Use a total differential to approximate the change in the values of f from P to Q. Compare your

Use a total differential to approximate the change in the...

Key Concepts

Linear Approximation

Linear approximation in the context of multivariable calculus uses the total differential to estimate the change in the function value near a point. This method simplifies complex functions into linear ones around the vicinity of the point of approximation.

Error Estimation

Error estimation involves comparing the linear approximation obtained using the total differential with the actual change in the function. This concept is critical as it highlights the limitations of using a linear model to estimate changes in nonlinear functions, especially when the change in variables is not infinitesimally small.

Total Differential

The total differential of a multivariable function is a linear approximation to the function near a specific point. It is computed by summing the products of each partial derivative and the corresponding change in the variable, which helps approximate how small changes in the input variables affect the function's output.

Partial Derivatives

Partial derivatives measure the rate at which a function changes with respect to one variable while keeping the others constant. They are fundamental in forming the total differential and provide insight into the sensitivity of the function to changes in each variable.

Transcript

00:01 In this problem, we will compute the change in f from p to q, and then we will use the differential of f at p to approximate that change.

00:12 So let's start by computing delta f, which is the change in f from p to q, and that's simply f evaluated at q minus f evaluated at p.

00:43 And then from here we just simply plug in the values for x, y, and z.

00:59 So this is 2 times .99 times minus 1 .02 squared times 2 .02 cubed, minus 2 times 1 times minus 1 squared times 2 cubed.

01:38 And when we evaluate this, we get that delta f is approximately 0 .97 -929.

02:01 So let's write that over here.

02:21 And so now we're going to use the differential of f at p to approximate the change in f from p to q.

02:30 So the differential of f, and this time we have three variables, but that's equal to the partial derivative of f with respect to x at p times d x plus the partial derivative of f with respect to y evaluated at p times d y and then finally the partial derivative of f with respect to z evaluated at p times d z and as we can see right away to compute this, we need to find dx, dy, and dz.

03:50 So to do so, we can let dx be the change in x, d.

04:00 Y, the change in y, and then dz to be the change in z.

04:15 So we get that dx is .99 minus 1, which is minus 0 .01.

04:29 And we get that dx is minus 1 .02 minus negative 1.

04:37 So that's minus 0 .02 and then d z which is 2 .02 minus 2 which is 0 .02 and now from here we want to compute our partial derivatives.

05:02 So we have the partial derivative of f with respect to x and our function is 2.

05:16 X times y squared times z cubed so we'll treat x as the only variable and when we do that we get that this partial derivative is two times y squared times z cubed and we still have to evaluate this at p so we get that this is two times negative one squared which is still two times two cubed which is two to the fourth which is 16.

06:09 And now we want the partial derivative of f with respect to y...

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