Understanding Normal Lines in Calculus

This video tutorial explains how to find the equation of a normal line to a curve. It begins with an introduction to the concept of normal lines and their relationship to tangent lines. The video then demonstrates how to calculate the slope of a normal line using derivatives and the negative reciprocal of the tangent line's slope. It provides a step-by-step guide to deriving the equation of a normal line using the point-slope formula. Two example problems are solved to reinforce the concepts, showing how to apply the methods to different curves and points.

What is the relationship between a tangent line and a normal line?

• They intersect at two points.

• They are perpendicular to each other.

• They are parallel to each other.

• They have the same slope.

If the slope of a tangent line is 4, what is the slope of the normal line?

• 4

• -4

• 1/4

• -1/4

How do you find the y-coordinate of a point on a curve?

• By using the point-slope formula.

• By finding the slope of the tangent line.

• By substituting the x-coordinate into the curve equation.

• By differentiating the curve equation.

What is the first step in finding the slope of the tangent line?

• Take the first derivative of the function.

• Use the point-slope formula.

• Find the y-intercept of the curve.

• Calculate the second derivative.

What is the slope of the normal line if the slope of the tangent line is -4?

• 1/4

• -1/4

• -4

• 4

Which formula is used to write the equation of a line given a point and a slope?

• Slope-intercept formula

• Standard form equation

• Point-slope formula

• Quadratic formula

In the second example, what is the y-coordinate when x is 3?

• 7

• 3

• 15

• 5

What is the first derivative of the function y = 3x^2 - 5x - 7?

• 3x^2 - 5

• 2x - 5

• 6x - 5

• 6x^2 - 5x

How is the slope of the normal line calculated from the slope of the tangent line?

• By multiplying the tangent slope by 2.

• By adding 1 to the tangent slope.

• By subtracting 1 from the tangent slope.

• By taking the negative reciprocal of the tangent slope.

What is the final form of the equation of the normal line in the second example?

• Slope-intercept form

• Standard form

• Quadratic form

• Point-slope form