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Wicomico High School Mrs. J. A. Austin 2009-2010 AP Calculus 1 AB Third Marking Term
Antiderivatives 4.1 Integration is the inverse operation to Differentiation Function F → Derivative f(x) uses differentiation Derivative f(x) → Function F uses integration F is used to represent the original function f (x) is used to represent its derivative
Leibnitz and Newton The Calculus Wars Two mathematicians were who was the “father of at odds for 10 years over calculus”. Issac Newton: Notation: Gottfied Leibnitz: Notation:
Antiderivatives 4.1 Many functions can have the SAME derivative: x 2 + 5 ; x 2 – 8 ; and x 2 + 10 ALL have the same derivative Because of this: A constant C MUST be attached to the antiderivative when using integration. The value may be found later using a given condition. The graph of the many functions that have the same derivative is called a SLOPE FIELD. A slope field shows all of the possible positions of the function that would have the graphed derivative. A slope field just shows little “tick marks” indicating the slope of the tangent line at that particular point. By connecting the tick marks you can see a sketch of the function.
Antiderivatives 4.1 Finding Antiderivatives: Write the derivative in differential form. Bring an integration symbol to each side of the equation. The integration symbol cancels out the dy or dx.
The Power Rule for Integration Differentiation brings the exponent down to multiply the coefficient and then takes the original power DOWN by one. Integration brings the power UP by one and the coefficient is divided by this NEW exponent.
Indefinite Integration 4.1 Indefinite Integration has no given bounds. Basic Integration Rules Constant Rule IF THE DERIVATIVE IS A CONSTANT, THE FUNCTION WAS OF DEGREE ONE. Constant Multiple Rule THE CONSTANT IF MOVED IN FRONT OF THE INTEGRAL AND WILL MULTIPLY THE FUNCTION ONCE IT IS DETERMINED Sum and difference Rule EACH PART OF THE SUM OR DIFFERENCE IS GIVEN ITS OWN INTEGRAL SIGN AND INTEGRATED SEPERATELY.
Indefinite Integration 4.1 Basic Integration Rules Product Rule AT THIS TIME YOU WILL NOT BE ABLE TO INTEGRATE A PRODUCT. YOU MUST THEREFORE EXPAND ANY MULTIPLICATION. Quotient Rule AT THIS TIME YOU WILL NOT BE ABLE TO INTEGRATE A QUOTIENT WITHOUT TRANSFORMING IT FIRST. A FUNCTION WITH ADD OR SUBT IN THE NUMERATION MUST BE SEPERATED INTO SEPARATE FRACTIONS. LONG DIVISION CAN BE POSSIBLY USED TO ELIMINATE THE DENOMINATOR OF POLYNOMIALS Power Rule INTEGRATION TAKES POWERS UP ONE! THEN DIVIDE BY THE NEW POWER
Indefinite Integration 4.2 Trig Rules MEMORIZE THE BASIC TRIG RULES. REMEMBER THAT THE INTEGRAND IS THE DERIVATIVE SO REVERSE YOUR THINKING TO FIND THE FUNCTION. TRIGONOMETRIC IDENTITIES CAN BE USED TO REWRITE INTEGRAND. DOUBLE ANGLE FORMULAS AND HALF ANGLE FORMULAS CAN BE USE TO REWRITE INTEGRANDS.
Indefinite Integration 4.1 Applying the Basic Integration Rules Using negative exponents Using fractional exponents Separating polynomials Separating fractions Using trig identities
Integration Initial Conditions & Particular Solutions General solution with + C Initial condition substitution Particular solution Slope Field: A graph of ALL the possibilities Tracing general solutions Finding a particular solution
The Sum of a Series 1 + 2 + 3 = 1+ 2 + 3 + 4 = 1+ 2+ 3+ 4+ 5 = 1 2 + 2 2 + 3 2 = 1 2 + 2 2 + 3 2 + 4 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 1 3 + 2 3 + 3 3 = 1 3 + 2 3 + 3 3 + 4 3 = 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = Sum of the first n terms: Sum of the first n squares: Sum of the first n cubes:
Setting Up the Area Formula Use Summation to evaluate: Write each using sigma notation:
∑ sigma Summation Area of Rectangles A = l w The area under a curve can be divided up into rectangles. Then the sum of the rectangles can be added to approximate the area under the curve. Summation Formulas are used to speed the process of adding a series of numbers. i = i 2, = i 3 =
Setting Up the Area Formula i represents the rectangles as a series. When i is 1, we are considering the first rectangle, When i is 2,we are looking at the second rectangle etc. ∆x represents the change or intervals between the rectangles. n represents the number of rectangles we are dividing the area into. A = (length) (width) l is the height of each rectangle under our curve. We can use the function of the curve to find this value. w is the width of each rectangle. We will use ∆x which equals the interval [a,b] divided by number of rectangles, n.
Setting Up the Area Formula Sigma is used first. i = 1 goes under it, n or the value of n goes on top and f(x). f(a + ∆xi ) (∆x) goes next. The represents the height time the width. ∆x is computed and replaced the symbol with a fraction. Note that i is attached to the inner fraction. ∑
Approximating the Area Under a Curve f(x) = -x 2 + 5 [0,2] n=5 Drawing the Rectangles Inscribed: These rectangles are under the curve. No part of them crosses above the function. These rectangles under estimate the true area under the curve. Circumscribed : These rectangles are formed over the function. These rectangles over estimate the true area under the curve. Choosing Which X’s to Use: Upper Sum: Here we will use the values for x which correspond to the f(x) points that create circumscribed rectangles. Lower Sum: Here we will use the values for x which correspond to the f(x) points that create inscribed rectangles. Right Sum: Here we are using the right side of the rectangle to determine the height or f(x). Left Sum: Here we are using the Left side of the rectangle to determine the height or f(x).
Approximating the Area Under a Curve f(x) = -x 2 + 5 [0,2] n=5 Using the given information we can set up a summation to represent adding the areas of 5 rectangles to approximate the area under the curve between 0 and 2.
Approximating the Area Under a Curve Simplify inside the parenthesis. The constant can be brought out to the front of the summation sign. Each term can now be given its own summation sign with the in front. All constant values must be brought out in front leaving ONLY a 1, i, i 2, or i 3 behind sigma. The i, i 2, or i 3 is now replaced with its formula. Pg 261 – 263 ( 4 – 64)
Using Midpoints to Estimate Area Finding a MIDPOINT
Riemann Sums Riemann developed a way to find the area under a curve with rectangles of Different Widths. Instead of a regular partition he used a general partition. His summations use the same idea: Add the area of all of the rectangles to get the approximate area under the curve. His method works well when only a table of values is given. Here we can set up rectangles with the given information and add them to estimate the area under the curve or the accumulation of change that occurred over the given interval.
Riemann Sums x01378 f(x)04187088 Four Rectangles can be created with the given data. f(0) (1) + f(1) (2) + f(3) (4) + f(7) (1) = (0)(1)+(4)(2)+(18)(4)+(70)(1) = 150 UNITS 2 Here we are using the left endpoints to calculate the heights and multiplying by the width. f(1) (1) + f(3) (2) + f(7) (4) + f(8) (1) = (4)(1)+(18)(2)+(70)(4)+(88)(1) = 408 UNITS 2 Here we are using the right endpoints to calculate the heights and multiplying by the width. How can the vast difference be accounted for?
Riemann Sums A Partition with Subintervals of Unequal Width Using the limit definition of summation Definition of a Riemann Sum Regular and general partitions ║ ∆ ║ denotes the norm of the partition Finding Area Using Riemann Sums Finding the “accumulation of change” Definition of Definite Integration Upper and lower limits of integration
Applying Riemann Sums Riemann Sums Applying Riemann Sums to find average rate of change
Definite Integrals The Definite Integral as the Area of a Region Areas of Common Geometric Figures Properties of Definite Integrals Definition of Two Special Definite Integral Evaluating definite integrals Using the additive interval property Preservation of Inequality
The FTC part one The Fundamental Theorem of Calculus Isaac Newton and Gottfried Leibniz Discovered independently Anti-differentiation and Integration Guidelines for Using the FTC Constant of integration C is not needed
The FTC The Fundamental Theorem of Calculus Evaluating a Definite Integral with Bounds With polynomials With fractional exponents Involving absolute value (total distance) The Mean Value Theorem for Integrals Connection to MVT of differentiation
The FTC part two The Fundamental Theorem of Calculus Average Value of a Function Finding the average value of a function on an interval The Second Fundamental Theorem of Calculus Accumulation function in respect to time Change of variable process, using 2 nd FTC Location of the derivative in an integral
Change of Variable Process When the derivative is in a different variable than the bounds.
Integration By Substitution Pattern Recognition U-substitution in integration Recognizing patterns in composite functions Multiplying and Dividing by a Constant Balancing the integral to accommodate du Rewriting the integrand entirely in terms of u
Trapezoidal Rule Area of a Trapezoid: The average of the two parallel sides times the width (|) The summation of the areas of trapezoids to estimate the total area under the curve or the total accumulation of change. Trapezoidal Rule n = number of trapezoids
Simpson’s Rule n must be EVEN Uses double subintervals Uses a polynomial of degree less than or equal to 2 to estimate heights Simpson’s Rule
Trapezoidal Error Analysis If the function has a continuous second derivative on [a,b] then the error E in approximating the integral using the Trapezoidal Rule is: a ≤ x ≤ b
Simpson’s Rule Error Analysis If the function has a continuous fourth derivative on [a,b] then the error E in approximating the integral using the Simpson’s Rule is: a ≤ x ≤ b
Chapter 4 Review Antiderivative Concept Integration Rules Summation Formulas Approximating the Area Under a Curve using: Rectangles: Upper Sums, Lower Sums, Right Hand Sums, Left Hand Sums Riemann Sums: Trapezoidal Rule: Simpson’s Rule: Change of Variable Definite Integration Absolute Value Functions Fundamental Theorem of Calculus