Home ## The Fundamental Theorem of CalculusThree Different ConceptsThe Fundamental Theorem of Calculus (Part 2) The Fundamental Theorem of Calculus (Part 1) More FTC 1 ## The Indefinite Integral and the Net ChangeIndefinite Integrals and Anti-derivativesA Table of Common Anti-derivatives The Net Change Theorem The NCT and Public Policy ## SubstitutionSubstitution for Indefinite IntegralsExamples to Try Revised Table of Integrals Substitution for Definite Integrals Examples ## Area Between CurvesComputation Using IntegrationTo Compute a Bulk Quantity The Area Between Two Curves Horizontal Slicing Summary ## VolumesSlicing and Dicing SolidsSolids of Revolution 1: Disks Solids of Revolution 2: Washers More Practice ## Integration by PartsIntegration by PartsExamples Integration by Parts with a definite integral Going in Circles Tricks of the Trade ## Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric FunctionsProduct of Sines and Cosines (mixed even and odd powers or only odd powers) Product of Sines and Cosines (only even powers) Product of Secants and Tangents Other Cases ## Trig SubstitutionsHow Trig Substitution WorksSummary of trig substitution options Examples Completing the Square ## Partial FractionsIntroductionLinear Factors Irreducible Quadratic Factors Improper Rational Functions and Long Division Summary ## Strategies of IntegrationSubstitutionIntegration by Parts Trig Integrals Trig Substitutions Partial Fractions ## Improper IntegralsType 1 - Improper Integrals with Infinite Intervals of IntegrationType 2 - Improper Integrals with Discontinuous Integrands Comparison Tests for Convergence ## Differential EquationsIntroductionSeparable Equations Mixing and Dilution ## Models of GrowthExponential Growth and DecayLogistic Growth ## Infinite SequencesApproximate Versus Exact AnswersExamples of Infinite Sequences Limit Laws for Sequences Theorems for and Examples of Computing Limits of Sequences Monotonic Covergence ## Infinite SeriesIntroductionGeometric Series Limit Laws for Series Test for Divergence and Other Theorems Telescoping Sums and the FTC ## Integral TestRoad MapThe Integral Test Estimates of Value of the Series ## Comparison TestsThe Basic Comparison TestThe Limit Comparison Test ## Convergence of Series with Negative TermsIntroduction, Alternating Series,and the AS TestAbsolute Convergence Rearrangements ## The Ratio and Root TestsThe Ratio TestThe Root Test Examples ## Strategies for testing SeriesStrategy to Test Series and a Review of TestsExamples, Part 1 Examples, Part 2 ## Power SeriesRadius and Interval of ConvergenceFinding the Interval of Convergence Power Series Centered at $x=a$ ## Representing Functions as Power SeriesFunctions as Power SeriesDerivatives and Integrals of Power Series Applications and Examples ## Taylor and Maclaurin SeriesThe Formula for Taylor SeriesTaylor Series for Common Functions Adding, Multiplying, and Dividing Power Series Miscellaneous Useful Facts ## Applications of Taylor PolynomialsTaylor PolynomialsWhen Functions Are Equal to Their Taylor Series When a Function Does Not Equal Its Taylor Series Other Uses of Taylor Polynomials ## Partial DerivativesVisualizing Functions in 3 DimensionsDefinitions and Examples An Example from DNA Geometry of Partial Derivatives Higher Order Derivatives Differentials and Taylor Expansions ## Multiple IntegralsBackgroundWhat is a Double Integral? Volumes as Double Integrals ## Iterated Integrals over RectanglesHow To Compute Iterated IntegralsExamples of Iterated Integrals Cavalieri's Principle Fubini's Theorem Summary and an Important Example ## Double Integrals over General RegionsType I and Type II regionsExamples 1-4 Examples 5-7 Order of Integration |
## Taylor PolynomialsProbably the most important application of Taylor series is to
We look at various Taylor polynomials: T_1(x)&=&f(a) + f'(a) (x-a)\\ T_2(x)&=&f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2\\ T_3(x)&=&f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3\\ &\vdots&\\ T_n(x)&=&f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n \end{eqnarray}$ You may recognize the first Taylor polynomial above. (Does it look familiar? Take a moment to
think about this.) It is the Taylor polynomials extend the idea of
linearization. To approximate $f$ at a given
value of $x$, we will use $T_n(x)$ for a value of $n$ that gives a
good enough approximation. We see from $T_n(x)$ above that we
will need to find an $x$-value $a$ at which we can evaluate the
derivatives of $f$. Our
approximations get better the more terms we include,
and they also allow us to get further from $a$ and still have a good
approximation.
$\begin{array}{lll} If you know a series,
finding an $n^{th}$ degree Taylor polynomial is easy.
If you don't have the series at hand, you can compute the polynomial
as we just did for sine.My calculator approximates $\sin(.5)=.4794255386$, and $T_7(.5)=.5-\frac{.5^3}{6}+\frac{.5^5}{120}-\frac{.5^7}{5040}=.4794255332$.
This video will compute approximations of $e^x$ and $\ln(x)$ near $a=0$, and $\sin x$ near $a=\frac{\pi}{4}$. |