Directional Derivatives and Gradients

This video tutorial introduces directional derivatives, explaining their significance in determining the slope of tangent lines in any direction for functions of two variables. It reviews partial derivatives and introduces the concept of the gradient. The video provides a formula for calculating directional derivatives using unit vectors and demonstrates this through two examples, highlighting the practical application of these concepts.

What does the first order partial derivative of a function of two variables represent?

• The slope of the tangent line in any direction

• The slope of the tangent line in the positive x or y direction

• The maximum value of the function

• The curvature of the surface

What is required to find the slope of a tangent line in a direction other than the positive x or y axis?

• Second order derivatives

• Integral calculus

• Partial derivatives

• Directional derivatives

Which of the following is a component of the working formula for directional derivatives?

• The sum of the function values

• The integral of the function

• The product of the function values

• The partial derivative with respect to x times cosine theta

What is the gradient of a function?

• The integral of the function

• A constant value

• A vector formed by the partial derivatives with respect to x and y

• A scalar value representing the function's maximum

How is the directional derivative of a function in the direction of a unit vector calculated?

• By adding the function values

• By multiplying the function values

• By taking the dot product of the gradient and the unit vector

• By integrating the function

In the first example, what is the directional derivative of the function at the point (3, 2) in the direction of theta equals pi over four?

• Positive one point four

• Negative square root of two

• Positive square root of two

• Zero

What does the red vector represent in the graphical representation of the first example?

• The direction of the maximum slope

• The direction of the directional derivative

• The direction of the minimum slope

• The direction of the integral

In the second example, what is the unit vector in the direction of the vector with components (3, 4)?

• (2/3, 1/3)

• (1/2, 1/2)

• (4/5, 3/5)

• (3/5, 4/5)

What is the directional derivative of the function in the second example at the point (1, 2, 3)?

• Seventeen fifth

• Nine fifth

• Zero

• Four fifth

Tags

CCSS.8.EE.B.5

In the second example, what does the steep upward movement of the line indicate?

• A negative slope

• A positive slope

• A constant slope

• A zero slope

Tags

CCSS.8.EE.B.5