Differential and True Change in Functions

The video tutorial explains how to find the differential of a function f(x, y) and compare it with the true change in z, delta z. It begins by introducing the concept of differentials and the tangent line. The tutorial then details the process of calculating partial derivatives with respect to x and y, treating the other variable as a constant. It proceeds to evaluate differential z using these partial derivatives and given delta x and delta y values. Finally, the tutorial calculates the true change in z, delta z, by substituting values into the original function and compares it with differential z, highlighting the approximation's limitations.

What is the main objective of the problem discussed in the video?

• To find the integral of a function

• To solve a quadratic equation

• To find the differential of a function and compare it to the true change

• To determine the maximum value of a function

What are the given changes in x and y for the calculation?

• Delta x = 0.5 and Delta y = -0.25

• Delta x = -0.25 and Delta y = 0.5

• Delta x = 0.25 and Delta y = 0.15

• Delta x = -0.5 and Delta y = 0.25

What is the ending point after applying the changes in x and y?

• (4.25, 1.5)

• (3.5, 1.75)

• (4.5, 1.25)

• (3.75, 1.15)

How is the partial derivative of f with respect to x calculated?

• By differentiating with respect to both x and y simultaneously

• By integrating with respect to x

• By treating y as a constant and differentiating with respect to x

• By treating x as a constant and differentiating with respect to y

What is the partial derivative of f with respect to y?

• 6x e^(3y)

• 2x e^(3y)

• 4x e^(3y)

• 3x e^(2y)

What is the formula for calculating differential z?

• Partial derivative of f with respect to x times delta y

• Partial derivative of f with respect to x times differential y plus partial derivative of f with respect to y times differential x

• Partial derivative of f with respect to y times delta x

• Partial derivative of f with respect to x times differential x plus partial derivative of f with respect to y times differential y

What is the approximate value of differential z calculated in the video?

• 80.45678

• 50.12345

• 62.26516

• 75.56864

What is the true change in z, denoted as delta z?

• 62.26516

• 75.56864

• 50.12345

• 80.45678

Why is differential z not a good approximation for delta z in this example?

• Because the function is linear

• Because the changes in x and y are too large

• Because the function is not continuous

• Because the changes in x and y are too small

What would improve the accuracy of using differential z to approximate delta z?

• Calculating the integral instead

• Using a different function

• Decreasing the changes in x and y

• Increasing the changes in x and y