the tangent plane of a general point that we gave at the start of these steps, we can see that the cartesian equation for the tangent plane to the graph of $f$ at the point $\\big( \\var{{a}} , \\var{{b}} , \\var{{c}} \\big)$ is

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$z =$

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Find a cartesian desciption for the tangent plane $\\Pi$ to the graph of $f$ at the point $\\big( \\var{{a}} , \\var{{b}} , \\var{{c}} \\big)$. (Press the \"Show steps\" button if you require a breakdown of the question into steps).

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$z =$ [[0]] .

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Let $Q$ be the horizontal plane passing through the point $(\\var{{a}} , \\var{{b}} , \\var{{c}})$, and let $l$ be the line of intersection of the plane $\\Pi$, defined in part b, and the plane $Q$. We will find the vector parametric description of the line $l$.

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We have already found, in part b, the cartesian description of the plane $\\Pi$. What is the cartesian description of the plane $Q$?

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$z =$ [[0]] .

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Since $l$ is the intersection of $\\Pi$ and $Q$, its cartesian description will consist of the cartesian equations for both $\\Pi$ and $Q$. We can obtain an equivalent cartesian description by substituting the cartesian equation of $Q$ into the cartesian equation of $\\Pi$. This gives:

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$y =$ [[1]] .

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Hence, the two equations
$y =$ [[1]] and
$z =$ [[0]]
also form a cartesian description for $l$.

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We can let $x$ play the role of the free variable; that is, let $x = t$. Then, $y$ can be expressed in terms of $t$ by using the first equation directly above. The second equation directly above shows us that $z$ has no dependence on $t$. Hence we can see that

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$\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} =$ [[2]]$t +$ [[3]] .

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This is the vector parametric equation for $l$.

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In the remainder of the question, we will look at the gradient of a function $f : \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$, and directional derivatives.

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What is the mathematical definition of $\\nabla f$, the gradient of $f$?
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What is the geometric interpretation of $\\nabla f$?
[[1]]

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Let $(a,b) \\in \\mathbb{R}^2$. What would be the rate of greatest increase of $f$ at the point $(a,b)$?
[[2]]

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Let $\\boldsymbol{u} \\in \\mathbb{R}^2$. What is the mathematical definition of $f_{\\boldsymbol{u}} (a,b)$?
[[3]]

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What is the geometric interpretation of $f_{\\boldsymbol{u}} (a,b)$?
[[4]]

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Please press the \"Show feedback\" button once you have completed all of this part of the question.

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$\\nabla f$ is a function from $\\mathbb{R}^2$ to $\\mathbb{R}^2$ defined by $\\nabla f (x,y) = \\begin{pmatrix} f_x (x,y) \\\\ f_y (x,y) \\end{pmatrix}$ for all $(x,y) \\in \\mathbb{R}^2$.

", "

$\\nabla f$ is a function from $\\mathbb{R}^2$ to $\\mathbb{R}^2$ defined by $\\nabla f (x,y) = \\Bigg\\| \\begin{pmatrix} f_x (x,y) \\\\ f_y (x,y) \\end{pmatrix} \\Bigg\\|$ for all $(x,y) \\in \\mathbb{R}^2$.

", "

$\\nabla f$ is a function from $\\mathbb{R}^2$ to $\\mathbb{R}^2$ defined by $\\nabla f (x,y) = \\frac{\\begin{pmatrix} f_x (x,y) \\\\ f_y (x,y) \\end{pmatrix}}{\\Bigg\\| \\begin{pmatrix} f_x (x,y) \\\\ f_y (x,y) \\end{pmatrix} \\Bigg\\|}$ for all $(x,y) \\in \\mathbb{R}^2$.

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For a point $(a,b) \\in \\mathbb{R}^2$ where $\\nabla f (a,b) \\neq \\boldsymbol{0}$, $\\nabla f (a,b)$ represents the direction of greatest increase of $f$ at the point $(a,b)$. That is, if a person where standing at the point $\\big(a,b, f(a,b) \\big)$ on the graph of $f$, and they tried walking in the direction $\\nabla f (a,b)$, then they would walk along the graph of $f$ in the steepest \"up-hill\" direction for the point $(a,b)$.

", "

For a point $(a,b) \\in \\mathbb{R}^2$ where $\\nabla f (a,b) \\neq \\boldsymbol{0}$,$\\nabla f (a,b)$ represents the direction of greatest descrease of $f$ at the point $(a,b)$. That is, if a person where standing at the point $\\big(a,b, f(a,b) \\big)$ on the graph of $f$, and they tried walking in the direction $\\nabla f (a,b)$, then they would walk along the graph of $f$ in the steepest \"down-hill\" direction for the point $(a,b)$.

", "

For a point $(a,b) \\in \\mathbb{R}^2$ where $\\nabla f (a,b) \\neq \\boldsymbol{0}$,$\\nabla f (a,b)$ represents a direction of no increase or decrease of $f$ at the point $(a,b)$. That is, if a person where standing at the point $\\big(a,b, f(a,b) \\big)$ on the graph of $f$, and they tried walking in the direction $\\nabla f (a,b)$, then they would walk along the graph of $f$ in the level (i.e. neither \"up-hill\" nor \"down-hill\") direction for the point $(a,b)$.

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$\\Bigg\\| \\begin{pmatrix} f_x (a,b) \\\\ f_y (a,b) \\end{pmatrix} \\Bigg\\|$

", "

$\\Bigg\\| \\begin{pmatrix} a \\\\ b \\end{pmatrix} \\Bigg\\|$

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$f_{x,y} (a,b)$

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$f_{\\boldsymbol{u}} (a,b) = \\bigg\\langle \\nabla f (a,b) , \\frac{\\boldsymbol{u}}{\\| \\boldsymbol{u} \\|} \\bigg\\rangle$.

", "

$f_{\\boldsymbol{u}} (a,b) = \\bigg\\langle \\nabla f (a,b) , \\boldsymbol{u} \\bigg\\rangle$.

", "

$f_{\\boldsymbol{u}} (a,b) = \\bigg\\langle \\frac{\\nabla f (a,b)}{\\| \\nabla f (a,b) \\|} , \\boldsymbol{u} \\bigg\\rangle$.

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$f_{\\boldsymbol{u}} (a,b)$ is the rate of change of $f$ at $\\big(a,b,f(a,b) \\big)$ in the direction $\\boldsymbol{u}$. That is, if we were \"standing\" at the point $\\big(a,b,f(a,b) \\big)$ and tried to \"walk\" along a path in the direction $\\boldsymbol{u}$, then the gradient of our path at that point would be $f_{\\boldsymbol{u}} (a,b)$.

", "

$f_{\\boldsymbol{u}} (a,b)$ is the angle between the direction $\\boldsymbol{u}$ and the direction of greatest increase at $\\big(a,b,f(a,b) \\big)$.

", "

$f_{\\boldsymbol{u}} (a,b)$ is the angle between the direction $\\boldsymbol{u}$ and the direction of greatest decrease at$\\big(a,b,f(a,b) \\big)$.

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Now suppose we are told that the function $f : \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$ is defined by $f(x,y) = 4000 - 0.001x^2 - 0.004 y^2$ for all $(x,y) \\in \\mathbb{R}^2$. The graph of this function represents the surface of a mountain.

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Suppose there is a mountain climber at the position $(\\var{{m}} , \\var{{n}} , \\var{{f_m_n}})$. We want to determine the direction they should travel in order to ascend at the greatest rate, and determine what this rate of ascent would be.

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We know the direcion of greatest ascent at $(\\var{{m}} , \\var{{n}} , \\var{{f_m_n}})$ is $\\nabla f (\\var{{m}} , \\var{{n}}) = \\begin{pmatrix} f_x (\\var{{m}} , \\var{{n}}) \\\\ f_y (\\var{{m}} , \\var{{n}}) \\end{pmatrix} =$ [[0]] .

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The rate of increase in this direction is $\\| \\nabla f (\\var{{m}} , \\var{{n}}) \\| =$ [[1]] . (If you wish to write something like $\\sqrt{2}$ then please write 2^(1/2) .)

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Finally, suppose $\\boldsymbol{u} := \\var{{u}}$. What would be the rate of increase if our climber, standing at $(\\var{{m}} , \\var{{n}} , \\var{{f_m_n}})$, tried to move in the direction $\\boldsymbol{u}$? That is, what is $f_{\\boldsymbol{u}} (a,b)$?

\n

$f_{\\boldsymbol{u}} (a,b) =$ [[2]] .

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This is the question for Lent Term week 5 of the MA100 course at the LSE. It looks at material from chapters 29 and 30.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "name": "MA100 LT Week 5", "type": "question", "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}]}], "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}