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Math 254

1. The document provides exercises involving calculating gradients, directional derivatives, and tangent planes. It asks the reader to: - Calculate the gradient of several multivariate functions. - Use the chain rule to find derivatives of composite functions. - Find directional derivatives and check if functions are increasing or decreasing. - Find equations of tangent planes to surfaces at given points.

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John Le Tourneux
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52 views2 pages

Math 254

1. The document provides exercises involving calculating gradients, directional derivatives, and tangent planes. It asks the reader to: - Calculate the gradient of several multivariate functions. - Use the chain rule to find derivatives of composite functions. - Find directional derivatives and check if functions are increasing or decreasing. - Find equations of tangent planes to surfaces at given points.

Uploaded by

John Le Tourneux
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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math 254

Section 14.5

2
1. Let f (x, y) = xey and c(t) = (1 + t , t2 ) .

(a) Calculate f and c 0 (t) .


d
(b) Use the Chain Rule for Paths to calculate f (c(t)) .
dt
(c) Write out the composite function f (c(t)) as a function of t and differentiate.
Check that the result agrees with part (b).

In Exercises 2-4, calculate the gradient.

y2
2. f (x, y) = sin(xy + y) 3. g(x, y) =
x+y


4. h(x, y, z) = x + yz

d
In Exercises 5-6, use the Chain Rule to calculate f (c(t)) .
dt

!
x
5. f (x, y) = sin , c(t) = (e2t , e5t ) , t=0
y


6. f (x, y) = ln(x2 + y) , c(t) = (tan(t) , et/4 ) , t= 4

1
In Exercises 7-9, calculate the directional derivative in the direction of v at the given point.
Remember to normalize the direction vector or use Eq. (4).

7. f (x, y) = cos(y x) , v = h1 , 2i, P = ( 6 , 2 )

8. f (x, y) = tan1 (xy) , v = h3 , 4i, P = (1, 2)

2
9. g(x, y, z) = y ex z , v = h1 , 1 , 1i, P = (2, 1, 3)

10. Suppose that fP = h1 , 4 , 2i . Is f increasing or decreasing at P in the direction


v = h2 , 3 , 4i ?

11. Let T (x, y) be the temperature at location (x, y) . Assume that T = hy 3 , x + 2 yi .


Let c(t) = (t3 , t) be a path in the plane. Find the values of t such that

d
T (c(t)) = 0
dt

In Exercises 12-13, find an equation of the tangent plane to the surface at the given point.

12. 2x2 + y 2 + 3z 2 = 20 , P = (2 , 3 , 1)

13. x2 2y 2 + z 2 + yz = 2 , P = (2 , 1 , 1)

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