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Estimating Directional Derivatives and the Gradient:

The contour plot of some function f(T, y) is shown above, with each contour line spaced evenly between adjacent contour lines. Estimate each of the following by showing your work on the graph above. (Hint: Recall that all 2D graphs use standard Calculus orientation: to the right and up.)

a) Estimate the value of the partial derivative: âˆ‚f/âˆ‚y(4,3)

b) Estimate the value of the directional derivative Du(f)(4, 3) in the direction ()

c) On the graph above, sketch an arrow indicating the approximate direction and length for Vf(2,4) – (If this vector is too long to fit the picture, state the vector’s length. If you sketch this answer on your own paper, state the length of your vector.)

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01:01

point) Use the contour diagram for f (x,y) shown below to estimate the directional derivative of f in the direction V at the point P(a) At the point P = (2,2) in the direction v = the directional derivative approximatelyAt the point P = (2,3) in the direction v = ~j, the directional derivative approximately(c) At the point P = (4, in the direction v = (i + J)VZ, the directional derivative is approximatelyAt the point P = (0,0) in the direction v = ~i , the directional derivative is approximately(Click on graph to enlarge)8

04:45

Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero.

Positive1. At the point (-1, 1) in the direction of (~i – jyv).Negative2. At the point (1, 0) in the direction of -j.3. At the point (-2, 2) in the direction of 7.Zero4. At the point (0, 2) in the direction of j.5. At the point (0, -2) in the direction of (i + 2jyv5).(Click graph to enlarge)6. At the point (-1, 1) in the direction of (~i + jyv).

06:58

Use the contour diagram for f(x), shown below, to estimate the directional derivative of f in the direction at the point P.(a) At the point (2,2) in the direction V = i, the directional derivative is approximately.At the point P = (2,3) in the direction V = -j, the directional derivative is approximately.At the point P = (4,1) in the direction V = (i + j)/âˆš2, the directional derivative is approximately.At the point P = (1,0) in the direction V = -i, the directional derivative is approximately.(Click on graph to enlarge)

03:22

Text: FF 6, H Av Read and Draw 8 Highlight Erase 9. (11 pts) Find all the critical points of f(s,0) = 1 + 317 6rv + 3 and classify them as maximums, local minimums, or saddle points. (11 pts) (a) Find the maximum and minimum values of J(z,9) = 1′ + v – Zy + 7 subject to the constraint 2 + v = 0. (b) The function also has a critical point at (0,1). Find the global minimum of f(I,y) {(I,v) |#+4 < 4} the set D. (8 pts) Below is the contour map of g(1,y). Use it to answer the following: Briefly explain and show your work. (a) Estimate gv (0,1). Is the directional derivative of g at (0, -1) in the < 1,1 > direction positive, negative, or zero? (c) Is V9(2,0) parallel to ~ij or _j?

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