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Ms. Bertsos knows the rate a population of bacteria is increasing ... 'Pulling out a konstant' Some General Rules. Differentiation Formula Integration Formula ...
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Use the integral to convert dy/dx to y = f(x) ... A $20,000 art collection is increases in. value at a rate of 300t dollars per year ...
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Antiderivatives. Definition. A function F is an antiderivative of f on an interval ... of f on I, then G is also an antiderivative of f on I if and only if ...
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tan x. ax / ln a. ax. sin x. cos x. ex. ex - cos x. sin x. ln |x| xr 1/(r 1) xr, r -1. Antiderivative ... Mika Sepp l : Antiderivatives ...
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A function F is the Antiderivative of f if ... Find all functions g such that: NOTE: see pg. 250 for more Antiderivative Rules Ex 3: Evaluate Ex 4: ...
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6 Integration Antiderivatives and the Rules of Integration Integration by Substitution Area and the Definite Integral The Fundamental Theorem of Calculus
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4.1 Antiderivatives and Indefinite Integration 4.1 Antiderivatives and Indefinite Integration 4.1 Antiderivatives and Indefinite Integration 4.1 Antiderivatives and ...
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The antiderivative of a function f is a function F such that F'=f. ... Solution: Differentiate the functions: ... Antiderivative of Power Function: ...
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k(x) = sec2(x) K(x) = tan(x) C. because K'(x) = k(x) Differential Equation ... Basic Integration Rules. Note the inverse nature of integration and differentiation ...
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To solve an initial value problem. Find the general solutions for antiderivatives ... Algebra Rules. 3) Algebra Rules. Sum rule. Difference Rule. Constant ...
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Just as represents finding a derivative, the ... Find the equation for y, given the derivative and. the indicated point on the curve. ...
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4.1: Antiderivatives and Indefinite Integration (Part II) Greg Kelly, Hanford High School, Richland, Washington Objectives Write the general solution of a ...
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5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington Page 269 gives rules for working with integrals, the most important ...
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Numerical Integration Approximating Definite Integral The Trapezoidal Rule Some elementary functions do not have antiderivatives that are elementary functions.
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Numerical Integration Approximating Definite Integral The Trapezoidal Rule Some elementary functions do not have antiderivatives that are elementary functions.
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Derivatives What do we mean by the derivative of a function f (x) at a point (a, f (a ... Limits Derivatives Recall Some Derivative Facts Antiderivatives Recall (?)
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Integral calculus: areas and volumes. antiderivatives. 13. Differential Calculus Review ... Solid integral across shape. See Mirtich,Eberly for more details ...
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We begin with two examples. Find any function whose derivative is f(x) = 9x2+4x b) f(x)=3cos2x. These are the antiderivatives. An antiderivative of the ...
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Riemann Sums and The Definite ... clarification regarding INDEFINITE INTEGRALS (with which we have worked before in finding antiderivatives) and DEFINITE INTEGRALS ...
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Fundamental Theorem of Calculus Part 2 Thurs Feb 5 Do Now Evaluate each definite integral 1) 2) Part 1 Review FTC Part 1 stated that we can use antiderivatives to ...
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... given by the FTC between antiderivatives and integrals, the ... (one antiderivative for each value of the constant C). INDEFINITE VS. DEFINITE INTEGRALS ...
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This gives us antiderivatives (integrals) of these functions ... Application. Find the area enclosed by. x = 1/2, x = 2, y = 0, and. Which pattern does this match? ...
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Going back to the function f(x) = x2, we see that the general antiderivative. of f is ?x3 C. ... f(x) = xn, n 0. f(x) = x-3. Example 1. ANTIDERIVATIVES. If F ...
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Title: Section 7.2* Author: Allen Fuller Last modified by: z Created Date: 6/20/2005 1:40:46 AM Document presentation format: On-screen Show (4:3) Other titles
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Apply concepts of indefinite integrals to a real world ... dx which is similar to rules we have seen ... be used to find an antiderivative that is not ...
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Consider the function whose derivative is given by. ... Notice that we say is an antiderivative and not the antiderivative. Why? ... HippoCampus. Slope Fields ...
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One technique is to inject a dye into a main vein near the. Heart. ... Definite Integral and Area. If an integrable function y = f(x) is nonpositive, the nonzero ...
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indefinite integral of f, and we introduce the. notation ... NOTE to find the indefinite integral is the. same as to find the most general antiderivative ...
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A function is increasing over an interval if the derivative is always ... A function is decreasing over an interval if the derivative is always negative. ...
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Title: Calculus 6.2 Subject: Integration by Substitution & Separable Differential Equations Author: Gregory Kelly Last modified by: St Stephen's School
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... Example Find the following derivatives Section 6.4 Second Fundamental Theorem of Calculus The 1st Fundamental Theorem of ... Document presentation format: On ...
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AP CALCULUS AB CHAPTER 6: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING SECTION 6.2: ANTIDIFFERENTIATION BY SUBSTITUTION Indefinite Integrals Leibniz Notation and ...
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We adopt the convention that, when a formula for ... We are guided by the fact that the slope of. y = F(x) is f (x). Graph Example 5 ...
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A car-leasing firm must decide how much to charge for maintenance on the cars it ... What total maintenance cost can the company expect for a 2-year lease? ...
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... limits changes the sign. 3. Constant multiples can be moved outside. 1. ... Constant multiples can be moved outside. 4. Integrals can be added and subtracted. ...
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Title: Calculus 6.2 Subject: Integration by Substitution & Separable Differential Equations Author: Gregory Kelly Last modified by: Mike Efram Created Date
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Trapezoid Rule. In general, for n trapezoids, Pattern: The ... Use the Trapezoid Rule to approximate the definite integral. (w/ n = 4) 6. Simpson's Rule ...
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Section 7.4* General Logarithmic and Exponential Functions GENERAL EXPONENTIAL FUNCTIONS NOTES ON f(x) = ax LAWS OF EXPONENTS DIFFERENTIATION OF GENERAL EXPONENTIAL ...
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Section 7.4* General Logarithmic and Exponential Functions GENERAL EXPONENTIAL FUNCTIONS NOTES ON f(x) = ax LAWS OF EXPONENTS DIFFERENTIATION OF GENERAL EXPONENTIAL ...
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new limit. new limit. We could have substituted back and used the original limits. New ... Don't forget to use the new limits. Until then, remember that half the AP exam and half ...
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The daily cost to manufacture x items is C = 5000 25x 2. How many items should ... minimize the daily. average cost. 4.10 Old problem. Given a function, find ...
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Title: Calculus 4.2 Subject: Mean Value Theorem Author: Gregory Kelly Last modified by: Gregory Kelly Created Date: 10/17/2002 4:22:00 AM Document presentation format
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REVIEW OF TRIGONOMETRY. BASIC IDENTITIES. DOUBLE ANGLE FORMULAS. DERIVATIVES OF BASIC TRIG FUNCTIONS. DERIVATIVES OF INVERSE TRIG FUNCTNS ...
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Mean Value Theorem for Derivatives 4.2 Teddy Roosevelt National Park, North Dakota If f (x) is a differentiable function over [a,b], then at some point c between a ...
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The derivative of is . Note that this only worked because of the 2x in the original. Many integrals can not be ... We can find new limits, ...
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Title: Calculus 6.2 Subject: Integration by Substitution & Separable Differential Equations Author: Gregory Kelly Last modified by: Michelle Honeycutt
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Section 14.2 Derivatives and Integrals of Vector Functions THE DERIVATIVE OF VECTOR FUNCTION TANGENT VECTORS AND TANGENT LINES THEOREM SMOOTH CURVES DIFFERENTIATION ...
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To add, we use Sigma Notation = S. GOAL: Find the area. under a curve ... and a LOWER SUM using the areas of rectangles of width 1. Summation / Sigma Notation ...
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Ch 6.2 Antidifferentiation by Substitution Calculus Graphical, Numerical, Algebraic by Finney, Demana, Waits and Kennedy Indefinite Integral Evaluating an Indefinite ...
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It doesn't matter whether the constant was 3 or -5, since when we take the ... AP Test Question. This is the actual question from the 2006 test ...
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We call the symbol an integral sign, f(x) the integrand, and C the constant of integration. ... For practice, evaluate each integral! x4 dx. 2u7 du (t2 3 t ...
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Definition: If a 0, we define the general exponential function ... 1. loge x = ln x. 2. THE CHANGE OF BASE FORMULA. For any positive number a (a 1), we have ...
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Title: 1 Author: HFJ Last modified by: Jack Created Date: 2/9/2006 12:25:36 PM Document presentation format: (4:3) Company: SYSU
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Slope fields are mostly used as a learning tool and are mostly done on a ... By following the slope field, you get a rough picture of what the curve looks like. ...
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Index. FAQ. Areas of Domains and Definite Integrals. Area under a Parabola. Definition of the Area of Certain Domains. Area under a Parabola Revisited ...
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December 2nd ... December 3rd. Go over any questions on previous nights homework (page 324, numbers 2-5) ... December 4th ...
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