Integration:

What two special options do you have when integrating?

Integration:

U-Substitution and Integration by Parts

Integration:

What is the formula for Integration by Parts?

Integration:

∫udv = uv - ∫vdu

Integration:

What is the hierarchy of picking a u value for U-Substitution and Integration by Parts?

Integration:

Log

Inverse trig

Algebra

Trig

Exponents

Integration:

When faced with a integral in which one of the limits of integration is ∞ (or -∞), what do you do?

Integration:

Replace the ∞ limit with t and take the limit as t → ∞ of integral. The final answer will either be a value or "divergent."

Integration:

What two items are easily forgotton that apply to all integrals? (Things to check before/after finishing a problem)

Integration:

For ∫f(x)dx

You should always check if f(x) is continuous between the limits of integration before beginning the problem (ex: 1/(3-x) is NOT continuous at x=3; if 3 is within the limits of integration, you cannot integrate normally).

Integration:

When faced with a non-continuous integral (when f(x) does not exist at some value between the limits of integration), what do you do?

Integration:

You must split the integral into two parts: one for x<# and one for x># (where # is the discontinuous value). Replace the # in the limits of integration with t and take the limit of the first part as t → #- and the second as t → #+. Your final answer will either be a number or "does not exist."

Integration Approximation:

If you are asked to find a value n (number of trapezoids, rectangles, trails, etc.) so that the output will have a certain accuracy, what must you use?

(You will be given that lf"(x)l ≤ K and an equation for lEl)

Integration Approximation:

If you are given ∫f(x)dx...

Find lf"(x)l and find the largest possible value for it between your limits of integration; this will be your K. Plug in your accuracy to E and plug your K into your lEl equation and solve for n. Your final answer should be n ≥ #)

Volumes Via Integrals:

What is the formula(s) for finding a volume using shells?

Volumes Via Integrals:

Shell volume = 2πrhΔx

r = x-(rotational x value)

h = (upper function)-(lower function)

Use ∫(shell volume)dx from (leftx) to (rightx) to find the volume of the entire shape.

Remember: Δx becomes dx when taking an integral.

Differential Equations:

Explain how to use Euler's Method to approximate f(x) at x when given a function of f'(x), a step size, and a point.

Differential Equations:

You can use a chart with the table headers: Point, Slope, and New Point. Use your given point as your first point. Use the function of f'(x) to find the slope. Multiply the slope with the given step size and add that value to your point's y value; then add your given step size to your x value to create a new point: (x+stepsize,y+slope(stepsize)) This new point will now become the Point value in the next row of your chart.

You will repeat this process until your output is a point with the x value you were asked to approximate.

Integral Applications:

What is the formula for work and what values can be used for each part?

Integral Applications:

W = Fd

W = work (joules[J])

F = force (pounds[lbs], mass*gravity[kgm/s²], newtons[N], etc.)

d = distance (feet[ft], yards[yd], meters[m], etc.)

Force will usually be given in the form of a mass or weight value.

Distance will be used as Δx or x when using integrals.

Integral Applications:

What is the formula for hydrostatic force and what values are used for each part?

Integral Applications:

HF = ΡgAd

Ρ = density of liquid (given value)

g = 9.8m/s²

A = Δx*(width)

d = x

Integral Applications:

To find the exact value of a ∑ sum (after finding the equation from the work or hydrostatic force problem) what do you do?

Integral Applications:

Take the limit as n → ∞ of the ∑ sum.

Differential Equations:

What is the auxilliary equation of a differential equation?

Differential Equations:

The auxilliary equation can be found by replacing x" with r², x' with r, and x with 1 in an equation and setting the entire equation equal to zero. Solving for r will then help you choose what general solution to use.

Differential Equations:

What are the three forms of complementary solutions and what values of r (from auxilliary equation) correspond to each?

Differential Equations:

Two unique r values: y = c₁e^r₁x + c₂e^r₂x

Only one r value: y = c₁e^rx + c₂e^rx

Two non-real r values of the form α±βi:

y = e^αx[c₁cos(βx) + c₂sin(βx)]

Differential Equations:

What are the three major forms of particular solutions and how are they chosen?

Differential Equations:

Chosen based upon G(x) if a differential equation equals G(x)

If G(x) is a polynomial of degree n, then the particular solution will be Axⁿ+Bx^n-1+...+C

If G(x) is sin(kx) or cos(kx), then the particular solution will be Asin(kx)+Bcos(kx)

If G(x) is e^kx, then the particular solution will be Ae^kx

If G(x) is a sum or product of these, the particular solution will also be a sum or product of the these.

Differential Equations:

How do you find a general solution?

Differential Equations:

To find a general solution to a differential equation, add the complementary solution and the particular solution to the differential equation.

(You'll have to find the values of A, B, C, D, etc for the particular solution by using G(x) as a guide first.)

Differential Equations:

If a differential equation already equals zero, how do you find the general solution?

Differential Equations:

The answer to a differential equation that equals zero is just the complementary solution. Solve for c₁ and c₂ if possible.

Differential Equations:

How do you find orthogonal trajectories?

Differential Equations:

Solve for dy/dx and then take the reciprocol and solve for y.

Differential Equations:

What is Newton's Law of cooling?

(Equation for dy/dx)

Differential Equations:

dy/dx = k(T_object - T_environment)

Series and Sequences:

What is the Divergence Test?

Series and Sequences:

The Divergence Test states that if the limit as n → ∞ of a_n is NOT zero, then ∑a_n diverges.

If the limit is zero, you cannot conclude anything from the test.

Series and Sequences:

What is the form of a geometric series and what can you conclude for a geometric seires?

Series and Sequences:

A geometric series is of the form:

∑arⁿ

1) If lrl ≥ 1, the seires diverges.

2) If lrl < 1, the series converges to a/(1-r)

Seires and Sequences:

What is a power series (p-series) and what can you conclude from it?

Series and Sequences:

A power series is of the form:

∑c_n(x-a)ⁿ

centered at a.

1) It could converge only at a (R=0)

2) It could converge at all x values (R=∞)

3) It could converge for lx-al < R

Taylor Series:

What is the form of a Taylor Series?

Taylor Series:

A Taylor series is of the form:

∑[f⁽ⁿ⁾(a)/n1]*[x-a]ⁿ

Taylor Series:

If you find a Taylor Series representation of a function that fits T(x) for only part of the n values, what must you do?

Taylor Series:

You must add the f⁽ⁿ⁾(a) values for every n value that does not fit the T(x) pattern to your Taylor Series representation.

Taylor Series:

What is R_n(x) and how do you find its value?

Taylor Series:

R_n(x) is the remainder term.

R_n(x) = f(x) - T_n(x)

lim_(n→∞) R_n(x) = f(x) - T(x)

lR_n(x)l ≤ M/(n+1)!*lx-alⁿ⁺¹

where d ≥ lx-al and M is the highest value of f⁽ⁿ⁺¹⁾(x) on the d interval.

Taylor Series:

How do you prove that a Taylor Series representation actually exists?

Taylor Series:

You must show that f(x)=T(x) by finding that the limit of R_n(x) as n → ∞ is zero.