PARAMETRIC EQUATIONS Sketch Translate to Translate from Finds rates of change i.e. Find slopes of tangent lines Find equations of tangent lines Horizontal and vertical tangent lines Singular Points

POLAR COORDINATES Sketch Translate to rectangular coordinate system Translate from rectangular coordinate system Definitions: Examples worth remembering

LENGTH OF AN ARC y = f(x) a b

The M.V.T. asserts : There exists an in the interval such that or Therefore Factor out from each term and then 

Or even better….

For functions defined parametrically…..

For Polar Functions….. Remember….. You do the rest and you will have the necessary formula for arc length

Rectangular Areas

Polar Areas

Maclaurin/Taylor series Tangent Line

To Quadratics and beyond Quadratic Approximation

SEQUENCES

Recall: Note: Theorem states for sequences what previous theorems said about functions. When the limit doesn’t exist, we say that the sequence diverges.

Theorem If For example: Note: L’Hôpital’s Rule applies just as it does with functions.(sort of)

Does the sequence converge or diverge?

MONOTONE SEQUENCES Strictly increasing: Increasing: Strictly decreasing: Decreasing: If then the sequence is _______________ If then the sequence is _______________ If then the sequence is______________ If then the sequence is______________

Show that the sequence is strictly increasing or strictly decreasing 1.Ratio Method 2.Difference Method 3.Derivative Method

INFINITE SERIES Partial Sums If the sequence of partial sums converges to a limit S then the infinite series converges and its sum is S. If this sequence diverges(i.e. the limit DNE) then the series diverges. There is no sum.

S =.1S = S -.1S = __________.9S = ________ S = _____

GEOMETRIC SERIES if __________

TELESCOPING SERIES

CONVERGENCE TESTS That is, in order for a series to converge the terms of the series must be heading toward 0. However, if the terms are heading to 0 that does not imply that the series converges. The Harmonic Series…… diverges

HARMONIC SERIES

CONVERGENCE TESTS(cont.)

For example…

CONVERGENCE TESTS(cont.) Integral test converges if converges and diverges if diverges.

For example…

CONVERGENCE TESTS(cont.) Comparison Test If are series with non-negative terms, and then will converge if converges. Similarly, will diverge if diverges.

CONVERGENCE TESTS(cont.) Limit Comparison Test If are series with positive terms and and is finite, then either both series converge or both diverge.

CONVERGENCE TESTS(cont.) Ratio Test is a series with positive terms If

EXAMPLES Comparison Test Limit Comparison Test Ratio Test

More ratio tests

CONVERGENCE TESTS(cont.) Alternating Series Test is an alternating series If and, then the series converges.

CONVERGENCE TESTS(cont.) Absolute Convergence A series converges absolutely if the series of absolute values converges. A series diverges absolutely if the series of absolute values diverges.

CONVERGENCE TESTS(cont.) If the series converges, then so does the series

CONVERGENCE TESTS(cont.) Absolute Convergence Ratio Test 1.If then the series converges absolutely and therefore converges 2.If then the series diverges 3. If the ratio is 1, then the test fails.

CONVERGENCE TESTS(cont.)

TAYLOR AND MACLAURIN SERIES Taylor Polynomial Taylor Series

MACLAURIN SERIES

TAYLOR SERIES about x = P(x) =

Interval of Convergence Compare with

Interval of Convergence (cont.)

Maclaurin Series

New Series from Old

Taylor Series Error Estimation Estimation Theorem- for some in the Interval

Estimate to 5 decimal places