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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
Tangent Lines Section 2.1.
Unit 6 – Fundamentals of Calculus Section 6
Sec 3.1: Tangents and the Derivative at a Point
Remember: Derivative=Slope of the Tangent Line.
Tangent Lines ( Sections 1.4 and 2.1 ) Alex Karassev.
2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.
Equations of Tangent Lines
Derivative and the Tangent Line Problem
Slope and Equation of a line How to find the slop of a line? (x 1, y 1 ) (x 2, y 2 ) How to find the equation of a line? Sec 2.1: Rates of Change and.
Equation of a Tangent Line
The derivative as the slope of the tangent line (at a point)
Homework Homework Assignment #11 Read Section 3.3 Page 139, Exercises: 1 – 73 (EOO), 71 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.
Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x
Rate of change and tangent lines
Rates of Change and Tangent Lines
Point Value : 20 Time limit : 2 min #1 Find. #1 Point Value : 30 Time limit : 2.5 min #2 Find.
Miss Battaglia AB Calculus. Given a point, P, we want to define and calculate the slope of the line tangent to the graph at P. Definition of Tangent Line.
Sec. 2.1: The Derivative and the Tangent Line
The Derivative Chapter 3:. What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another.
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