Differentials and Partial Derivatives

This video tutorial introduces differentials for functions of two variables, explaining how to calculate the total differential Z using partial derivatives with respect to X and Y. It demonstrates the process of comparing the true change in a function to the approximate change using differentials. The tutorial includes practical examples and calculations, emphasizing the usefulness of differentials in approximating changes in functions. The video concludes with additional practice on finding total differentials.

What is the total differential of a function of two variables?

• The sum of the derivatives of the function with respect to each variable.

• The product of the derivatives of the function with respect to each variable.

• The sum of the partial derivatives of the function with respect to each variable, each multiplied by the change in that variable.

• The difference between the partial derivatives of the function with respect to each variable.

How can differentials be used in calculus?

• To approximate the change in a function when variables change.

• To determine the maximum value of a function.

• To solve differential equations.

• To find the exact value of a function.

What is the first step in calculating the total differential of a function?

• Finding the limit of the function.

• Calculating the integral of the function.

• Determining the partial derivatives of the function.

• Finding the second derivative of the function.

What is the purpose of rewriting the function as X squared plus Y squared raised to the 1/2 power?

• To simplify the calculation of partial derivatives.

• To make the function easier to integrate.

• To solve the function for specific values of X and Y.

• To find the maximum value of the function.

When calculating differential Z, what is the role of partial derivatives?

• They are used to solve the function for a specific value.

• They are used to determine the rate of change of the function with respect to each variable.

• They are used to find the maximum and minimum values of the function.

• They are used to integrate the function over a given interval.

What is the significance of the comparison between differential Z and delta Z?

• It proves that delta Z is more accurate than differential Z.

• It shows that differential Z is always greater than delta Z.

• It highlights that differential Z can be a good approximation of delta Z.

• It demonstrates that differential Z is a perfect approximation of delta Z.

In the context of differentials, what does DX represent?

• The integral of X over a given interval.

• The change in the variable X.

• The derivative of X with respect to time.

• The maximum value of X.

What is the result of simplifying the expression for the partial derivative of Z with respect to X?

• X divided by the square root of X squared plus Y squared.

• Y divided by the square root of X squared plus Y squared.

• The square root of X squared plus Y squared.

• X times Y divided by the square root of X squared plus Y squared.

What is the final value of differential Z in the example provided?

• 5.3009

• 3.009

• 0.3

• 0.3009

What mathematical rule is applied when finding the partial derivative of a function with respect to X, given that both functions have X in them?

• The chain rule.

• The power rule.

• The product rule.

• The quotient rule.