Derivatives and Tangent Lines 2

This lesson builds upon the previous lesson on derivatives by introducing tangent lines to curves. The concept of secant lines is used to approximate the slope of the tangent line at a given point, which can be made more precise by taking the limit as the distance between the two points approaches zero. The formula for the slope of the tangent line is introduced, and an example problem is solved step-by-step to find the slope of the tangent line to a given curve at a specific point. By the end of the lesson, viewers will have a solid understanding of how to calculate the slope of tangent lines to curves.

An introduction to derivatives and tangent lines to curves.

• How do you find the slope of the tangent line of f(x) = x^2 - 3 at (1, -2)?
• How can you use limits and the formula for slope to find the exact slope of the tangent line at a point on a curve?
• If you have a graph y = f(x), how do you find the slope between (c, f(c)) and (c + h, f(c + h)) if c is very small?
• How do you derive the formal definition of a derivative?
• What is the formal definition of a derivative?
• What is the definition of a tangent line with slope m?
• How do you find the limit as h goes to 0 of (f(1 + h) - f(1))/h?
• How do you find the slope of a tangent line by using the formal limit definition of a derivative?
• What is the limit as h goes to 0 of 2 + h?