2.7/2.8 Tangent Lines & Derivatives

Slides:

Advertisements

Similar presentations

Sec 3.1: Tangents and the Derivative at a Point

Advertisements

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.

Slope and Equation of a line How to find the slop of a line? (x 1, y 1 ) (x 2, y 2 ) How to find the equation of a line? Sec 2.1: Rates of Change and.

Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.

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

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.

Meanings of the Derivatives. I. The Derivative at the Point as the Slope of the Tangent to the Graph of the Function at the Point.

 Find an equation of the tangent line to the curve at the point (2, 1).

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.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

DO NOW: Use Composite of Continuous Functions THM to show f(x) is continuous.

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.

The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.

1.4 – Differentiation Using Limits of Difference Quotients

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,

 Total Change/Length of the interval  Δy/Δx  Average rate of change is the slope of the secant line.

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))

Motion in One Dimension Average Versus Instantaneous.

Mrs. Rivas International Studies Charter School.Objectives: slopes and equations 1.Find slopes and equations of tangent lines. derivative of a function.

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 =

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

GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE, INSTANTANEOUS VELOCITY AT A POINT. 3.1 Definition of Derivative.

Velocity and Other Rates of Change Notes: DERIVATIVES.

Chapter 3.1 Tangents and the Derivative at a Point.

The Derivative Objective: We will explore tangent lines, velocity, and general rates of change and explore their relationships.

December 3, 2012 Quiz and Rates of Change Do Now: Let’s go over your HW HW2.2d Pg. 117 #

§3.2 – The Derivative Function October 2, 2015.

2.1 Position, Velocity, and Speed 2.1 Displacement  x  x f - x i 2.2 Average velocity 2.3 Average speed  

position time position time tangent!  Derivatives are the slope of a function at a point  Slope of x vs. t  velocity - describes how position changes.

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.

Instantaneous Velocity The velocity at an instant of time. For a curved graph, use very small intervals of time.

Solve and show work! State Standard – 4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line.

MAT 1234 Calculus I Section 2.1 Part I Derivatives and Rates of Change

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.

2.1 The Derivative and The Tangent Line Problem Slope of a Tangent Line.

DO NOW:  Find the equation of the line tangent to the curve f(x) = 3x 2 + 4x at x = -2.

Derivatives Limits of the form arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry.

Section 2.4 – Calculating the Derivative Numerically.

Motion and Motion Graphs

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.

Meanings of the Derivatives. I. The Derivative at the Point as the Slope of the Tangent to the Graph of the Function at the Point.

Section 2.1 – Average and Instantaneous Velocity.

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.

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

Ch. 2 – Limits and Continuity

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.

Section 11.3A Introduction to Derivatives

Velocity and Speed Graphically

Rate of Change.

Pg 869 #1, 5, 9, 12, 13, 15, 16, 18, 20, 21, 23, 25, 30, 32, 34, 35, 37, 40, Tangent to a Curve.

2.4 Rates of Change and Tangent Lines Day 1

2.1A Tangent Lines & Derivatives

2.4 Rates of Change & Tangent Lines

Sec 2.7: Derivative and Rates of Change

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

The Tangent and Velocity Problems

2.2C Derivative as a Rate of Change

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

2.3B Higher Derivatives.

Packet #4 Definition of the Derivative

Tangent Line Recall from geometry

Rates of Change and Tangent Lines

Section 2.1 – Average and Instantaneous Velocity

30 – Instantaneous Rate of Change No Calculator

Graphical Analysis – Uniform Acceleration

Sec 2.7: Derivative and Rates of Change

Presentation transcript:

2.7/2.8 Tangent Lines & Derivatives

P = (a, f(a)) with a slope of, Tangent Line The line through P = (a, f(a)) with a slope of,

The derivative of f at a is f(a):

Find the slope of the tangent line at x = 4. Ex 1: Find the slope of the tangent line at x = 4. Find the slope of the tangent line at x = a.

Ex 2: Find f (x) and use it to find the equation of the tangent line to f (x) at x = 9.

Average Velocity: the slope of the secant line between two points, P & Q, on a position curve. Don’t Forget!

Instantaneous Velocity: the slope of the tangent line to a point P on a position curve. Don’t Forget!

2.7 pg. 154 # 1, 2, 3, 5, 11, 13, 15, 25

NOTE: f(x) = the slope of the tangent line at x s(a) = the velocity at t = a | s(a)| = the speed at t = a

Ex 3: The displacement (in meters) of a particle moving in a straight line is given where t is seconds.

a) Find the average velocity on the interval [4, 4 a) Find the average velocity on the interval [4, 4.5] b) Find the instantaneous velocity at t = 4 c) Find the speed at t = 5

Let P(t) be the population of the United States at time t, in years. Ex 4: Let P(t) be the population of the United States at time t, in years. What are the units of P(t)? What does P(1992) = 2.7 million mean?

Ex 5: Sketch the graph of the function g for which g(0) = 0, g(0) = 3, g(1) = 0, & g(2) = 1.

Ex 6: The limit represents the derivative of f at some number a. State f and a.

The limit represents the derivative of f at some Ex 7: The limit represents the derivative of f at some number a. State f and a.

2.8 pg. 161 # 1, 3, 4, 5, 13, 15, 19 – 24 all, 25, 27, 28, 31