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derivatives, and the gradient

The Maple commands for computing partial derivatives are `D`
and `diff`. The **Getting Started** worksheet has examples
of how to use these commands to compute partial derivatives.

Maple doesn't have a simple command for computing directional
derivatives. There is a command in the `tensor` package that
can be used, but it is a little confusing unless you know something
about tensors. Fortunately, the method described above and the method
using the gradient described below are both easy to implement in
Maple. Examples are given in the `Getting Started` worksheet.

- The gradient can be used to compute the directional derivative as follows.
- The gradient points in the direction of
maximum increase of the value of
*F*at . - The gradient is perpendicular to the
level curve of
*F*that passes through the point . - The gradient can be easily generalized to apply to functions of three or more variables.

Maple has a fairly simple command `grad` in the `linalg`
package (which we used for curve computations). Examples of computing
gradients, using the gradient to compute directional derivatives, and
plotting the gradient field are all in the `Getting Started`
worksheet.

- 1.
- Compute the four second order partial derivatives of
the following functions using the indicated Maple command.
- (a)
- , using
`diff`. - (b)
- , using
`D`.

- 2.
- Consider
*F*(*x*,*y*) =*x*-^{2}*y*+3^{3}*y*. Compute the directional derivative of*F*at the point (1,-1) in the direction of the vector using each of the two methods demonstrated in the`Getting Started`worksheet. - 3.
- Consider the simple function
*F*(*x*,*y*) = 5-*x*-2^{2}*y*. It is easy to discover by plotting this function that it has a maximum value of 5 at the origin. Demonstrate, with plots of the gradient field and an explanation, how the gradient field can also be used to locate (approximately) where the maximum occurs.^{2} - 4.
- Compute the gradient of the function and plot the gradient field over the domain and . Use your plot to find the approximate location of the maximum value of this function over this domain.
- 5.
- In a previous lab, you used contour plots to get information about the
behavior of the graph of a function of two variables. Use the
technique demonstrated in the
`Getting Started`worksheet to plot the gradient field and a contour plot on the same graph for the function*H*(*x*,*y*) from the previous exercise. Use the same domain of and . Which plot (contour plot or gradient field plot) do you think gives you more information about the behavior of the function? Explain your answer. - 6.
- Suppose
*F*(*x*,*y*) is a differentiable function. Explain why the directional derivative can be computed using the equation (Hint - Let and use the definition of the directional derivative to show that )

2/7/2000