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Activity 11.3.6.

Shown below in Figure 11.3.11 is a contour plot of a function $f\text{.}$ The values of the function on a few of the contours are indicated to the left of the figure.

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Figure 11.3.11. A plot of $f(t)=M(0.03,t)$
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(a)

Estimate the partial derivative $f_x(-2,-1)\text{.}$ (Hint: How can you find values of $f$ that are of the form $f(-2+h)$ and $f(-2-h)$ so that you can use a symmetric difference quotient?)

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(b)

Estimate the partial derivative $f_y(-2,-1)\text{.}$

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(c)

Estimate the partial derivatives $f_x(-1,2)$ and $f_y(-1,2)\text{.}$

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(d)

Locate, if possible, one point $(x,y)$ where $f_x(x,y)= 0\text{.}$

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(e)

Locate, if possible, one point $(x,y)$ where $f_x(x,y)\lt 0\text{.}$