Wicomico High SchoolMrs. J. A. Austin

2009-2010

AP Calculus 1 ABThird 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 NewtonThe Calculus Wars

Two mathematicians were who was the “father of

at odds for 10 years overcalculus”.

Issac Newton:Notation:

Gottfied Leibnitz:Notation:

Antiderivatives 4.1Many functions can have the SAME derivative:

x2 + 5 ; x2 – 8 ; and x2 + 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 4.1Finding 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 RuleIF 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 RuleEACH PART OF THE SUM OR DIFFERENCE IS GIVEN

ITS OWN INTEGRAL SIGN AND INTEGRATED SEPERATELY.

Indefinite Integration 4.1 Basic Integration Rules

Product RuleAT THIS TIME YOU WILL NOT BE ABLE TO INTEGRATE A

PRODUCT. YOU MUST THEREFORE EXPAND ANY MULTIPLICATION.

Quotient RuleAT 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 exponentsUsing fractional exponentsSeparating polynomialsSeparating fractionsUsing trig identities

IntegrationInitial Conditions & Particular

SolutionsGeneral solution with + CInitial condition substitutionParticular solution

Slope Field: A graph of ALL the possibilitiesTracing general solutionsFinding a particular solution

The Sum of a Series1 + 2 + 3 =1+ 2 + 3 + 4 =1+ 2+ 3+ 4+ 5 =

12 + 22 + 32 =12 + 22 + 32 + 42 =12 + 22 + 32 + 42 + 52=

13 + 23 + 33 =13 + 23 + 33 + 43 =13 + 23 + 33 + 43 + 53

=

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 SummationArea of Rectangles

A = l wThe 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 =

i2, =

i3 =

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 Curvef(x) = -x2 + 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) = -x2 + 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, i2, or i3 behind sigma.

The i, i2, or i3 is now replaced with its formula.

Pg 261 – 263 ( 4 – 64)

Using Midpoints to Estimate AreaFinding a MIDPOINT

Riemann SumsRiemann 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 Sumsx 0 1 3 7 8

f(x) 0 4 18 70 88

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 UNITS2

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 UNITS2

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 WidthUsing the limit definition of summation

Definition of a Riemann SumRegular and general partitions ║∆║ denotes the norm of the partition

Finding Area Using Riemann SumsFinding the “accumulation of change”

Definition of Definite IntegrationUpper and lower limits of integration

Applying Riemann Sums Riemann Sums

Applying Riemann Sums to find average rate of change

Definite IntegralsDefinite IntegralsThe Definite Integral as the Area of a

RegionAreas of Common Geometric Figures

Properties of Definite IntegralsDefinition of Two Special Definite IntegralEvaluating definite integralsUsing the additive interval propertyPreservation of Inequality

The FTC part oneThe Fundamental Theorem of CalculusIsaac Newton and Gottfried Leibniz

Discovered independentlyAnti-differentiation and Integration

Guidelines for Using the FTCConstant of integration C is not needed

The FTCThe Fundamental Theorem of CalculusEvaluating a Definite Integral with

BoundsWith polynomialsWith fractional exponentsInvolving absolute value (total distance)

The Mean Value Theorem for IntegralsConnection to MVT of differentiation

The FTC part twoThe Fundamental Theorem of CalculusAverage Value of a Function

Finding the average value of a function on an interval

The Second Fundamental Theorem of CalculusAccumulation function in respect to timeChange of variable process, using 2nd FTCLocation of the derivative in an integral

Change of Variable ProcessWhen the derivative is in a different variable

than the bounds.

Integration By SubstitutionPattern Recognition

U-substitution in integrationRecognizing patterns in composite functions

Multiplying and Dividing by a ConstantBalancing the integral to accommodate du

Rewriting the integrand entirely in terms of u

Trapezoidal RuleArea 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 Rulen = number of trapezoids

Simpson’s Rulen must be EVENUses double subintervalsUses 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 AnalysisIf 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 ReviewAntiderivative ConceptIntegration RulesSummation FormulasApproximating 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 VariableDefinite Integration

Absolute Value FunctionsFundamental Theorem of Calculus