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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 a point on the curve Point A is the point of tangency
Tangent Line Problem How to find slope of a curve at a point? xx + Δx Secant Line Tangent Line
xx + Δx Setting up a limit! Slope of the Tangent Line
1.) Find slope of the secant line x x + Δx Secant Line
xx + Δx Called the difference quotient Conclusion:
For a function f(x) the average rate of change along the function is given by: Which is called the derivative of f Definition of the Derivative
Notation of the Derivative The derivative of a function at x is given by: **Provided the limit exists Notation:
2.) Find the slope of the tangent line to the curve at (2,6) First, find the Slope at any point
Terminology Differentiation (Differentiate) – the process of finding the derivative Differentiable – when a functions derivative exists at x
When Derivatives Fail 1.Cusp or sharp point: cusp
When Derivatives Fail 2.Vertical asymptotes: 3.When one sided limits fail
When Derivatives Fail 4.Removable discontinuity
When Derivatives Fail 5. Corners or vertical tangents
3.) Differentiate (if possible)
4.) Differentiate (if possible)
5.) Differentiateif possible
6.) Find the derivative of
HOMEWORK Page 104 # 5 – 21 (odd), 61 and 62, 83-88 (all). Find where f(x) is not differentiable and state the type of discontinuity
DERIVATIVE OF A FUNCTION 1.5. DEFINITION OF A DERIVATIVE OTHER FORMS: OPERATOR:,,,
Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.
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SECTION 3.1 The Derivative and the Tangent Line Problem.
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1 The Derivative and the Tangent Line Problem Section 2.1.
2.1 The Derivative and The Tangent Line Problem The Definition of a Derivative.
THE DERIVATIVE AND THE TANGENT LINE PROBLEM Section 2.1.
Sec 3.1: Tangents and the Derivative at a Point The difference quotient of ƒ at x 0 with increment h. Example: Find the difference quotient of ƒ at x0=2.
1.4 – Differentiation Using Limits of Difference Quotients The Difference Quotient is used to find the average rate of change between two points. The Difference.
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Differentiable vs. Continuous The process of finding the derivative of a function is called Differentiation. A function is called Differentiable at x if.
Pre-Calculus Limits Calculus. Objectives: 1.Discuss slope and tangent lines. 2.Be able to define a derivative. 3.Be able to find the derivative of various.
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.
1.5 Cusps and Corners When we determine the derivative of a function, we are differentiating the function. For functions that are “differentiable” for.
Sec 3.2: The Derivative as a Function If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION.
Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.
AP Calculus Review Definition(s) of the Derivative.
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,
AP Calculus AB/BC 3.2 Differentiability, p. 109 Day 1.
Warm Ups. AP Calculus 3.1 Tangent Line Problem Objective: Use the definition to find the slope of a tangent line.
Limit Definition of the Derivative. Objective To use the limit definition to find the derivative of a function. TS: Devoloping a capacity for working.
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Differentiability and Rates of Change. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical.
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2.1 The Derivative and the Tangent Line Problem Main Ideas Find the slope of the tangent line to a curve at a point. Use the limit definition to find the.
Section 3.2 Calculus Fall, Definition of Derivative We call this derivative of a function f. We use notation f’(x) (f prime of x)
Differentiability and Piecewise Functions. What are the three things that make a function not differentiable ? Not continuous at the point Vertical tangent.
Chapter 2 Section 2 The Derivative!. Definition The derivative of a function f(x) at x = a is defined as f’(a) = lim f(a+h) – f(a) h->0 h Given that a.
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The Tangent Line Problem “And I dare say that this is not only the most useful and general problem in geometry that I know, but even that I ever desire.
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park 2.1 The Derivative and the Tangent Line Problem (Part.
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Mrs. Rivas International Studies Charter School.Objectives: slopes and equations 1.Find slopes and equations of tangent lines. derivative of a function.
Warmup 1)describe the interval(s) on which the function is continuous 2) which of the following points of discontinuity of are not removable? a)0c) -2e)
Differentiability 2.1 Day HWQ The slope of the curve at any point is: Slope at any point on the graph of a function:
Section 3.2 The Derivative as a Function AP Calculus September 24, 2009 Berkley High School, D2B2.
2.1 The Derivative and the Tangent Line Problem Devil’s Tower, Wyoming.
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.
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