Calc 2 final

Diverge

A sequence that does not approach a real number (inf, -inf).

Converge

A sequence that approaches a real number (0, 1, 2).

Oscillation

A type of diverging sequence that alternates between values like (0, 7, 0, 7, 0, 7, 0).

Sequence approaches (inf, -inf)

A type of diverging sequence that tends towards infinity or negative infinity.

Value of series = limit of its sequence of partial sums

The sum of a series is the limit of the sequence of partial sums.

Constant multiple rule

Σcan = CΣan, simplifying by factoring out a constant.

Sum rule for series

Σ(an + bn) = Σan + Σbn.

Nth term test

A test for convergence or divergence where if the limit of a sequence is not 0, the series is divergent.

Ratio test

Test for convergence where a(n+1)/an, multiplying by the inverse.

Area

Calculated as ∫top-bottom dx, finding the area under a curve.

To find x bounds

Set equations equal to each other to determine the x values.

Washers rotating over x-axis

y = x, for finding the volume of revolution using washers.

Washers rotating over y-axis

x = y, for finding the volume of revolution using washers.

Shells rotating over x-axis

x = y, for finding the volume of revolution using shells.

Shells rotating over y-axis

y = x, for finding the volume of revolution using shells.

Washer formula

∫pi(outer² - inner²) dx

Shells formula

∫2pi x (outer - inner) dx, formula for finding the volume using shells.

Moment of Y

∫x(x1 - x2) dx

Moment of X

½ ∫(x1² - x2²) dx, calculation for the moment of inertia about the x-axis.

Arc length

sqrt(1 + f’x) dx, formula for finding the length of a curve.

Conical tank

density∫pi(rx/h)²(h-x) dx, formula for the volume of a conical tank.

Vectors dot product

ab/sqrt(a) x sqrt(b), calculation for the dot product of two vectors.

Finding angle between vectors

θ = cos^-1(dot product), formula for finding the angle between two vectors.

Area of a parallelogram

||v|| ||w|| sin θ, formula for finding the area of a parallelogram given two vectors.

How to evaluate integrals

Integrate by parts, use u-substitution, and add solutions together.

Taylor sequence

f^n’(a)/n! (x-a)^n, formula for the Taylor series expansion.

Maclaurin sequence

(f^n(0)x)’/n!, formula for the Maclaurin series expansion.