Differential Equations: Growth and Decay Calculus 5.6.

Slides:

Advertisements

Similar presentations

Ch 6.4 Exponential Growth & Decay Calculus Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy.

Advertisements

Exponential Growth and Decay

DIFFERENTIAL EQUATIONS: GROWTH AND DECAY Section 6.2.

AP AB Calculus: Half-Lives. Objective To derive the half-life equation using calculus To learn how to solve half-life problems To solve basic and challenging.

Exponential Functions Functions that have the exponent as the variable.

Differential Equations Definition A differential equation is an equation involving derivatives of an unknown function and possibly the function itself.

6.2 Growth and Decay Law of Exponential Growth and Decay C = initial value k = constant of proportionality if k > 0, exponential growth occurs if k < 0,

Diff EQs 6.6. Common Problems: Exponential Growth and Decay Compound Interest Radiation and half-life Newton’s law of cooling Other fun topics.

Section 6.2 – Differential Equations (Growth and Decay)

Warm Up. 7.4 A – Separable Differential Equations Use separation and initial values to solve differential equations.

1Chapter 2. 2 Example 3Chapter 2 4 EXAMPLE 5Chapter 2.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Exponential Growth and Decay Section 6.4.

Chapter 6 AP Calculus BC.

6.4 Exponential Growth and Decay Greg Kelly, Hanford High School, Richland, Washington Glacier National Park, Montana Photo by Vickie Kelly, 2004.

AP Calculus Ms. Battaglia. Solve the differential equation.

Exponential Growth and Decay 6.4. Exponential Decay Exponential Decay is very similar to Exponential Growth. The only difference in the model is that.

CHAPTER 5 SECTION 5.6 DIFFERENTIAL EQUATIONS: GROWTH AND DECAY

Exponential Growth and Decay

BC Calculus – Quiz Review

Section 7.4: Exponential Growth and Decay Practice HW from Stewart Textbook (not to hand in) p. 532 # 1-17 odd.

Math 3120 Differential Equations with Boundary Value Problems Chapter 2: First-Order Differential Equations Section 2-2: Separable Variables.

Differential Equations. Definition A differential equation is an equation involving derivatives of an unknown function and possibly the function itself.

Using calculus, it can be shown that when the rate of growth or decay of some quantity at a given instant is proportional to the amount present at that.

C2: Exponential Functions Learning Objective: to be able to recognise a function in the form of f(x) = a x.

AP Calculus Ms. Battaglia. The strategy is to rewrite the equation so that each variable occurs on only one side of the equation. This strategy is called.

2.2 Unconstrained Growth. Malthusian Population Model The power of population is indefinitely greater than the power in the earth to produce subsistence.

Differential Equations Copyright © Cengage Learning. All rights reserved.

6 Differential Equations

Differential Equations Objective: To solve a separable differential equation.

Unit 9 Exponential Functions, Differential Equations and L’Hopital’s Rule.

Section 8.2 Separation of Variables.  Calculus,10/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights.

The Random Nature of Radioactive Decay

Exponential Growth and Decay 6.4. Separation of Variables When we have a first order differential equation which is implicitly defined, we can try to.

Differential Equations: Growth & Decay (6.2) March 16th, 2012.

AP CALCULUS AB Chapter 6:

7.4 B – Applying calculus to Exponentials. Big Idea This section does not actually require calculus. You will learn a couple of formulas to model exponential.

Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050.

GrowthDecay. If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by.

Exponential Growth and Decay 6.4. Slide 6- 2 Quick Review.

Any population of living creatures increases at a rate that is proportional to the number present (at least for a while). Other things that increase or.

7.3B Applications of Solving Exponential Equations

Chapter 2 Solutions of 1 st Order Differential Equations.

Warm Up Dec. 19 Write and solve the differential equation that models the verbal statement. Evaluate the solution at the specified value. The rate of change.

Aim: Growth & Decay Course: Calculus Do Now: Aim: How do we solve differential equations dealing with Growth and Decay Find.

1.3 Exponential Functions. Slide 1- 2 Exponential Function.

Chapter 5 Applications of Exponential and Natural Logarithm Functions.

6.2 Growth and Decay Obj: set up and solve growth and decay problems.

Ch. 7 – Differential Equations and Mathematical Modeling 7.4 Solving Differential Equations.

2/18/2016Calculus - Santowski1 C Separable Equations Calculus - Santowski.

6.4 Exponential Growth and Decay Objective: SWBAT solve problems involving exponential growth and decay in a variety of applications.

Chapter 6 Integration Section 3 Differential Equations; Growth and Decay.

6.4 Exponential Growth and Decay. The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep present.

6.4 Applications of Differential Equations. I. Exponential Growth and Decay A.) Law of Exponential Change - Any situation where a quantity (y) whose rate.

6.4 Exponential Growth and Decay Law of Exponential Change Continuously Compounded Interest Radioactivity Newton’s Law of Cooling Resistance Proportional.

Drill If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the.

Homework Homework Assignment #8 Read Section 5.8 Page 355, Exercises: 1 – 69(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.

AP Calculus AB 6.3 Separation of Variables Objective: Recognize and solve differential equations by separation of variables. Use differential equations.

6.4 Exponential Growth and Decay Greg Kelly, Hanford High School, Richland, Washington Glacier National Park, Montana Photo by Vickie Kelly, 2004.

Table of Contents 1. Section 5.8 Exponential Growth and Decay.

Section 6.6: Growth and Decay Model Theorem 6.33: If y is a differentiable function of t such that y > 0 and, for some constant c, then where y 0 is the.

Differential Equations

Calculus II (MAT 146) Dr. Day Monday, Oct 23, 2017

6-2 Solving Differential Equations

Section Indefinite Integrals

6.2 Differential Equations: Growth and Decay (Part 1)

Solving Equations with Variables on Both Sides

Solving Equations with Variables on Both Sides

Section Indefinite Integrals

5.2 Growth and Decay Law of Exponential Growth and Decay

Presentation transcript:

Differential Equations: Growth and Decay Calculus 5.6

2 Differential equations so far.. Calculus 5.6

3 In general Separate variables Each variable is only on one side of the equation Then integrate Calculus 5.6

4 Examples Calculus 5.6

5 Exponential growth and decay If y is a differentiable function of t such that y > 0 and y´ = ky, for some constant k, then C is the initial value of y k is the proportionality constant k > 0 means growth, k < 0 means decay See proof on page 363 Calculus 5.6

6 Example Write and solve the differential equation. The rate of change of P with respect to t is proportional to 10 – t. Calculus 5.6

7 Example Find the exponential function that passes through the points (0, 4) and (5, 2). Calculus 5.6

8 Radioactive decay Uses half-life Exponential decay Use the initial amount to find C Use the half-life to find k Use the equation to find whatever else you need. Calculus 5.6

9 Examples Carbon-14 has a half-life of 5,730 years. If the initial quantity is 3.0 g, how much is left after 10,000 years? Plutonium-239 has a half-life of 24, 360 years. After 10, 000 years, 0.4 g remains. What was the initial amount? Calculus 5.6