Differentiating “Combined” Functions Deriving Product Rule for Differentiation.

Slides:

Advertisements

Similar presentations

The Important Thing About By. The Important Thing About ******** The important thing about ***** is *****. It is true s/he can *****, *****, and *****.

Advertisements

2.4 The Chain Rule If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given.

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

Section 3C Dealing with Uncertainty Pages

Sample size computations Petter Mostad

Level 1 Laboratories University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Handling & Propagation of Errors : A simple approach 1.

IntegersIntegers IntegersIntegers Integers. Integers can be represented on a number line Whole numbers Integers can either be negative(-), positive(+)

Monday, April 2, 2012MAT 121. Monday, April 2, 2012MAT 121.

1 The student will learn about: the derivative of ln x and the ln f (x), applications. §3.5 Derivatives of Logarithmic and Exponential Functions. the derivative.

Logarithmic and Exponential Functions

Electric Field You have learned that two charges will exert a force on each other even though they are not actually touching each other. This force is.

1.7 Exponential Growth and Decay Math 150 Spring 2005.

Maximum Likelihood See Davison Ch. 4 for background and a more thorough discussion. Sometimes.

Investment. A Simple Example In a simple asset market, there are only two assets. One is riskfree asset offers interest rate of zero. The other is a risky.

The Scientific Method 1. Using and Expressing Measurements Scientific notation is written as a number between 1 and 10 multiplied by 10 raised to a power.

Chapter 8 Introduction to Inference Target Goal: I can calculate the confidence interval for a population Estimating with Confidence 8.1a h.w: pg 481:

A measured value Number and unit Example 6 ft.. Accuracy How close you measure or hit a true value or target.

Exponents. Definition: Exponent The exponent of a number says how many times to use that number in a multiplication. It is written as a small number to.

Divisibility rules.

Unit 1 Chapter 2. Common SI Units SI System is set-up so it is easy to move from one unit to another.

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

8.2 Negative and Zero Exponents I love exponents!.

Treatment of Uncertainties

The Quotient Rule. Objective  To use the quotient rule for differentiation.  ES: Explicitly assessing information and drawing conclusions.

Differentiating “Combined” Functions Deriving the Product Rule for Differentiation.

1.6 – Differentiation Techniques The Product and Quotient Rules.

Инвестиционный паспорт Муниципального образования «Целинский район»

DIFFERENTIATING “COMBINED” FUNCTIONS PRODUCTS. PRODUCTS: A GAMBIER EXAMPLE LARGE RATS!!!!

Confidence Intervals and Tests of Proportions. Assumptions for inference when using sample proportions: We will develop a short list of assumptions for.

Adding Integers. Zero Pair = 0 Why it works… __________ Property says you can add or subtract zero without changing the value of an expression.

(x – 8) (x + 8) = 0 x – 8 = 0 x + 8 = x = 8 x = (x + 5) (x + 2) = 0 x + 5 = 0 x + 2 = x = - 5 x = - 2.

Other Means. The Geometric Mean The Harmonic Mean.

Inference for 2 Proportions Mean and Standard Deviation.

Complete the pattern… 1 =2=3=4=5= 6=7=8=9=10=. Proportionality If two quantities change by a related amount they are said to be in proportion or PROPORTIONALto.

3.6The Chain Rule. Nothing we have done so far could help us take the derivative of this: But let’s try a function we know using a different approach:

The Product Rule Complete the warm-up on your notesheet.

5.3 Definite Integrals and Antiderivatives Objective: SWBAT apply rules for definite integrals and find the average value over a closed interval.

Sampling distributions rule of thumb…. Some important points about sample distributions… If we obtain a sample that meets the rules of thumb, then…

Integral of the Reciprocal Function:  6-1—Why we need it.  6-2—What it can’t be.  6-2—What it could be.  6-3—By definition it is:  How we can use.

Basic Differentiation Rules The CONSTANT Rule: The derivative of a constant function is 0.

Section 2.3 The Derivative Function. Fill out the handout with a partner –So is the table a function? –Do you notice any other relationships between f.

Lesson 4-Remainder Theorem 17 December, 2015ML4 MH Objectives : - The remainder and Factor theorems - It’s used to help factorise Polynomials - It’s used.

For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point. We call the equation of the.

The Chain Rule Composite Functions When a function is composed of an inner function and an outer function, it is called a “composite function” When a.

Let’s start with a definition: Work: a scalar quantity (it may be + or -)that is associated with a force acting on an object as it moves through some.

Significant Figures. Significant Digits or Significant Figures We must be aware of the accuracy limits of each piece of lab equipment that we use and.

Techniques of Differentiation. We now have a shortcut to find a derivative of a simple function. You multiply the exponent by any coefficient in front.

Electric Fields Montwood High School AP Physics C R. Casao.

The Product Rule. Do Now  Find the derivative of f(x) = x(x 2 + 2x – 1).  What is the derivative of sinx? of cosx?

照片档案整理一、照片档案的含义二、照片档案的归档范围三、卷内照片的分类、组卷、排序与编号四、填写照片档案说明五、照片档案编目及封面、备考填写六、数码照片整理方法七、照片档案的保管与保护.

공무원연금관리공단 광주지부 공무원대부등 공적연금 연계제도 공무원연금관리공단 광주지부. 공적연금 연계제도 국민연금과 직역연금 ( 공무원 / 사학 / 군인 / 별정우체국 ) 간의 연계가 이루어지지 않고 있 어 공적연금의 사각지대가 발생해 노후생활안정 달성 미흡 연계제도 시행전.

Жюль Верн ( ). Я мальчиком мечтал, читая Жюля Верна, Что тени вымысла плоть обретут для нас; Что поплывет судно громадней «Грейт Истерна»; Что.

UNIT 2 LESSON 9 IMPLICIT DIFFERENTIATION 1. 2 So far, we have been differentiating expressions of the form y = f(x), where y is written explicitly in.

Differentiating “Combined” Functions Deriving the Sum and Product Rules for Differentiation.

9-1 Larry’s Lawn Service provides lawn care in a planned community where all lawns are approximately the same size. At the end of May, Larry prepared his.

Basic derivation rules We will generally have to confront not only the functions presented above, but also combinations of these : multiples, sums, products,

Percentages Revision.

Population Growth Rate

Population Math.

PARTIAL DIFFERENTIATION 1

3.11: Derivatives of Inverse Functions

The 2nd Derivative.

3.2: Differentiability.

Differential Equations

Topic connection Help Homework

5.5: Linearization and Newton’s Method

PARTIAL DIFFERENTIATION 1

Exponent Rules.

Prairie Dogs By: Jayden.

Presentation transcript:

Differentiating “Combined” Functions Deriving Product Rule for Differentiation.

Products: A Gambier Example LARGE RATS!!!!

A Gambier Example gobble The rate at which the deer gobble my hostas is proportional to the product of the number of deer and the number of hostas. So we have a gobble function: What can we say about the rate of change of this function? What if over a short period of time  t, from t to t +  t the deer population increases by a small amount by  R L and the hosta population increases by  H. How much does the gobble rate change between time t and time t +  t ?

A Gambier Example

The change in g has three relevant pieces: Old Rats eating poor baby hostas “Cute” baby rats eating vulnerable old hostas “Cute” baby rats eating poor baby hostas

A Gambier Example 0

SO, the rate of change of gobble is given by... The rate at which the number of hostas is changing times the number of large rats. The rate at which the number of large rats is changing times the number of hostas.

The Product Rule for Derivatives So the rate at which a product changes is not merely the product of the changing rates. The nature of the interaction between the functions, causes the overall rate of change to depend on the size of the quantities themselves.

In general, we have…

In the course of the calculation above, we said that Is this actually true? Is it ALWAYS true? Continuity Required! xx + h

The Limit of the Product of Two Functions Would be zero if g were continuous at x=a. Is it?

The Limit of the Product of Two Functions g is continuous at x=a because g is differentiable at x=a.