Rate of change and tangent lines

Slides:

Advertisements

Similar presentations

Unit 6 – Fundamentals of Calculus Section 6

Advertisements

The Derivative.

Sec 3.1: Tangents and the Derivative at a Point

The Derivative 3.1. Calculus Derivative – instantaneous rate of change of one variable wrt another. Differentiation – process of finding the derivative.

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.

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.

2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

Equation of a Tangent Line

The Derivative and the Tangent Line Problem

Welcome To Calculus (Do not be afraid, for I am with you) The slope of a tangent line By: Mr. Kretz.

Equations of Tangent Lines

The derivative as the slope of the tangent line (at a point)

2.4 Rates of Change and Tangent Lines. What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited.

Warmup describe the interval(s) on which the function is continuous

The Tangent Line Problem Part of section 1.1. Calculus centers around 2 fundamental problems: 1)The tangent line -- differential calculus 2) The area.

Copyright © 2011 Pearson Education, Inc. Slide Tangent Lines and Derivatives A tangent line just touches a curve at a single point, without.

DERIVATIVES 3. DERIVATIVES In this chapter, we begin our study of differential calculus.  This is concerned with how one quantity changes in relation.

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

Calculus 2.1 Introduction to Differentiation

Rates of Change and Tangent Lines

2.4 RATES OF CHANGE & TANGENT LINES. Average Rate of Change  The average rate of change of a quantity over a period of time is the slope on that interval.

AP CALCULUS 1005: Secants and Tangents. Objectives SWBAT determine the tangent line by finding the limit of the secant lines of a function. SW use both.

Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

Derivatives in physics. Why Do We Need Derivatives? In physics things are constantly changing. Specifically, what we’ll be interested with is how physical.

Rates of Change and Tangent Lines Section 2.4. Average Rates of Change The average rate of change of a quantity over a period of time is the amount of.

1.4 – Differentiation Using Limits of Difference Quotients

16.3 Tangent to a Curve. (Don’t write this! ) What if you were asked to find the slope of a curve? Could you do this? Does it make sense? (No, not really,

2.4 Rates of Change and Tangent Lines

Kinematics with Calculus: dervitives

RATES OF CHANGE & TANGENT LINES DAY 1 AP Calculus AB.

+ Section Average velocity is just an algebra 1 slope between two points on the position function.

Unit 1 Limits. Slide Limits Limit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily.

2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.

2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.

2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change.

Chapter 2 Review Calculus. Quick Review 1.) f(2) = 0 2.) f(2) = 11/12 3.) f(2) = 0 4.) f(2) = 1/3.

4.2 Mean Value Theorem Objective SWBAT apply the Mean Value Theorem and find the intervals on which a function is increasing or decreasing.

1.8 The Derivative as a Rate of Change. An important interpretation of the slope of a function at a point is as a rate of change.

AIM : How do we find tangent lines to a graph? Do Now: Find the slope of the line shown below that goes through points (0,6) and (3,0). HW2.1: p66-67 #9,10,19,22,23.

Warm Up Determine a) ∞ b) 0 c) ½ d) 3/10 e) – Rates of Change and Tangent Lines.

AP CALCULUS 1006: Secants and Tangents. Average Rates of Change The AVERAGE SPEED (average rate of change) of a quantity over a period of time is the.

OBJECTIVES: To introduce the ideas of average and instantaneous rates of change, and show that they are closely related to the slope of a curve at a point.

2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming.

Objectives Determine tangent lines to functions for a given point Compute the slope of curves Compute Instantaneous rate of change.

Section 1.4 The Tangent and Velocity Problems. WHAT IS A TANGENT LINE TO THE GRAPH OF A FUNCTION? A line l is said to be a tangent to a curve at a point.

Section 2.4 Rates of Change and Tangent Lines Calculus.

Rates of Change and Tangent Lines Devil’s Tower, Wyoming.

1 10 X 8/30/10 8/ XX X 3 Warm up p.45 #1, 3, 50 p.45 #1, 3, 50.

Warm Up a) What is the average rate of change from x = -2 to x = 2? b) What is the average rate of change over the interval [1, 4]? c) Approximate y’(2).

2-4 Rates of change & tangent lines

2.1 Tangents & Velocities.

Rate of Change.

Pg 869 #1, 5, 9, 12, 13, 15, 16, 18, 20, 21, 23, 25, 30, 32, 34, 35, 37, 40, Tangent to a Curve.

2.1A Tangent Lines & Derivatives

Rate of change and tangent lines

Sec 2.7: Derivative and Rates of Change

Definition of the Derivative

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

2.2C Derivative as a Rate of Change

2 Differentiation 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE

Section 3.2 Calculus AP/Dual, Revised ©2017

2.7/2.8 Tangent Lines & Derivatives

Tangent Line Recall from geometry

2.4 Rates of Change & Tangent Lines

30 – Instantaneous Rate of Change No Calculator

Presentation transcript:

Rate of change and tangent lines

The average rate of change of a function over an interval is the amount of change divided by the length of the interval. On a graph this is equal to the slope of a secant line

This graph shows the temperature of a cup of coffee over a 30 minute period. What is the average rate the coffee cools during the 1st 20 minutes?

When the coffee was 1st made the temperature was After 20 minutes the temperature was

The line connecting these two points is a secant line.

Find the slope of this line Find the slope of this line. This will be the average change in temperature.

The slope of the tangent line gives the instantaneous rate of change. Find the instantaneous rate of change of the temperature of the coffee at 5 min.?

A tangent line in geometry is a line that touches a circle in exactly one point. This is not always the same in calculus. Think of it as a line that goes in the direction of the curve if you zoom really close to the curve. When you zoom in close enough to any curve it will appear to be a straight line. This line is the same as the tangent line at that point.

Click Here to see an example of tangent lines

The slope of the tangent line at x=5 will give the instantaneous rate of change

The slope of the tangent line will give the instantaneous rate of change

Find the average rate of change of Over the interval

Find the slope of the secant

Find the slope of the secant The average rate of change from [-3,2] is 3.

Lets find the instantaneous rate of change at x=3 That is find the slope of the tangent line at x=3

In the last example I drew in the tangent line and found some points on it to find the slope. This time we will find it a little more exact.

The point (3,0) is on the graph A generic point on the graph is Plug 3 into the equation The point (3,0) is on the graph A generic point on the graph is

Find the slope of the secant line between the point (3,0) and the generic point

What happens when we slide the generic point closer to the point (3,0)? Click on when to see

You can graph this on your calculator to find the limit. If we take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You should have gotten -12 You should have gotten -12. This is the instantaneous rate of change at x=3, the slope of the tangent at x=3 and the derivative at x=3

Now lets find the derivative another way. Graph the original function and not the slope of the secant like we did previously

Lets Review Find the average rate of change over the given interval.

Find the derivative at x = 3 using the limit then check with the calculator. 7