Section 9.4 Slope of a Tangent Line & Compensating for Change

IdeaIf the output value (z) needs to be held at some constant

value, and one of the input variables changes, by how much must the other input variable change in order to compensate?

We can estimate this by using the slope of the tangent line to the K contour curve at the given input values.

IdeaConsider a specific K contour curve for some function and a tangent line to this contour curve at the point of interest.

(a, b)

IdeaThe slope of the tangent line (in green on the previous slide)

at the point (a, b) is , that is, the derivative of a 2-D function.

The question is, how do we find the derivative of a 2-D contour curve of a 3-D function given the formula of the 3-D function?

FactThe derivative of the tangent line discussed above can be

found with the formula:

Proving this is beyond the scope of this class, but we will use the results nonetheless.

Example 1aFind the slope of the indicated tangent line to the 117-contour curve of the function at the point (4, 5).

Example 1bFind the slope of the indicated tangent line to the 1-contour curve of the function at the point .

Compensating for ChangeIf we want to stay on some contour curve (hold z constant at that value), and we know how much x (or y) changes, we can use the derivative to estimate by how much y (or x) must change to compensate.

Example 2aReturning to example 1a, if we want to stay at a z value of 117, and x is increased by 0.2, by about how much must y be changed to compensate?

Example 2bReturning to example 1b, if we want to stay at a z value of 1, and y is increased by 0.4, by how much must x be changed to compensate?

Example 3aA model for the apparent temperature (what temperature it feels like) when it is T F and the relative humidity is 100h% is

F.

Evaluate and interpret .

Example 3bA model for the apparent temperature (what temperature it feels like) when it is T F and the relative humidity is 100h% is

F.

Find a formula for .

Example 3cA model for the apparent temperature (what temperature it feels like) when it is T F and the relative humidity is 100h% is

F.

If the conditions are as in part (a), and the apparent temperature is to remain the same, estimate the needed change in humidity required to compensate for a 2F increase in temperature.

Example 4aA person’s body-mass index is modeled by the function

points where h is height measured in inches and w is weight measured in pounds.

Find the body-mass index for a person who is 5 feet 7 inches tall and weighs 129 pounds.

Example 4bA person’s body-mass index is modeled by the function

points where h is height measured in inches and w is weight measured in pounds.

Find the formula for .

Example 4cA person’s body-mass index is modeled by the function

points where h is height measured in inches and w is weight measured in pounds.

For the person in part (a), if his body-mass index is to remain the same, and he grows 2 inches, estimate the change in his weight needed to compensate.