Understanding Tangent Lines and Derivatives

This video tutorial explains how to find the slope and equation of a tangent line at a point using derivatives and limits. It covers the power rule for derivatives, provides examples, and demonstrates how to use limits to derive the slope of a tangent line. The video also discusses using the average rate of change to approximate tangent line slopes and explains how to find the equation of a normal line, which is perpendicular to the tangent line.

What is the primary method introduced for finding the slope of a tangent line?

• Using algebraic equations

• Using trigonometry

• Using integrals

• Using derivatives

According to the power rule, what is the derivative of x^5?

• 5x^5

• 5x^4

• x^4

• 4x^5

What is the derivative of a constant, such as 5 or -7?

• One

• Zero

• The constant itself

• The constant times x

How do you find the slope of the tangent line at a specific point using derivatives?

• By using the quadratic formula

• By integrating the function

• By finding the derivative and substituting the x-value

• By solving the function for y

What is the slope of the tangent line for the function f(x) = 3x^2 + 5 at x = 2?

• 6

• 12

• 9

• 15

What is the limit definition of a derivative?

• The limit as h approaches zero of [f(x+h) - f(x)]/h

• The slope of a secant line

• The integral of a function

• The average rate of change

How can the average rate of change be used in calculus?

• To find the integral of a function

• To approximate the instantaneous rate of change

• To determine the maximum value of a function

• To solve quadratic equations

What is the relationship between the slope of a secant line and a tangent line?

• They are unrelated

• The tangent line slope is always greater

• The secant line slope approximates the tangent line slope as the interval decreases

• They are always equal

What happens to the secant line as the two points get closer to the tangent point?

• It becomes parallel to the tangent line

• It intersects the tangent line at multiple points

• It becomes perpendicular to the tangent line

• It moves away from the tangent line

What is the slope of the tangent line for f(x) = x^3 - 2x + 4 at x = 1?

• 2

• 1

• 4

• 3