Gradient Vector and Partial Derivatives

The video tutorial explains how to determine the gradient vector field of a function with three variables and calculate the gradient vector at a specific point. It covers the components of the gradient vector field, the use of partial derivatives, and the application of the chain rule. The tutorial also demonstrates how to find the direction of maximum increase of the function using the gradient vector.

What is the x component of the gradient vector field?

• Partial derivative of f with respect to z

• Partial derivative of f with respect to y

• Partial derivative of f with respect to x

• Partial derivative of f with respect to a constant

Which rule is applied to find each partial derivative of the function?

• Quotient rule

• Chain rule

• Power rule

• Product rule

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

• 3z squared times e raised to the power of the function

• 4y times e raised to the power of the function

• 4x times e raised to the power of the function

• e raised to the power of the function

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

• e raised to the power of the function

• 4 times e raised to the power of the function

• 4x times e raised to the power of the function

• 3z squared times e raised to the power of the function

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

• 4 times e raised to the power of the function

• e raised to the power of the function

• 3z squared times e raised to the power of the function

• 4x times e raised to the power of the function

What is the exponent on e when calculating the gradient vector at the point (2, -1, -1)?

• Two

• Negative three

• Positive three

• Zero

What is the x component of the gradient vector at the point (2, -1, -1)?

• 2 e to the third

• 3 e to the third

• 8 e to the third

• 4 e to the third

What is the y component of the gradient vector at the point (2, -1, -1)?

• 4 e to the third

• 8 e to the third

• 2 e to the third

• 3 e to the third

What is the z component of the gradient vector at the point (2, -1, -1)?

• 3 e to the third

• 8 e to the third

• 2 e to the third

• 4 e to the third

What does the gradient vector indicate in the context of the function?

• Direction of minimum increase

• Direction of maximum increase

• Direction of maximum decrease

• Direction of no change