32 – Applications of the Derivative No Calculator

Slides:

Advertisements

Similar presentations

Copyright © 2011 Pearson Education, Inc. Slide Tangent Lines and Derivatives A tangent line just touches a curve at a single point, without.

Advertisements

 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.

1 Basic Differentiation Rules and Rates of Change Section 2.2.

Linear Motion III Acceleration, Velocity vs. Time Graphs.

Differentiating the Inverse. Objectives Students will be able to Calculate the inverse of a function. Determine if a function has an inverse. Differentiate.

Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.

1 Instantaneous Rate of Change  What is Instantaneous Rate of Change?  We need to shift our thinking from “average rate of change” to “instantaneous.

Assigned work: pg 74#5cd,6ad,7b,8-11,14,15,19,20 Slope of a tangent to any x value for the curve f(x) is: This is know as the “Derivative by First Principles”

Chapter 2 Motion Along a Straight Line. Linear motion In this chapter we will consider moving objects: Along a straight line With every portion of an.

Chapter Acceleration Non-uniform motion – more complex.

Warm Up 10/3/13 1) The graph of the derivative of f, f ’, is given. Which of the following statements is true about f? (A) f is decreasing for -1 < x

The derivative of a function f at a fixed number a is In this lesson we let the number a vary. If we replace a in the equation by a variable x, we get.

Section 6.1 Polynomial Derivatives, Product Rule, Quotient Rule.

Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Do Now: Aim: To memorize more stuff about differentiation: Product/quotient rules and.

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.

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

Objectives: 1.Be able to find the derivative of functions by applying the Product Rule. Critical Vocabulary: Derivative, Tangent Daily Warm-Up: Find the.

2.1b- Limits and Rates of Change (day 2). 3) Substitution Sometimes the limit of f(x) as x approaches c doesn’t depend on the value of f at x =c, but.

Two kinds of rate of change Q: A car travels 110 miles in 2 hours. What’s its average rate of change (speed)? A: 110/2 = 55 mi/hr. That is, if we drive.

Basic Differentiation Rules and Rates of Change Section 2.2.

 The derivative of a function f(x), denoted f’(x) is the slope of a tangent line to a curve at any given point.  Or the slope of a curve at any given.

AP Calculus AB Exam 3 Multiple Choice Section Name:_____________ 2. An equation of the line tangent to the graph of f( x ) = x ( 1 – 2x) 3 at the point.

Review Problems Integration 1. Find the instantaneous rate of change of the function at x = -2 _ 1.

SECTION 3.2 Basic Differentiation Rules and Rates of Change.

2.2 Differentiation Techniques: The Power and Sum-Difference Rules 1.5.

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.

Equations of Motion Review of the 5 Equations of Motion.

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.

3.5: Derivatives of Trig Functions Objective: Students will be able to find and apply the derivative of a trigonometric function.

1 10 X 8/30/10 8/ XX X 3 Warm up p.45 #1, 3, 50 p.45 #1, 3, 50.

1D Kinematics Equations and Problems. Velocity The rate at an object changes position relative to something stationary. X VT ÷ x ÷

Graphs of a falling object And you. Objective 1: Graph a position –vs- time graph for an object falling from a tall building for 10 seconds Calculate.

Finding the Derivative/Rate of Change.  The derivative of a constant is 0. That is, if c is a real number, then 1. Sketch a graph to demonstrate this.

2.2 Basic Differentiation Rules and Rate of Change

Mean Value Theorem.

PRODUCT & QUOTIENT RULES & HIGHER-ORDER DERIVATIVES (2.3)

22. Use this table to answer the question.

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).

Warm-Up- AP free response (No Calculator )

12.3 Tangent Lines and Velocity

Rate of Change.

Rate of change and tangent lines

Sec 2.7: Derivative and Rates of Change

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE (2.2)

Chapter 2 – Limits and Continuity

Unit 6 – Fundamentals of Calculus Section 6

Unit 6 – Fundamentals of Calculus Section 6

Lesson 7: Applications of the Derivative

Lesson 7: Applications of the Derivative

Tangent Lines and Derivatives

The Tangent and Velocity Problems

Daily Warm-Up: Find the derivative of the following functions

Click to see each answer.

2.2C Derivative as a Rate of Change

Derivatives and Graphs

2.7/2.8 Tangent Lines & Derivatives

Drill: Find the limit of each of the following.

2.2: Formal Definition of the Derivative

2.4 Rates of Change & Tangent Lines

30 – Instantaneous Rate of Change No Calculator

Section 9.4 – Solving Differential Equations Symbolically

§2.7. Derivatives.

Find the derivative of the following function: {image} .

VELOCITY, ACCELERATION & RATES OF CHANGE

Click to see each answer.

Presentation transcript:

32 – Applications of the Derivative No Calculator Derivative Investigations 32 – Applications of the Derivative No Calculator

Equation of tangent line to f(x) at x = a: Slope of a normal line to f(x) at x = a: 1. Find the equation of the line tangent to f(x) at x = 1. 2. Find the equation of the line normal to f(x) at x = 2.

3. Find the equation of the line tangent to g(x) at x = 2. 4. Find the equation of the line normal to g(x) at x = –1.

5. Find the equation of the line tangent to h(x) at x = 0. 6. Find the equation of the line normal to h(x) at x = 1.

DERIVATIVE POSITION s(t) VELOCITY v(t) ACCELERATION a(t) SLOPE AREA X

A ball is thrown straight down from the top of a 100 foot tall building with an initial velocity of Use the position function for free-falling objects: 7. Determine the position function of the ball. 8. Find v(2). Include units in your answer. 9. Find a(3). Include units in your answer.

Given 10. Find the initial position. 11. Find the initial velocity. 12. Find the average velocity on the interval [0, 2]. 13. Find a(1).

Given 14. Find a(1). 15. Find the average acceleration on the interval [0, 3].

Given 16. Find the average velocity on [0, 2]. 17. Find the velocity at t = 2. 18. Find the average acceleration on [0, 2]. 19. Find the acceleration at t = 2.