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Activity 11.3.2.

In this activity, we will be examining the function $\displaystyle f(x,y) = \frac{x y^2}{x+1}$ at the point $(1,2)\text{.}$ Specifically, we will be looking how the traces of this function will be related to the values of the partial derivatives.

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(a)

Write a formula for the trace $f(x,2)$ as a function of $x\text{.}$ On the axes provided, draw the graph of the trace with $y=2$ around the point where $x=1\text{.}$ Then sketch the tangent line at the point with $x=1$ on your graph.

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(b)

Compute the partial derivative $f_x(1,2)$ and relate its value to the sketch you just made.

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(c)

Write a formula for the trace $f(1,y)$ as a function of $y\text{.}$ On the axes provided, draw the graph of the trace with $x=1\text{.}$ Then sketch the tangent line at the point with $y=2$ on your graph.

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(d)

Compute the partial derivative $f_y(1,2)$ and relate its value to the sketch you just made.

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Activity 11.3.2.

(a)

(b)

(c)

(d)