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Calculus 2

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All Calculus 2Volumes of Solids of RevolutionIntegration by PartsTrigonometric IntegralsTrigonometric substitutionPartial fractionsImproper integralsStrategy for integrationArc lengthArea of a surface of revolutionIntroduction to differential equationsSeparable differential equationsLinear differential equationsParametrized curvesPolar coordinatesSequencesSeries and the integral testComparison testsAlternating series and absolute convergenceRatio and root testsPower series and representations of functionsTaylor and Maclaurin seriesApplications of Taylor polynomials
Reparameterizing Parametric Functions

Show that a parametric function tracing a curve can be re-parameterized to alter the rate at which the curve is traced without changing its shape.

Parametrized curvesMediumVideoNeeds Attention
Slope of Tangent Line to Parametrized Curve

Find the slope of the tangent line to a parameterized curve given functions x(t)x(t) and y(t)y(t).

Parametrized curvesMediumVideoNeeds Attention
Second Derivative of a Parameterized Curve

Find the second derivative of a parameterized curve given functions x(t)x(t) and y(t)y(t).

Parametrized curvesMediumVideoNeeds Attention
Graphing Polar Coordinates

Graph polar coordinates with given radius and angle.

Polar coordinatesEasyVideoNeeds Attention
Solving First Order Differential Equations with Separation of Variables

Solve a first order differential equation using the method of separation of variables.

Separable differential equationsMediumVideoNeeds Attention
Using the Integral Test for Series Convergence

For a series represented with a corresponding function over an interval, use the integral test to determine convergence.

Series and the integral testMediumVideoNeeds Attention
Applying the Divergence Test to Logarithmic Series

Apply divergence test to a series where the sequence does not converge to zero, such as one involving logarithmic terms.

Series and the integral testMediumVideoNeeds Attention
Convergence of Series Using Comparison Tests

For series that resemble quotient forms, like nn3+1\frac{n}{n^3 + 1}, determine convergence using the comparison or limit comparison test.

Comparison testsMediumVideoNeeds Attention
Comparing Series with NonStandard Terms

Attempt comparing series with non-standard terms using the limit comparison test.

Comparison testsMediumVideoNeeds Attention
Applying the Root Test to Series with Power Terms

Utilize the root test for series with terms that include powers like (something)n(\text{something})^n.

Ratio and root testsMediumVideoNeeds Attention
Applying the Ratio Test to Factorial Series

Apply the ratio test to series involving factorial terms and powers, such as those with n!n! or similar structures.

Ratio and root testsMediumVideoNeeds Attention
Improper Integrals and Series Convergence

If you look at the improper interval and the improper integral converges or diverges, whatever it does, the same is true of the series.

Improper integralsMediumVideoNeeds Attention
Convergence of Alternating Series

Take the series (−1)n−1Bn(-1)^{n-1} B_n, where BnB_n is positive. Determine if the series is convergent or divergent based on if Bn+1≤BnB_{n+1} \leq B_n and lim⁡n→∞Bn=0\lim_{{n \to \infty}} B_n = 0.

Alternating series and absolute convergenceMediumVideoNeeds Attention
Alternating Series Convergence Test for Series Involving Fractions

Apply the alternating series test to different series to determine convergence or divergence: (−1)n⋅3n−12n+1(-1)^n \cdot \frac{3n-1}{2n+1} and (−1)n+1⋅n2n3+4(-1)^{n+1} \cdot \frac{n^2}{n^3+4}.

Alternating series and absolute convergenceMediumVideoNeeds Attention
Convergence of n to the n over n Factorial Using Ratio Test

Use the ratio test to determine the convergence of the series nnn!\frac{n^n}{n!}.

Ratio and root testsMediumVideoNeeds Attention
Basic Comparison Test for Series

Suppose that the two series ana_n (the series we care about) and bnb_n (the series we will use for comparison) have positive terms. If the series bnb_n is convergent and the terms an≤bna_n \leq b_n for all nn, then the series ana_n converges.

Comparison testsEasyVideoNeeds Attention
Convergence of Series Using the Direct Comparison Test

Using the direct comparison test, determine whether a series converges or diverges when one series is bounded by another, given that both sequences are positive.

Comparison testsMediumVideoNeeds Attention
Exploring Differential Equations with Initial Conditions

For our first actual example, we're going to pretend this random differential equation with initial conditions that I just got from my textbook yields a solution that's very important to an engineer or scientist.

Introduction to differential equationsMediumVideoNeeds Attention
Taylor Polynomial of Degree n at a Specific x Value

Find the Taylor polynomial of degree n at x = C.

Taylor and Maclaurin seriesMediumVideoNeeds Attention
Maclaurin Series for Cosine of Double Angle

Using the Maclaurin series for cos⁡(x)\cos(x), rewrite the series to accommodate cos⁡(2x)\cos(2x), and simplify the expression as necessary.

Taylor and Maclaurin seriesMediumVideoNeeds Attention