Values of f(x, y) are in Table 14.3 . Assuming they exist, decide whether you expect the followi

Values of f(x, y) are in Table 14.3 . Assuming they exist,...

Key Concepts

Partial Derivatives

A partial derivative measures the rate at which a function changes as one variable changes while all other variables are held constant. It is a fundamental concept in multivariable calculus that helps in understanding how a function behaves locally in each direction of its input variables.

Finite Difference Approximation

When a function is provided in tabular form rather than with an explicit formula, the partial derivatives can be approximated using finite differences by computing the ratio of the change in the function value to the change in the variable of interest, while keeping the other variable constant.

Monotonicity Analysis

To determine the sign of a partial derivative from a table, one studies the pattern of function values as one variable increases. If the function consistently increases with an increasing variable, the partial derivative is positive; if it decreases, the derivative is negative. This analysis of how function values change informs the expected behavior of the derivative.

Transcript

00:01 And we want to determine if these partial derivatives at these specified points are positive or negative.

00:08 So let's take a little look at this table and remember how to read this table.

00:13 If you look over here at this x slash y, the x values are in this column and the y values are in this row.

00:26 And so if we want to find a point negative two negative member order pairs x comma y here's negative 2 for x here's negative 1 for y so the function value at the point negative 2 negative 1 would be this 7 so that's how you use this table well let's go ahead and change this color and remember you can estimate the partial derivative of f with respect to x by looking at the change in f over the change in x.

01:09 And since we're only looking to determine if these partial derivatives at these points are positive or negative, we just have to look at how the function is behaving, is it increasing or decreasing, and how the x values are behaving, are they increasing or decreasing? so from the point negative to negative one, what's going to have, is we want to look at the change in x so we're not going to move to decide where the y values are changing because if we want partial derivative of f with respect to x we only want to move in the x direction so since x is this column we're going to be moving down so we're starting from the seven because the function value at the point negative 2 negative 1 equals 7 as we move down the column, okay, the x values are increasing, negative 2 to 0 to 2 to 4.

02:11 So x is going to be increasing.

02:16 But what about the function values, okay? at the point negative 2, negative 1, the function value is 7.

02:25 At the point 0 comma negative 1, the function value is 8.

02:30 It increased.

02:31 At the point 2 common negative 1, the function value is 10 still increasing.

02:39 At the point 4 common negative 1, the function is 13 still increasing.

02:47 So as we move down in the direction of the x column, okay, because here's the x column here, as the xs are increasing, the function values are increasing.

03:01 And so change in f over change in x positive divided by positive is going to be a positive and so the partial derivative of f with respect to x at the point negative 2 negative 1 is going to be positive all right how about the partial derivative of f with respect to y at the point 2 comma 1 first let's find a point 2 comma 1 so x is 2.

03:32 Y is 1.

03:33 So we're looking at this little square right here because the function itself takes on the value of 7 at the point 2 comma 1.

03:46 All right.

03:47 So now, since we're looking for the partial derivative of f with respect to y, we want to move along the y values.

03:57 So the y values are moving to the right here...

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