Ch. 8 – Applications of Definite Integrals 8.1 – Integral as Net Change.

Slides:

Advertisements

Similar presentations

Position, Velocity and Acceleration

Advertisements

7.1 – 7.3 Review Area and Volume. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Determine when the particle is moving.

Graphing motion. Displacement vs. time Displacement (m) time(s) Describe the motion of the object represented by this graph This object is at rest 2m.

Particle Straight Line Kinematics: Ex Prob 2 This is an a = f(t) problem. I call it a “total distance” problem. Variations: v = f(t), s = f(t)…. A particle.

Homework Homework Assignment #4 Read Section 5.5

6036: Area of a Plane Region AB Calculus. Accumulation vs. Area Area is defined as positive. The base and the height must be positive. Accumulation can.

Section 2.2 – Basic Differentiation Rules and Rates of Change.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Chapter 6 The Integral Sections 6.1, 6.2, and 6.3

CALCULUS II Chapter 5. Definite Integral Example.

Acceleration. Changing Motion Objects with changing velocities cover different distances in equal time intervals.

Motion in One Dimension Average Versus Instantaneous.

Warmup: YES calculator 1) 2). Warmup Find k such that the line is tangent to the graph of the function.

1 5.4 – Indefinite Integrals and The Net Change Theorem.

CALCULUS II Chapter 5.

Section 5.3: Evaluating Definite Integrals Practice HW from Stewart Textbook (not to hand in) p. 374 # 1-27 odd, odd.

Warmup 1) 2). 5.4: Fundamental Theorem of Calculus.

Integrals in Physics. Biblical Reference An area 25,000 cubits long and 10,000 cubits wide will belong to the Levites, who serve in the temple, as their.

AP CALCULUS AB PRACTICE EXAM. 1)Multiply by clever form of 1 3 and 1/3.

1 When you see… Find the zeros You think…. 2 To find the zeros...

SECTION 4-4 A Second Fundamental Theorem of Calculus.

4-4 THE FUNDAMENTAL THEOREM OF CALCULUS MS. BATTAGLIA – AP CALCULUS.

Miss Battaglia AB/BC Calculus.  Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous.

 (Part 2 of the FTC in your book)  If f is continuous on [a, b] and F is an antiderivative of f on [a, b], then **F(b) – F(a) is often denoted as This.

4.3 Copyright © 2014 Pearson Education, Inc. Area and Definite Integrals OBJECTIVE Find the area under a curve over a given closed interval. Evaluate a.

SECT. 3-A POSITION, VELOCITY, AND ACCELERATION. Position function - gives the location of an object at time t, usually s(t), x(t) or y(t) Velocity - The.

A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee?

Functions Defined by Integrals

1 5.d – Applications of Integrals. 2 Definite Integrals and Area The definite integral is related to the area bound by the function f(x), the x-axis,

Section 6.1 Antiderivatives Graphically and Numerically.

Section 7.1: Integral as Net Change

8.1 A – Integral as Net Change Goal: Use integrals to determine an objects direction, displacement and position.

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

Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions.

Friday, October 23, 2015MAT 145 Please review TEST #2 Results and see me with questions.

5.2 Definite Integrals. When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. subinterval partition The width.

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.

“THE MOST MISSED” 1.Average rate of change does not mean average value –Average rate of change deals with derivatives It is the slope between the two endpoints.

V ECTORS AND C ALCULUS Section 11-B. Vectors and Derivatives If a smooth curve C is given by the equation Then the slope of C at the point (x, y) is given.

Ch. 6 – The Definite Integral

AP CALC: CHAPTER 5 THE BEGINNING OF INTEGRAL FUN….

3023 Rectilinear Motion AP Calculus. Position Defn: Rectilinear Motion: Movement of object in either direction along a coordinate line (x-axis, or y-axis)

A particle moves on the x-axis so that its acceleration at any time t>0 is given by a(t)= When t=1, v=, and s=.

When you see… Find the zeros You think…. To find the zeros...

5.3 Definite Integrals and Riemann Sums. I. Rules for Definite Integrals.

By: Hawa Soumare.  Question   Part A: Finding zeros  Part B: Integration  Part C: Interpreting graphs  Part D: Finding derivative and comparison.

Net Change as the Integral of a Rate Section 5.5 Mr. Peltier.

C.1.5 – WORKING WITH DEFINITE INTEGRALS & FTC (PART 1) Calculus - Santowski 6/30/ Calculus - Santowski.

If the following functions represent the derivative of the original function, find the original function. Antiderivative – If F’(x) = f(x) on an interval,

Wednesday, April 6, 2016MAT 145 Please review TEST #3 Results and see me with questions!

9/26/2016HL Math - Santowski1 Lesson 49 – Working with Definite Integrals HL Math- Santowski.

Sect. 3-A Position, Velocity, and Acceleration

Average Value Theorem.

Calculus I (MAT 145) Dr. Day Monday November 27, 2017

Ch. 6 – The Definite Integral

Unit 6 – Fundamentals of Calculus Section 6

Rate of Change and Accumulation Review

Graphing Motion Walk Around

The Fundamental Theorem of Calculus (FTC)

AP Calc: Chapter 5 The beginning of integral fun…

A function F is the Antiderivative of f if

Kinematics Formulae & Problems Day #1

Warm Up 1. Find 2 6 2

Total Distance Traveled

Section Net Change in Position/Distance Traveled

Section Net Change in Position/Distance Traveled

The integral represents the area between the curve and the x-axis.

Warm Up 1. Find 2 6 2

Velocity-Time Graphs for Acceleration

The Amazing Power of the Derivative in

Presentation transcript:

Ch. 8 – Applications of Definite Integrals 8.1 – Integral as Net Change

Recall: Given a velocity function v(t)… –The position function s(t) is the antiderivative of v(t) Antiderivative means integral, which means area underneath curve of v(t) –The acceleration function a(t) is the derivative of v(t) Derivative means slope of the curve of v(t) Ex: A particle is moving along the x-axis with a velocity of: The initial position of the particle is s(0) = 9. –Find the particle’s position at t = 1. First, find an expression for s(t) using the initial position given. Second, evaluate for s(1).

Ex: A particle is moving along the x-axis with a velocity of: The initial position of the particle is s(0) = 9. –Find the particle’s displacement from t = 3 to t = 7. Recall that displacement is stop minus start! –Find the total distance traveled by the particle over [0, 5]. Recall that total distance is the absolute value of total displacement, since we count every displacement (positive or negative) as distance To do this by hand, we would need to find the total positive area over [0, 5] and add it to the total negative area over [0, 5]. We’ll use the calculator for this one, though:

Ex: A particle is moving along the x-axis with a velocity of m/s. The initial position of the particle is s(0) = 2 m. –When is the particle moving to the right in the first 5 seconds? “moving to the right” = velocity is positive We must find x-intercepts of v(t) and make a sign chart to see where v(t) > 0. Since t is on [0, 5], then Now make a sign chart for v(t): –Answer: v(t) is moving to the right on [0, π/2] and [3π/2, 5] on the interval [0, 5]. ++ -

Ex: Below is a graph of the velocity, v(t), of a particle moving along the x-axis. The particle is at x = 2 when t = 0. Using the graph, find… –…the location of the particle at t = 10. Position  antiderivative of v(t)  integral (area) from t = 0 to t = 10. Find areas of each section! – 12 = -4.5 Since we started at x = 2, we end at 2 + (-4.5) = -2.5 –…the total distance traveled by the particle over [0, 10]. Distance = positive sum of areas of each section =