The Derivative and Tangent Line Problem Section 2

  • Slides: 20
Download presentation
The Derivative and Tangent Line Problem Section 2. 1 AP Calc

The Derivative and Tangent Line Problem Section 2. 1 AP Calc

Given two points on f(x) how do you find the slope of the secant

Given two points on f(x) how do you find the slope of the secant line between them?

Definition: Tangent Line with slope m If f is defined on an open interval

Definition: Tangent Line with slope m If f is defined on an open interval containing c, and if the limit, exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at point (c, f(c)).

Find the slope of the tangent line of y=3 x-4 at the point (6,

Find the slope of the tangent line of y=3 x-4 at the point (6, 14).

Find the slopes of the tangent lines to the graph g(x)=5 -x².

Find the slopes of the tangent lines to the graph g(x)=5 -x².

Find the equation of the tangent line of at (-1, -2).

Find the equation of the tangent line of at (-1, -2).

Definition: Derivative of a Function The derivative of f at x is given by

Definition: Derivative of a Function The derivative of f at x is given by provided the limit exists. For all x for which the limit exists, f’ is a function of x.

The process of finding a derivative = differentiation

The process of finding a derivative = differentiation

Find the derivative using the limit process: f(x)=1 - x²

Find the derivative using the limit process: f(x)=1 - x²

Find for

Find for

Find the derivative with respect to t for the function y = t² +

Find the derivative with respect to t for the function y = t² + 2 t + 1

Alternative limit form of the derivative:

Alternative limit form of the derivative:

The function must be differentiable from the left and the right: (these limits must

The function must be differentiable from the left and the right: (these limits must both exist and be equal)

f is differentiable on the closed interval [a, b] if it is differentiable on

f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a and from the left at b both exist.

If a function is not continuous at x=c, then it is not differentiable at

If a function is not continuous at x=c, then it is not differentiable at x=c.

Use the alternate form to find the derivative at x=c for , when c=-5.

Use the alternate form to find the derivative at x=c for , when c=-5.

Look at the greatest integer function on page 100

Look at the greatest integer function on page 100

Given Is the function continuous at x=4? Find Vertical Tangent, not differentiable

Given Is the function continuous at x=4? Find Vertical Tangent, not differentiable

Thm 2. 1 Differentiability Implies Continuity If f is differentiable at x=c, then f

Thm 2. 1 Differentiability Implies Continuity If f is differentiable at x=c, then f is continuous at x=c. Contrapositive also true: If f is not continuous, then it is not differentiable.

It is possible to have a function that is continuous, but not differentiable.

It is possible to have a function that is continuous, but not differentiable.

The derivative and the tangent line problem 2The derivative and the tangent line problem 2
Derivative and the Tangent Line Problem The beginningsDerivative and the Tangent Line Problem The beginnings
2 72 8 TANGENT LINES DERIVATIVES TANGENT LINE2 72 8 TANGENT LINES DERIVATIVES TANGENT LINE
Tangent lines n Recall tangent line is theTangent lines n Recall tangent line is the
Section 2 1 The Derivative Tangent Lines andSection 2 1 The Derivative Tangent Lines and
Tangent and Cotangent Graphs Reading and Drawing TangentTangent and Cotangent Graphs Reading and Drawing Tangent
Tangent and Cotangent Graphs Reading and Drawing TangentTangent and Cotangent Graphs Reading and Drawing Tangent
10 1 Tangent Ratios Obj Use tangent to10 1 Tangent Ratios Obj Use tangent to
11 1 Tangent Lines Chapter 11 Circles Tangent11 1 Tangent Lines Chapter 11 Circles Tangent
The Tangent Ratio The Tangent using Angle TheThe Tangent Ratio The Tangent using Angle The
Tangent Line EXDetermine the slope of the lineTangent Line EXDetermine the slope of the line
Econom Line Slim Line Standart Line Capri LineEconom Line Slim Line Standart Line Capri Line
Definition of the Derivative Lesson 3 4 TangentDefinition of the Derivative Lesson 3 4 Tangent
Derivatives and its variants Derivative Definition A derivativeDerivatives and its variants Derivative Definition A derivative
Section 3 9 Differentials Tangent Line Approximations ASection 3 9 Differentials Tangent Line Approximations A
Hedging Risk Derivative Securities Derivative Markets l lHedging Risk Derivative Securities Derivative Markets l l
3 1 Derivatives Derivative A derivative of a3 1 Derivatives Derivative A derivative of a
2 5 Derivative Markets Derivative AssetContingent Claim Security2 5 Derivative Markets Derivative AssetContingent Claim Security
Lecture 6 More On The Derivative Rules DerivativeLecture 6 More On The Derivative Rules Derivative
Extremas First Derivative Test Second Derivative Test CalculusExtremas First Derivative Test Second Derivative Test Calculus
Derivatives and the Tangent Line Blast from theDerivatives and the Tangent Line Blast from the
RESOURCE 10 EXTERNAL TANGENT LINE FROM CIRCLE TORESOURCE 10 EXTERNAL TANGENT LINE FROM CIRCLE TO
Application of Differentation Equation of the tangent lineApplication of Differentation Equation of the tangent line
Find the slope of the tangent line toFind the slope of the tangent line to
Example Helical Surface of Helical Tangent Line AdditionalExample Helical Surface of Helical Tangent Line Additional
Using a tangent line approximation of the functionUsing a tangent line approximation of the function
Line Drawing by Algorithm Line Drawing Algorithms LineLine Drawing by Algorithm Line Drawing Algorithms Line
Line Tag Data Index Set Line Line TagLine Tag Data Index Set Line Line Tag
Measure Line Segments Unlike a line a lineMeasure Line Segments Unlike a line a line
Model Validation For Derivative Pricing The First LineModel Validation For Derivative Pricing The First Line
October 5 2013 Proportional Integral Derivative Line FollowingOctober 5 2013 Proportional Integral Derivative Line Following
Section 2 1 Tangent Functions Copyright all rightsSection 2 1 Tangent Functions Copyright all rights
Section 1 Problem Section 2 Experiment Design SectionSection 1 Problem Section 2 Experiment Design Section
Chapter 10 Limits and the Derivative Section 7Chapter 10 Limits and the Derivative Section 7
Chapter 10 Limits and the Derivative Section 1Chapter 10 Limits and the Derivative Section 1
Section 4 6 The Derivative in Graphing andSection 4 6 The Derivative in Graphing and
Section 4 3 The Derivative in Graphing andSection 4 3 The Derivative in Graphing and
Chapter 3 Limits and the Derivative Section 1Chapter 3 Limits and the Derivative Section 1
Section 4 5 The Derivative in Graphing andSection 4 5 The Derivative in Graphing and
Section 4 1 The Derivative in Graphing andSection 4 1 The Derivative in Graphing and