Directional Derivatives and Partial Derivatives

The video tutorial explains how to find the directional derivative of a function f(x, y) = x^3y^2 in the direction of a given vector at a specific point. It begins with a graphical representation to help visualize the problem, followed by a mathematical derivation of the directional derivative formula. The tutorial then calculates the partial derivatives of the function at the given point and finds the unit vector in the direction of the given vector. Finally, it combines these elements to compute the directional derivative and provides a decimal approximation of the result.

What is the function given in the problem for which the directional derivative needs to be found?

• f(x, y) = x^2 y^3

• f(x, y) = x^3 y^2

• f(x, y) = x^2 + y^2

• f(x, y) = x^3 + y^3

In which direction is the directional derivative to be calculated?

• Vector (1, 2)

• Vector (3, -2)

• Vector (-1, 1)

• Vector (2, 3)

What does the directional derivative represent graphically?

• The area under the curve

• The height of the surface

• The slope of the tangent line

• The volume of the solid

What is the formula for the directional derivative in terms of partial derivatives and trigonometric functions?

• Partial derivative with respect to x times sine theta minus partial derivative with respect to y times cosine theta

• Partial derivative with respect to x times cosine theta plus partial derivative with respect to y times sine theta

• Partial derivative with respect to x times cosine theta minus partial derivative with respect to y times sine theta

• Partial derivative with respect to x times sine theta plus partial derivative with respect to y times cosine theta

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

• 2x^3 y

• 3x y^2

• 3x^2 y^2

• x^3 y^2

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

• 1

• 2

• 4

• 3

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

• 2x^3 y

• 3x^2 y^2

• x^2 y^3

• x^3 y

What is the magnitude of the vector (3, -2)?

• 5

• √10

• 4

• √13

What is the x-component of the unit vector in the direction of (3, -2)?

• 2/√13

• 4/√13

• 1/√13

• 3/√13

What is the approximate value of the directional derivative to four decimal places?

• 3.6056

• 2.6056

• 4.6056

• 1.6056