We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byMary Jaquess
Modified over 2 years ago
u-du : Integrating Composite Functions AP Calculus
Integrating Composite Functions (Chain Rule) Remember: Derivatives Rules Remember: Layman’s Description of Antiderivatives *2 nd meaning of “du” du is the derivative of an implicit “u”
u-du Substitution Integrating Composite Functions (Chain Rule) Revisit the Chain Rule If let u = inside function du = derivative of the inside becomes
Development from the layman’s idea of antiderivative “The Family of functions that has the given derivative” must have the derivative of the inside in order to find the antiderivative of the outside
A Visual Aid USING u-du Substitution a Visual Aid REM: u = inside function du = derivative of the inside let u = becomes now only working with f, the outside function
Working With Constants: Constant Property of Integration With u-du Substitution REM: u = inside function du = derivative of the inside Missing Constant? Worksheet - Part 1 u = du =
Example 1 : du given Ex 1:
Example 2: du given Ex 2:
Example 3: du given Ex 3:
Example 4: du given Ex 4:
Example 5: Regular Method Ex 5:
Working with Constants Constant Property of Integration ILL. let u = du = and becomes = Or alternately = =
Example 6 : Introduce a Constant - my method
Example 7 : Introduce a Constant
Example 8 : Introduce a Constant >
Example 9 : Introduce a Constant - extra constant
Example 10 : Polynomial
Example 11: Separate the numerator
Formal Change of Variables > Solve for x in terms of u ILL: Let Solve for x in terms of u then and becomes
Formal Change of Variables > Rewrite in terms of u - du
Formal Change of Variables > Solve for x in terms of u - du > - could divide or multiply by
Complete Change of Variables > At times it is required to even change the du as the u is changed above. We will solve this later in the course.
Development must have the derivative of the inside in order to find the antiderivative of the outside *2 nd meaning of “dx” dx is the derivative of an implicit “x” more later if x = f then dx = f /
4012 u-du : Integrating Composite Functions AP Calculus.
FUNCTIONS – Composite Functions RULES :. FUNCTIONS – Composite Functions RULES : Read as f at g of x.
6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002.
The Rational Zero Theorem. The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives.
Differential Equations By Johnny Grooms and Ryan Barr AP Calculus BC Mrs. Miller, 2 nd Period.
3.1 Derivatives. Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is.
4.5 Integration by Substitution Outside Function Inside Function Derivative of Inside Function.
The Quotient Rule Brought To You By Tutorial Services The Math Center.
6.2 Antidifferentiation by Substitution If y = f(x) we can denote the derivative of f by either dy/dx or f(x). What can we use to denote the antiderivative.
Differentiation Revision for IB SL. Type of function Rule used to differentiate Polynomial Constant Always becomes zero Remember that, e, ln(3), are still.
Solving Systems of Equations using Substitution * Best method when one variable is already solved for or if a variable has a coefficient of 1.
© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 26: Integration by Substitution Part 1 Part 1.
Introducing: common denominator least common denominator like fractions unlike fractions. HOW TO COMPARE FRACTIONS.
Solving Systems of three variables Use the answers in the original equations to solve the remaining variables. Solve the new system of equations by elimination.
4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.
Lesson 7-1 Integration by Parts. Derived from the Product Rule in Differentiation D(uv) = v × du + u × dv ∫ d(uv) = ∫v du + ∫u dv uv = ∫v du + ∫u dv ∫u.
Related Rates Finding the rates of change of two or more related variables that are changing with respect to time.
Combining Like Terms and Distributive Property Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.
Differentiation – Product, Quotient and Chain Rules Department of Mathematics University of Leicester.
Objective SWBAT simplify rational expressions, add, subtract, multiply, and divide rational expressions and solve rational equations SWBAT simplify rational.
Exponential & Logarithmic Equations Exponential Equations with Like Bases Exponential Equations with Different Bases Logarithmic Equations.
5.5 Differentiation of Logarithmic Functions By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial, life,
2.3 Rules for Differentiation Colorado National Monument Vista High, AB Calculus. Book Larson, V9 2010Photo by Vickie Kelly, 2003.
QUESTION # 1 If =, this is an example of the ______ Property. QUESTION # 2 Mr. Pearson DO NOW Tuesday, September 3, 2013 Selby Lane is an example of the.
Roots & Zeros of Polynomials How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related. 2.5 Zeros of Polynomial Functions.
3.3 Differentiation Rules Colorado National Monument Photo by Vickie Kelly, 2003 Created by Greg Kelly, Hanford High School, Richland, Washington Revised.
The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential.
SYSTEMS OF EQUATIONS. Substitution Method Ex. 1:2x + y = 15 y = 3x We have been using the Substitution Property in our proofs. Now we are going to use.
6.4 day 1 Separable Differential Equations Jefferson Memorial, Washington DC Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly,
2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover (www.mrgover.com) Objectives: Find limits using direct substitution Find limits when substitution.
© 2016 SlidePlayer.com Inc. All rights reserved.