Calculus 2413 Ch 3 Section 1 Slope, Tangent Lines, and Derivatives

Objectives Find the slope of a secant line Find the slope of a tangent line Find the equation of a line tangent to a curve at a point Find the derivative of an equation

Slope of a line The slope of the line between points (a,f(a)) and (b,f(b)) of the function is:

Example 1 Slope between x = 3 and x =5 for the function: f(x) = x 2 – 4

Secant Line A line that goes through two points on a curve.

Example 2 Find an equation of the secant to: f(x) = x 2 – 4 when x = -1 and x = 3. Points: (-1,-3) and (3,5) Slope: Equation:

Generic Secant Line For any function f(x) find the slope of the secant line through: (x,f(x)) and (x+h,f(x+h) (x,f(x)) (x+h,f(x+h)) h

Generic Secant Line Points: (x,f(x)) and (x+h,f(x+h) (x,f(x)) (x+h,f(x+h)) Slope:

When the two points move very close together we have h->0. Write that limit. (x,f(x)) This is the slope of the tangent line – also known as the derivative

Example 3 Find the slope of the line tangent to f(x) = x + 1 at (1,2) Slope:

Example 4 Find the derivative of f(x) = x 2 Derivative:

Example 5 Find the derivative of: Derivative:

Example 6 Find the equation of the line tangent to f(x) = x 2 when x = 3 Slope = Derivative: Point on Curve: Equation:

Derivatives and graphs a b c d e Derivative Graph: a b c d e

Omit from Assignment: #4, 5, 8, 11, 14, 18, 19, 21