Tangent Planes and Partial Derivatives

The video tutorial explains the concept of tangent planes to surfaces, comparing it to tangent lines in two dimensions. It covers the derivation of the tangent plane equation using partial derivatives and provides a step-by-step example of calculating a tangent plane to a given surface. The tutorial concludes with a brief mention of further derivation in another video.

What is the equation of a tangent line in 2D at a point (x1, f(x1))?

• y - f(x1) = f'(x1) * (x - x1)

• y = f(x1) + f'(x1) * (x - x1)

• y + f(x1) = f'(x1) * (x + x1)

• y = f(x1) - f'(x1) * (x - x1)

In 3D, what does the partial derivative of f with respect to x represent?

• Slope of the tangent line in the positive x direction

• Slope of the tangent line in the positive y direction

• Slope of the tangent plane in the z direction

• Slope of the tangent line in the negative x direction

What is the z-coordinate of the point of tangency in the example problem?

• 3

• 4

• 5

• 6

How do you find the partial derivative of f with respect to x?

• Differentiate f with respect to y, treating x as a constant

• Differentiate f with respect to x, treating y as a constant

• Differentiate f with respect to both x and y

• Differentiate f with respect to x, treating z as a constant

What is the value of the partial derivative of f with respect to x at the point (-2, 1)?

• 0

• -1

• 1

• 2

What is the value of the partial derivative of f with respect to y at the point (-2, 1)?

• 0

• -1

• 1

• 2

What is the equation of the tangent plane at the point (-2, 1, 5)?

• z = x + 2y - 1

• z = -x - 2y + 1

• z = x - 2y + 1

• z = -x + 2y + 1

What is the constant term in the final equation of the tangent plane?

• 0

• 1

• 2

• 3

Which of the following is true about the tangent plane equation?

• It is valid only for linear functions.

• It is only valid if the function is not differentiable at the point.

• It is valid only if the function is differentiable at the point.

• It is valid for any point on the surface.

What will be covered in another video according to the transcript?

• A different method to find tangent planes

• The derivation or proof of the tangent plane formula

• Applications of tangent planes in physics

• The history of calculus