f(a+h) Slope of the line = Average Rate of Change = f(a+h) – f(a) h

Slides:

Advertisements

Similar presentations

Definition of the Derivative Using Average Rate () a a+h f(a) Slope of the line = h f(a+h) Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) h.

Advertisements

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.

Difference Quotient (4 step method of slope) Also known as: (Definition of Limit), and (Increment definition of derivative) f ’(x) = lim f(x+h) – f(x)

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.

4.2 The Mean Value Theorem.

Chapter 3 The Derivative Definition, Interpretations, and Rules.

CHAPTER Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h  0 ( f(a + h) – f(a))

The derivative as the slope of the tangent line

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.

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?

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

1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent =

Tangent Lines 1. Equation of lines 2. Equation of secant lines 3. Equation of tangent lines.

Objectives: 1.Be able to find the derivative using the Constant Rule. 2.Be able to find the derivative using the Power Rule. 3.Be able to find the derivative.

Lesson 2-4 Tangent, Velocity and Rates of Change Revisited.

By: Jane Kim, Period 4, 2007 Source: Mr. Wiencek’s Noted from Class.

Lesson 2-8 Introduction to Derivatives. 2-7 Review Problem For a particle whose position at time t is f(t) = 6t 2 - 4t +1 ft.: b. Find the instantaneous.

Tangents. The slope of the secant line is given by The tangent line’s slope at point a is given by ax.

Math 1304 Calculus I 2.7 – Derivatives, Tangents, and Rates.

Powerpoint Templates Page 1 Powerpoint Templates Review Calculus.

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.

Derivatives Using the Limit Definition

Aaron Barker DEFINITION OF THE DERIVATIVE.  The Definition of the Derivative can be used to find the derivative of polynomials and exponential functions.

Chapter 3.2 The Derivative as a Function. If f ’ exists at a particular x then f is differentiable (has a derivative) at x Differentiation is the process.

Warmup  If y varies directly as the square of x and inversely as z and y =36 when x =12 and z =8, find x when y=4, and z=32.

Tangents, Velocities, and Other Rates of Change Definition The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope.

TANGENT LINES Notes 2.4 – Rates of Change. I. Average Rate of Change A.) Def.- The average rate of change of f(x) on the interval [a, b] is.

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

UNIT 1B LESSON 7 USING LIMITS TO FIND TANGENTS 1.

MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.2 The Derivative as a Function Copyright © 2010 by Ron Wallace, all rights reserved.

If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives.

Section 2.1 – Average and Instantaneous Velocity.

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

Calc Tangents, Velocities and Other Rates of Change Looking to calculate the slope of a tangent line to a curve at a particular point. Use formulas.

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

Calculating Derivatives From first principles!. 2.1 The Derivative as a Limit See the gsp demo demodemo Let P be any point on the graph of the function.

4.2 The Mean Value Theorem.

3.2 Rolle’s Theorem and the

Lesson 3.2 Rolle’s Theorem Mean Value Theorem 12/7/16

Mean Value Theorem.

Ch. 2 – Limits and Continuity

MTH1150 Tangents and Their Slopes

Ch. 11 – Limits and an Introduction to Calculus

Find the equation of the tangent line for y = x2 + 6 at x = 3

Warm-Up: October 2, 2017 Find the slope of at.

Sec 2.7: Derivative and Rates of Change

The Derivative and the Tangent Line Problems

3.2 Rolle’s Theorem and the

Average Rate vs. Instantaneous Rate

Tangent line to a curve Definition: line that passes through a given point and has a slope that is the same as the.

Exam2: Differentiation

Prep Book Chapter 5 – Definition of the Derivative

Tangent line to a curve Definition: line that passes through a given point and has a slope that is the same as the.

Definition of a Derivative

2.7/2.8 Tangent Lines & Derivatives

Derivatives: definition and derivatives of various functions

Prep Book Chapter 5 – Definition pf the Derivative

2.2: Formal Definition of the Derivative

Section 2.1 – Average and Instantaneous Velocity

30 – Instantaneous Rate of Change No Calculator

MATH 1314 Lesson 6: Derivatives.

The Intermediate Value Theorem

What is a limit?.

The derivative as the slope of the tangent line

§2.7. Derivatives.

Differentiation from first principles

Difference Quotient (4 step method of slope)

Differentiation from first principles

Miss Battaglia AB Calculus

Presentation transcript:

Definition of the Derivative Using Average Rate (Page 129 - 133 and 160 in the book) f(a+h) Slope of the line = Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) f(a) h h a a+h

Now, Watch what happens when: Point a is fixed and the size of the interval h shrinks a+h h a a+h h a a+h h a a h

h f(a+h) f(a) a+h a As h shrinks and approaches zero (but not = 0), the line becomes a Tangent Line h f(a+h) f(a) a+h a Slope of the line = Average Rate of Change = f(a+h) – f(a) h Slope of the Tangent line = f(a+h) – f(a) h As h approaches zero

The slope of the Tangent Line at a is the Derivative, f ' (a) As h approaches zero, or: f(a+h) – f(a) h = f(a) a f(a+h) – f(a) h = lim h 0 lim: Limit, as h approaches zero The slope of the Tangent Line at a is the Derivative, f ' (a) f(a+h) – f(a) h lim h 0 f ' (a) =

If f(x) = -2x + 3, then f ' (x) = -2 Example: Use the definition of the derivative to obtain the following result: If f(x) = -2x + 3, then f ' (x) = -2 f(x+h) – f(x) h f ' (x) = lim h 0 Solution: Using the definition f (x + h) = -2(x + h) + 3 = (-2x - 2h + 3) f (x + h) – f (x) h f ' (x) = lim h 0 = (-2x - 2h + 3) – (-2x + 3) h lim h 0 = (-2h) h lim h 0 = -2

If f(x) = x2 - 8x + 9, then f ' (x) = 2x - 8 Example: Use the definition of the derivative to obtain the following result: If f(x) = x2 - 8x + 9, then f ' (x) = 2x - 8 f(x+h) – f(x) h f ' (x) = lim h 0 Solution: Using the definition f (x + h) = (x + h)2 - 8(x + h) + 9 = (x2 + 2xh + h2 - 8x -8h + 9) f (x + h) – f (x) h f ' (x) = lim h 0 = (x2 + 2xh + h2 - 8x - 8h + 9) – ( x2 - 8x + 9) h lim h 0 = (2xh + h2 - 8h) h lim h 0 = h (2x + h - 8) h lim h 0 = (2x + h - 8) lim h 0 = 2x - 8