Acceleration and Deceleration Section 7.4. Acceleration Acceleration is the rate of change of the rate of change. In other words, acceleration = f’’(x)

Slides:

Advertisements

Similar presentations

Physics Quicky A Stone Thrown Downward

Advertisements

Free falling …. ACCELERATION DUE TO GRAVITY All bodies in free fall (that is no air resistance) near the Earth's surface have the same downward acceleration.

5-Minute Check on Activity 4-1 Click the mouse button or press the Space Bar to display the answers. 1.Which direction does y = 3x 2 open? 2.Which function’s.

A rocket test projectile is launched skyward at 88 m/sec (198 mi/hr). How high does it go? v o = ? v final = ? a = ? time to reach peak, t = ? height.

3.4 Velocity and Rates of Change

Velocity and Acceleration. Definitions of Velocity and Acceleration.

MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.4 The Derivative as a Rate of Change Copyright © 2010 by Ron Wallace, all rights.

FOR APPLIED CALCULUS MTH 271 ARE YOU READY?. ARE YOUR READY The following Power Point will consist of10 math problems that every student should be accustomed.

9-4 Quadratic Equations and Projectiles

The Height Equation. h= ending height g = gravity constant (32 if feet, 9.8 if meters) v 0 = initial velocity h 0 = initial height t = time.

10.4 Projectile Motion Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002 Fort Pulaski, GA.

Projectile Motion. We are going to launch things, and then find out how they behave.

10.4 Projectile Motion Fort Pulaski, GA. One early use of calculus was to study projectile motion. In this section we assume ideal projectile motion:

Tangent Lines and Derivatives. Definition of a Tangent Line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.

Chapter 3: Derivatives 3.1 Derivatives and Rate of Change.

SECTION 3.4 POLYNOMIAL AND RATIONAL INEQUALITIES POLYNOMIAL AND RATIONAL INEQUALITIES.

Sec. 4.1 Antiderivatives and Indefinite Integration By Dr. Julia Arnold.

Free Fall and Terminal Velocity CH4 Physics A Winter,

4.1 Antiderivatives and Indefinite Integration. Suppose you were asked to find a function F whose derivative is From your knowledge of derivatives, you.

AP Calculus AB Chapter 4, Section 1 Integration

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.

Warm Up 1.What is the absolute minimum of f(x) = 2x 4 – 17x x 2 – 7x – 15 2.A ball is thrown into the air. It’s height above the ground in feet is.

Warm UpMar. 5 th 1.What is the absolute minimum of f(x) = 2x 4 – 17x x 2 – 7x – 15 2.A ball is thrown into the air. It’s height above the ground.

Review: 6.5g Mini-Quiz 1. Find 2 consecutive positive integers whose product is Find 2 consecutive positive odd integers whose product is 99.

1.6 Acceleration Near Earth’s Surface

Lesson Velocity PVA, derivatives, anti-derivatives, initial value problems, object moving left/right and at rest.

An Application Activity

Review Topic A toy rocket is launched and has the trajectory that can be modeled by s(t) = t – 16t 2 a)What is the height at 2 seconds? b)What is.

So that’s what he's thinking about 3.4. Original Function 1 st Derivative 2 nd Derivative 3 rd Derivative.

5.3: Position, Velocity and Acceleration. Warm-up (Remember Physics) m sec Find the velocity at t=2.

Derivative Notation and Velocity. Notation for the Derivative.

13.2 More on Solving Quadratic Equations. To Solve in the form ax² =k Divide both sides by a Take the square root of both sides Remember to use + Simplify.

13.2 More on Solving Quadratic Equations Goals: -To solve ax² =k --To solve by factoring into a binomial square -To solve using (x +a)² =k.

Problem of the Day If f(x) = -x 3 + x + 1, then f '(-1) = x A) 3 B) 1 C) -1 D) -3 E) -5.

Rate of Change. What is it? A slope is the rate at which the y changes as the x changes Velocity is the rate the position of an object changes as time.

Physics 12 Newtonian Physics Projectiles III Mr. Jean.

Chapter 3 sections 3.3 and 3.4 What is differentiation and what are the methods of differentiation?

Using Graphs In Physics. Find the acceleration of the object for the first 10 seconds of travel Slope = rise/run = (40 m/s)/(10 s) = 4.0 m/s 2.

Instantaneous Rate of Change The (instantaneous) rate of change of f with respect to x at a is the derivative: provided the limit exists.

Warm - up 1) Enter the data into L1 and L2 and calculate a quadratic regression equation (STAT  calc Quadreg). Remember: time – x distance – y. 2) Find.

3-4 VELOCITY & OTHER RATES OF CHANGE. General Rate of Change The (instantaneous) rate of change of f with respect to x at a is the derivative! Ex 1a)

2.2 Basic Differentiation Rules and Rate of Change

Physics 11 Mr. Jean September 20th, 2011.

Today Kinematics: Description of Motion Position and displacement

Calculus Section 3.7 Find higher ordered derivatives.

Sec. 4.1 Antiderivatives and Indefinite Integration

Solving Inequalities Algebraically

1-7 functions Goals: Identify a function. Find function values.

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE (2.2)

5-1 Solving Quadratic Equations by Graphing and Factoring SWBAT

Section 6.1 Slope Fields.

The Tangent and Velocity Problems

Explain the effect of gravity and air resistance.

Explain the effect of gravity and air resistance.

9-4 Quadratic Equations and Projectiles

Vertical Motion Problems

A function F is the Antiderivative of f if

Free-Fall acceleration

1.6 Acceleration Due to Gravity.

Introduction to Calculus

Derivatives Jeopardy Rates of Change Q $100 Q $100 Q $100 Q $100

Today Kinematics: Description of Motion Position and displacement

Bouncing Ball Physics – Velocity / Acceleration / Displacement

Algebra 1 Section 12.4.

VELOCITY, ACCELERATION & RATES OF CHANGE

Calculus 3-4 Rates of Change.

1-7 functions Goals: Identify a function. Find function values.

Presentation transcript:

Acceleration and Deceleration Section 7.4

Acceleration Acceleration is the rate of change of the rate of change. In other words, acceleration = f’’(x) = second derivative of the function *If f’’(x) is negative, the object is… decelerating!

Example 1 Find the instantaneous vertical acceleration at time t of a projectile whose height at time t is given by h(t) = t - 16 t 2. h’(t) = 55 – 32t feet per second h’’(t) = -32 feet per second per second

Example 2 A ball is dropped from the Leaning Tower of Pisa, 56 m above the ground. The height (in meters) of the ball above the ground at time t seconds is given by h(t) = –4.9t a. What is the instantaneous velocity of the ball at time t = 3 sec.? b. What is the acceleration of the ball at time t = 3 seconds? c. At what time t does the ball hit the ground? d. What is the instantaneous velocity of the ball at the moment it hits the ground? h’(t) = -9.8th’(3) = -9.8(3)h’(3) = m/second h’’(t) = -9.8 h’’(3) = -9.8h’’(3) = -9.8 m/sec / sec 0= -4.9t = -4.9t = t 2 T = 3.38 sec h’(3.38) = -9.8(3.38) h’(3.38) = m/sec

Homework Pages 442 – – 6, 8 – 9, 12