Full YearReview

MTH 251/252/253 – Calculus

MTH251 – Differential CalculusLimitsContinuityDerivativesApplications

MTH251 – Differential CalculusLimits

Evaluate limits at a point.

Evaluate end behavior limits.

Evaluate limits of quotients.

L’Hôpitals Rule

lim ( )x a

f x

lim ( )x

f x

If lim ( ) and lim ( ) are both 0 or both ,

( ) '( )then lim lim

( ) '( )

x a x a

x a x a

f x g x

f x f x

g x g x

( )lim

( )x a

f x

g x

MTH251 – Differential CalculusContinuity

Where does discontinuity occur?Jumps (piecewise functions)AsymptotesHolesIntervals where the function is

undefinedGiven a graph or equation of a

function; indicate where the function is discontinuous

MTH251 – Differential CalculusDerivatives

PolynomialsTrigonometric and Inverse

TrigonometricExponentialLogarithmic (esp. Natural Logs)ProductsQuotientsChain Rule

MTH251 – Differential CalculusApplications

Optimization Problems (i.e. max/min problems)

Equations of tangent lines

Rectangular

Parametric

0 0( , )x y

dym

dx

0

dydt

dxdt t

m

0 0( )y y m x x

MTH252 – Integral Calculus

Indefinite IntegralsDefinite IntegralsMethods of IntegrationApplications

MTH252 – Integral Calculus

Indefinite Integrals Antiderivative

F(x) is an antiderivative of f(x) iff F’(x) = f(x)

Indefinite IntegralThe set of all antiderivatives of a

function.( ) ( )

where ...

'( ) ( )

is any constant

f x dx F x c

F x f x

c

MTH252 – Integral Calculus

Definite IntegralsThe limit of a Riemann Sum

The Fundamental Theorem of Calculus

1

( ) lim ( )

where ... &

nb

ka nk

k

f x dx f x x

b ax x a k x

n

( ) ( ) ( )

where ... '( ) ( )

b

af x dx F b F a

F x f x

MTH252 – Integral Calculus

Methods of IntegrationRecognition: poly, trig, exp, log, …Algebraic Manipulation(Simple) SubstitutionPartsPowers of Trigonometric FunctionsTrigonometric SubstitutionsPartial Fractions

MTH252 – Integral Calculus

ApplicationsArea Problems

MTH253 – Analytic Geometry, Sequences, & Series ConicsPolar Coordinates & EquationsSequencesSeries w/ Tests for ConvergenceTaylor & Maclaurin Series

Conics

Rotate (eliminate Bxy)

Translate (eliminate Dx and Ey) … complete the squares

ParabolaEllipseCircleHyperbolaPolar Forms

2 2 0Ax Bxy Cy Dx Ey F 11

2 tanB

A C

2 4x py 2 4y px

2 2

2 21

x y

a b

2 2 2x y r 2 2

2 21

x y

a b

2 2

2 21

y x

a b

2 2 2c a b

2 2 2c a b

1 sin

ekr

e

1 sin

ekr

e

1 cos

ekr

e

1 cos

ekr

e

Foci:

Eccentricity:

Directrixes:

MTH253 – Analytic Geometry, Sequences, & Series

2 2

2 21

x y

b a

cae

2a ae cd

c

Polar Coordinates & EquationsLocate & Identify PointsGraph Equations

Circles, Flowers, Limaçons/Cardiods, Leminiscates, & Spirals

Convert to and from Rectangular Coordinates & Equations

Tangent Lines

0

0 0 0 0 0 0

0 0

( , ) ( cos , sin )

sin cos

cos sin

( )

drd

drd

x y r r

rm

r

y y m x x

MTH253 – Analytic Geometry, Sequences, & Series

SequencesAn Ordered Set or List of NumbersNotationGiven the first several terms, find the

nth term.Evens 2nOdds 2n+1 or 2n-1Alternating (–1)n or (–1)n+1 Constant Increment kn + aFactorials n! or (n + a)! or (n – a)!

Convergence

1, , ... s

n r r sn ra a a a

lim nna

MTH253 – Analytic Geometry, Sequences, & Series

SeriesThe sum of the terms of a sequence.NotationSequence of partial sumsTest for Convergence

Divergence TestGeometric Series & P-SeriesIntegralComparison & Limit ComparisonRatio & RootAlternating Series and Absolute

Convergence

1 21

... ... n nn

a a a a

MTH253 – Analytic Geometry, Sequences, & Series

Taylor & Maclaurin SeriesTaylor Series at x = a

Maclaurin Series

Three Important Maclaurin Series

( )2''( ) ( )

( ) ( ) '( )( ) ( ) ( ) 2! !

nnf a f a

f x f a f a x a x a x an

( )2''(0) (0)

( ) (0) '(0) 2! !

nnf f

f x f f x x xn

2

2 2

3 2 1

1 2! !

cos 1 ( 1) 2! (2 )!

sin ( 1) 3! (2 1)!

nx

nn

nn

x xe x

n

x xx

n

x xx x

n

MTH253 – Analytic Geometry, Sequences, & Series