Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park 2.1 The Derivative and the Tangent Line Problem (Part.

Slides:

Advertisements

Similar presentations

3.2 Differentiability Photo by Vickie Kelly, 2003 Arches National Park Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry.

Advertisements

6.4 day 1 Separable Differential Equations

DERIVATIVE OF A FUNCTION 1.5. DEFINITION OF A DERIVATIVE OTHER FORMS: OPERATOR:,,,

2.1 Tangent Line Problem. Tangent Line Problem The tangent line can be found by finding the slope of the secant line through the point of tangency and.

2.1 The derivative and the tangent line problem

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

Rolle’s Theorem and the Mean Value Theorem3.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by.

1 The Derivative and the Tangent Line Problem Section 2.1.

The Derivative As A Function. 2 The First Derivative of f Interpretations f ′(a) is the value of the first derivative of f at x = a. f ′(x) is.

3.9: Derivatives of Exponential and Logarithmic Functions Mt. Rushmore, South Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

3.2 Differentiability Arches National Park.

3.6 A Summary of Curve Sketching Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1995 Old Faithful Geyser, Yellowstone National.

4.3 Using Derivatives for Curve Sketching Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1995 Old Faithful Geyser, Yellowstone.

TI89 and Differentiability Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park.

Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2.

2.3 Continuity Grand Canyon, Arizona Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.

Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.

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.

10.2 day 2 Vector Valued Functions Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006 Everglades National Park, FL.

Differentiability Arches National Park- Park Avenue.

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 London Bridge, Lake Havasu City, Arizona 2.3 Product and Quotient Rules.

2.1 Day Differentiability.

3.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

3.8 Derivatives of Inverse Trig Functions Lewis and Clark Caverns, Montana Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly,

Limits Calculus 2.1b part 2 of 2. 9/13/2012 – Today’s Learning Objective: Differentiability and Continuity Find the derivative with respect to t for the.

10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National.

Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

5.5 Bases Other than e and Applications (Part 1) Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008 Acadia National Park,

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 London Bridge, Lake Havasu City, Arizona 3.4 Derivatives of Trig Functions.

2.1 The Derivative and the Tangent Line Problem.

Bell-ringer 11/2/09 Suppose that functions f and g and their derivatives with respect to x have the following values at x=0 and x=1. 1.Evaluate the derivative.

Lesson 3.2 Differentiability AP Calculus Mrs. Mongold.

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity.

The Second Derivative Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Photo by Vickie Kelly, 2003 Arches.

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

2.1 The Derivative and the Tangent Line Problem (Part 1) Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto.

3.3 Differentiation Rules Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

2.1 The Derivative and the Tangent Line Problem Objectives: -Students will find the slope of the tangent line to a curve at a point -Students will use.

2.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

2.3 The Product and Quotient Rules (Part 1)

London Bridge, Lake Havasu City, Arizona

2.7 and 2.8 Derivatives Great Sand Dunes National Monument, Colorado

2.1 Tangent Line Problem.

Mean Value Theorem for Derivatives

3.2 Differentiability Arches National Park

The Derivative as a Function

3.2 Differentiability.

2.1 The Derivative and the Tangent Line Problem (Part 2)

5.1 (Part II): The Natural Logarithmic Function and Differentiation

London Bridge, Lake Havasu City, Arizona

3.2 Differentiability Arches National Park - Park Avenue

5.4: Exponential Functions: Differentiation and Integration

3.2: Differentiability.

5.3 Inverse Function (part 2)

The Derivative (cont.) 3.1.

2.1B Derivative Graphs & Differentiability

2.2 Basic Differentiation Rules and Rates of Change (Part 1)

5.1 (Part II): The Natural Logarithmic Function and Differentiation

3.2 Differentiability Arches National Park

2.4 The Chain Rule (Part 2) Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

4.2 Area Greenfield Village, Michigan

8.8 Improper Integrals Greg Kelly, Hanford High School, Richland, Washington.

2.2 Basic Differentiation Rules and Rates of Change (Part 1)

5.1 (Part I): The Natural Logarithmic Function and Differentiation

3.2 Differentiability Arches National Park

2.1 The Derivative and the Tangent Line Problem (Part 2)

5.5 Bases Other than e and Applications (Part 2)

5.3 Inverse Function (part 2)

London Bridge, Lake Havasu City, Arizona

Presentation transcript:

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park 2.1 The Derivative and the Tangent Line Problem (Part 2)

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park

Objectives Understand the relationship between differentiability and continuity.

If f is differentiable at x = c, then f is continuous at x = c. Differentiability implies continuity. If a function is NOT continuous at x=c, then it is NOT differentiable. Is the converse true? No Theorem 2.1

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

Graph with a Sharp Turn f '(2) DNE (the tangent line is not unique)

Graph with a Sharp Turn The graph is continuous at x=0, but f ' (0) DNE.

Most of the functions we study in calculus will be differentiable. So f ' (x) does NOT exist (or f is not differentiable) if the graph has a sharp corner or turn, a vertical tangent line, or a discontinuity.

Homework 2.1 (page 102) #33, all, 61, odd