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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 11.7.2 .
In this activity, we will use
equation (11.7.2) to calculate and interpret directional derivatives for the function
\(f(x,y) = 3xy-x^2y^3\text{.}\)
(a) Calculate
\(f_x(x,y)\) and
\(f_y(x,y)\text{.}\)
(b) Use
equation (11.7.2) to determine
\(D_{\vi} f(x,y)\) and
\(D_{\vj} f(x,y)\text{.}\) Write a couple of sentences to describe what familiar functions
\(D_{\vi} f\) and
\(D_{\vj} f\) are.
Hint .
Remember that
\(\vi\) is the unit vector in the positive
\(x\) -direction and
\(\vj\) is the unit vector in the positive
\(y\) -direction.
(c) Use
equation (11.7.2) to find the derivative of
\(f\) in the direction of the vector
\(\vv = \langle 2, 3 \rangle\) at the point
\((1,-1)\text{.}\)
Hint .
Remember that a
unit direction vector is needed.
(d) Find the derivative of
\(f\) in the direction of the vector
\(\vv = \langle 4, 6 \rangle\) at the point
\((1,-1)\text{.}\)
(e) Use
equation (11.7.2) to find the derivative of
\(f\) in the direction of the vector
\(\vv = \langle -2, -3 \rangle\) at the point
\((1,-1)\text{.}\) Write a couple of sentences to explain why this result is different from your answer to the previous two tasks, even though the direction vectors are parallel.
