Practice for Test #3

Practice for Test #3

MTH 153 Name:________________________

03/05/2014

Signature:_________________________

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Give your answers in exact form (e.g.

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To receive credit you must show all of your work and make your work easy to follow. A correct answer without supporting work or work that is not clear will not receive credit.

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2 or π / 3 ) except where noted in particular problems.

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Place a box around your answer to each problem, and do your work in pencil!

GOOD LUCK!!!

GOOD LUCK!!!

GOOD LUCK!!!

1. Use the level curves of the function z = f (x, y) to determine the sign (positive or negative) of each of the following partial derivatives at the point P . Provide an explanation for each part.

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a)

∂f (P) ∂x

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b)

∂f (P) ∂y

c)

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€ d)

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∂2 f (P) ∂x 2

e)

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∂2 f (P) ∂x∂y

∂2 f (P) ∂y 2

2. The graphs below show the level curves for a function z = f (x, y) , x as a function of t , and y as a

dz is positive or negative. dt t= 2 € € €

function of t . Use these graphs to determine if the derivative

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3. Suppose f is a differentiable function of two variables and that f x (2,−1) = 5 and f y (2,−1) = π . a) Find a vector in the xy − plane that is orthogonal to the level curve f (x, y) = 3 at the point (2,−1) .

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b) Find the equation of the tangent plane to the surface z = f (x, y) at the point (2,−1, 3)

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c) Is there a direction such that the rate of change of f at the point (2,−1) in that direction is equal to 35 ? If so, find the direction, and if not, explain why no such direction exists.

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4. Suppose that a bug moves along the helix x = cos 3t , y = sin 3t , and z = 3t . Meanwhile, the temperature at a point (x, y,z) in space is given by T(x, y,z) = 2x 2 + sin xy − 8z . Suppose that x , y , and z are measured in meters, t is measured in seconds, and T is measured in degrees Celsius.

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€ the point P(−1,0, π ) in degrees Celsius per meter. a) Find the rate of change in the temperature at €

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b) Find the rate of change in the temperature at the point P(−1,0, π ) in degrees Celsius per second.

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c) Suppose now that the bug wants to change course at the point P(−1,0, π ) and move in the direction in which the temperature is decreasing most rapidly. Which direction should the bug move in, and what is the rate of change in the temperature in this direction?

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d) If the bug moves a distance of ds = 0.2 meters in the direction from part c, by how about much will the temperature change?

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5. Suppose a person in your class is wearing a lot of cologne and that at each point ( x, y, z ) in space the 2

function ϑ (x, y, z) = e ( ) + z 2 ( x + y) measures the odor of the cologne. Furthermore, suppose there’s a fly in the classroom that always flies in the direction in which the odor decreases the fastest. If the fly has a constant speed of 3 m/s, find the fly’s velocity vector at the point (2,−2,1 €). − x+ y

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6. Consider the vector field F (x, y, z) = P(x, y, z)i + Q(x, y, z) j + R(x, y, z)k , where the functions P , Q , and  R have continuous second partial derivatives. We define the curl of F to be the vector field

 $ ∂ R ∂Q '  $ ∂ P ∂R '  $ ∂ Q ∂P '  curl ( F ) = & − − )k )i + & − ) j + & % ∂y ∂z ( % ∂z ∂x ( €% ∂x ∂y (

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

and the divergence of F as the function

 ∂P ∂Q ∂R div( F ) = + + . ∂x ∂y ∂z

€ € Prove the following two identities.

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

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( ( )) = 0 .

a) The divergence of the curl of F is the zero function, i.e. show that div curl F

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b) If f has continuous second partial derivatives, show that the curl of the gradient of f is the zero  vector, i.e. show that curl (∇f ) = 0 .

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