Partial Derivatives and Directional Derivatives

The video tutorial explains how to find the directional derivative of a function at a given point in the direction of a specified vector. It begins with an introduction to the problem, followed by a graphical representation to aid understanding. The tutorial then details the process of calculating partial derivatives with respect to x and y, evaluating these derivatives at the given point, and finally determining the unit vector to compute the directional derivative. The result is both an exact value and a decimal approximation, with a graphical explanation of why the derivative is negative.

What is the function f(x, y) given in the problem?

• Natural log of x times y

• Natural log of x squared plus y squared

• Natural log of x plus y

• x squared plus y squared

In which direction is the directional derivative to be calculated?

• Vector with components 1, 3

• Vector with components 2, -2

• Vector with components 3, 1

• Vector with components -2, 2

What is the purpose of finding the z-coordinate in the graphical explanation?

• To find the slope of the tangent line

• To determine the height of the surface at the point

• To evaluate the function at the origin

• To calculate the directional vector

What is the partial derivative of f with respect to x at the point (3, 1)?

• 1/5

• 3/5

• 2/5

• 6/10

How is the partial derivative with respect to y evaluated at the point (3, 1)?

• 2/5

• 3/5

• 6/10

• 1/5

What is the magnitude of the vector with components -2, 2?

• 2

• Square root of 4

• 4

• Square root of 8

What is the x-component of the unit vector in the direction of the given vector?

• -1 divided by square root of 2

• 1 divided by square root of 2

• -2 divided by square root of 2

• 2 divided by square root of 2

What is the value of cosine theta for the unit vector?

• 1 divided by square root of 2

• -1 divided by square root of 2

• 2 divided by square root of 2

• -2 divided by square root of 2

What is the final value of the directional derivative?

• -3 divided by 5 square root of 2

• -2 divided by 5 square root of 2

• 3 divided by 5 square root of 2

• 2 divided by 5 square root of 2

Why is the directional derivative negative in this example?

• The line is slanted upwards

• The line is slanted downwards

• The vector is in the negative direction

• The vector is in the positive direction