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

Slides:

Advertisements

Similar presentations

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

Advertisements

The Derivative.

Sec 3.1: Tangents and the Derivative at a Point

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.

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

The Derivative and the Tangent Line Problem

Chapter 2  2012 Pearson Education, Inc. Section 2.4 Rates of Change and Tangent Lines Limits and Continuity.

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.

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

Rate of change and tangent lines

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

Rates of Change and Tangent Lines

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

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.4 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.

The Derivative Chapter 3:. What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another.

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.

Business Calculus Rates of Change Types of Change  Average rate of change: the average rate of change of y with respect to x is a ratio of.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

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.

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

Everything is in motion … is changing the-Universe.jpg.

2.1 The Derivative and the Tangent Line Problem

3.2 Continuity JMerrill, 2009 Review 3.1 Find: Direct substitution causes division by zero. Factoring is not possible, so what are you going to do?

1 Discuss with your group. 2.1 Limit definition of the Derivative and Differentiability 2015 Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland,

Differentiability and Rates of Change. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical.

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.

The Tangent and Velocity Problem © John Seims (2008)

Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.

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

Chapter 3.1 Tangents and the Derivative at a Point.

11-2 Key to evens 2a) -5 2b) -3 2c) 0 4a) 0 4b) 1 4c) -2 6) -1/10 8) -5 10) 27 12) - 7/14 14) 1/8 16) 1/16 18) 0 20) 1/4 22) -1/6 24) 4 26) -1/4 28) 1.

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

§3.2 – The Derivative Function October 2, 2015.

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

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.

Section 2.1 How do we measure speed?. Imagine a ball being thrown straight up in the air. –When is that ball going the fastest? –When is it going the.

Rates of Change and Tangent Lines Chapter 2.4. Average Rates of Change 2.

§3.1 – Tangent Lines, Velocity, Rate of Change October 1, 2015.

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

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.

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

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

Ch. 2 – Limits and Continuity 2.4 – Rates of Change and Tangent Lines.

Ch. 2 – Limits and Continuity

2.4 Rates of Change and Tangent Lines

Rates of Change and Tangent Lines

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.

2.7 and 2.8 Derivatives Great Sand Dunes National Monument, Colorado

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

2.7 Derivatives and Rates of Change

Sec 2.7: Derivative and Rates of Change

Lesson 2-4: Rates of Change

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

Rates of Change and Tangent Lines

The Tangent and Velocity Problems

2.1 Limits, Rates of Change, and Tangent Lines

2.4 Rates of Change and Tangent Lines

2.7/2.8 Tangent Lines & Derivatives

2.4 Rates of Change and Tangent Lines

Sec 2.7: Derivative and Rates of Change

2.4 The Derivative.

Presentation transcript:

Warmup describe the interval(s) on which the function is continuous 2) which of the following points of discontinuity of are not removable? 0 c) -2 e) -5 6 d) 4

2.4 Rates of Change and Tangent Lines

The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).

The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?

slope slope at The slope of the curve at the point is:

The slope of the curve at the point is: is called the difference quotient of f at a. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use. Sometimes, you will see the problem already in the difference quotient, and have to figure out the limit.

The slope of a curve at a point is the same as the slope of the tangent line at that point. In the previous example, the tangent line could be found using . If you want the normal line (perpendicular line), use the negative reciprocal of the slope. (in this case, )

Example 4: Let b Where is the slope ?

Review: velocity = slope These are often mixed up by Calculus students! average slope: slope at a point: average velocity: (slope) So are these! instantaneous velocity: (slope at 1 point) If is the position function: velocity = slope

(slope of the tangent line to graph at the point

Suggested Review Problems for Ch.2 from the book: The End Day 1: p. 87 (1-5 odd, 7, 9-23 odd) Day 2: p. 87 (2-6 even, 12-24 even, 29, 30) Suggested Review Problems for Ch.2 from the book: p. 91 (6, 7, 9, 21-32, 39-43, 52)