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

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

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

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

Slope between x = 3 and x =5 for the function: f(x) = x2 – 4 Example 1

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

Example 2 Points: (-1,-3) and (3,5) Find an equation of the secant to: f(x) = x2 – 4 when x = -1 and x = 3. 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+h,f(x+h)) h (x,f(x))

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

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

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

Example 4 Derivative: Find the derivative of f(x) = x2

Example 5 Derivative: Find the derivative of:

Example 6 Slope = Derivative: Find the equation of the line tangent to f(x) = x2 when x = 3 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

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