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Math 241 Chapter 13

Dr
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Wyss-Gallifent

§13
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Definition: A function like f (x, y), f (x, y, z), g(s, t) etc
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Definition of the graph of a function of two variables and classic examples like: Plane, paraboloid,
cone, parabolic sheet, hemisphere
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Definition of level curve for f (x, y) and level surface for f (x, y, z)
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Graphs of surfaces which are not necessarily functions: Sphere, ellipsoid, cylinder sideways
parabolic sheet like y = x2 , double-cone
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2 Limits and Continuity
1
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§13
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Defn: We can define the partial derivative of f (x, y) with respect to x, denoted ∂f or fx , as the
∂x
derivative of f treating all variables other than x as constant
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2
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A picture can clarify
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Higher derivatives will also be used but there are some points to note:
(a) fxy means (fx )y so first take the derivative with respect to x and then y
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means x both times
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§13
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Consider: For example if f is a function of x and y which are both functions of s and t then really
f is a function of s and t and so ∂f and ∂f make sense
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The chain rule says:
(a) First draw the tree diagram
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(c) Add the paths
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The chain rule is good for related rates problems when multiple rates are given and one rate is
needed
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5 Directional Derivative
1
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Defn: If u = a ˆ + b  is a unit vector then the directional derivative of f in the direction of u is
¯
ı
ˆ
¯
Du f = afx + bfy
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Sometimes we use the term “directional
¯
derivative” when the direction is not a unit vector so we must make it a unit vector first
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A good analogy is that f (x, y, z) is temperature and Du f gives us temperature change (slope) in
¯
a specific direction
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6 The Gradient
ˆ
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ı
ˆ
2
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¯
¯
¯
(b) Since Du f = u · ∇f = ||¯||||∇f || cos θ = ||∇f || cos θ we see that the directional derivative is
¯
u
¯
maximum when θ = 0 which shows that the gradient points in the direction of maximum
directional derivative
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(d) In the 2D case ∇f is perpendicular to the level curve for f (x, y) at (x, y)
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(e) In the 3D case ∇f is perpendicular to the level surface for f (x, y, z) at (x, y, z)
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§13
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Defn: Relative maximum/minimum/extremum for f (x, y)
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Those are the critical points
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• If D(x, y) > 0 and fxx (x, y) < 0 then (x, y) is a relative maximum
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Good examples: f (x, y) = x2 + 2y 2 − 6x + 8y + 1 and f (x, y) = 3x2 − 3xy 2 + y 3 + 3y 2
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Defn: Absolute m/m/e of f (x, y) on a closed and bounded region R
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Take f of those
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Usually this involves
combining f with the equation for the region (sometimes part by part) and then getting f
in a form where we can see what the max and min would be
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Good examples: f (x, y) = x2 − y 2 with
vertices (0, 0), (0, 3) and (6, 0)
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9 Lagrange Multipliers
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Thm: If a max/min occurs at (x, y) then ∇f = λ∇g at that point so the method is:
(a) We set those equal and solve those along with the constraint
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(b) The result are potential winners
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Good Examples: f (x, y) = 2x + 3y with x2 + y 2 = 9, f (x, y) = xy with (x − 1)2 + y 2 = 1 and
f (x, y) = x2 + y 2 with 2x + 6y = 10

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