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# sn9_1 - -19.84 b-12.32 c-16.16 d-11.36 e 11.68 15.9...

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Math 2433 – Week 9 Popper 009 1. Find the point(s) on the surface at which the tangent plane is horizontal. a) b) c) d) e) 2. What is today? a) Tuesday b) Wednesday c) Thursday 15.7 - Maxima and Minima with Side Conditions Examples: 1. Maximize xy on the ellipse 2 2 4 9 36 x y + = .

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2. Minimize 2 4 x y z + + on the sphere 2 2 2 7 x y z + + = . 3. Find the points on the sphere 2 2 2 1 x y z + + = that are closest to and farthest from the point (3, 1, 3).

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Hint for #20: 15.8 – Differentials Recall from Calc I: Now, for ( , ) f x y , the differential ( ) df f x h = ∇ h h i or And for ( , , ) f x y z

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Examples: 1. Find the differential df a. b.
2. 3. Use differentials to approximate 4 125 17

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Popper009 3. Find the differential df . a) b) c) d) e) 4. Use differentials to approximate the value of f at the point P . a)

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Unformatted text preview: -19.84 b) -12.32 c) -16.16 d) -11.36 e) 11.68 15.9 - Reconstructing a Function from its Gradient The gradient of a function ( , ) f x y is ( , ) f x y ∇ = So if we have a (gradient) vector in the form ( , ) P x y i + ( , ) Q x y j We can begin reconstruct the function by setting P = / f x ∂ ∂ and Q = / f y ∂ ∂ Now when integrating / f x ∂ ∂ with respect to x , remember that y is a constant! Example: xy 2 i + x 2 y j Note: Sometimes the given vector could not be the gradient of a function. More examples: Popper009 5. Determine whether or not the vector function is the gradient ∇ f ( x , y ) of a function everywhere defined. If so, find all the functions with that gradient. a) b) c) d) e) 6. D...
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sn9_1 - -19.84 b-12.32 c-16.16 d-11.36 e 11.68 15.9...

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