Level 599
Level 601

#### 67 words 0 ignored

Ready to learn
Ready to review

## Ignore words

Check the boxes below to ignore/unignore words, then click save at the bottom. Ignored words will never appear in any learning session.

**Ignore?**

∫k f(u) du=

k ∫f(u) du

∫[f(u) ± g(u)] du=

∫f(u) du ± ∫g(u) du

∫kdu=

ku+C

∫u^n du =

(u^(n+1))/(n+1) +C , n≠ -1

∫cosu du=

sinu + C

∫sinu du=

-cosu + C

∫sec^2 u du=

tanu + C

∫secu tanu du=

secu + C

∫csc^2 u du=

-cotu + C

∫cscu cotu du=

-cscu + C

∫e^u du=

e^u + C

∫a^u du=

(a^u/ ln a) + C

∫tanu du=

-ln |cosu| + C

∫cotu du=

ln |sinu| + C

∫secu du=

ln |secu + tanu| + C

∫cscu du=

-ln |cscu + cotu| + C

∫1/u du=

ln |u| + C

∫lnu du=

ulnu-u + C

arcsin(u/a) + C

∫du/ √(a^2 - u^2 ) =

(1/a)arctan(u/a) + C

∫du/ (a^2 + u^2) =

(1/a)arcsec(|u|/a) + C

∫ du/u√u²-a²

(f(b)-f(a))/(b-a)

Slope of secant line of f(x) on [a,b].

Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that

intermediate value theorem (IVT)

suppose f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b). Then there exists a number "e" such that f(e)=N.

Definition of Continuity

At a point:

Critical Number

Let f be defined at c. If f'(c)=0 or if f' is undefined at c, then c is a critical number of f.

First Derivative test

If the first derivative changes from negative to positive at c then f has a relative minimum at the point (c,f(c)) If it goes from positive to negative it is a relative maximum

Second derivative test

If the second derivative is greater than zero then the function has a relative minimum

concavity

a graph is concave up is the tangent lines are below the graph; a graph is concave down if the tangent lines are above the graph

Test for Concavity

Let f be a function whose second derivative exists on an open interval I.

Inflection Point

a point in the domain of the function at which a tangent line exists and the concavity of the function changes.

First Fundamental Theorem of Calculus

int from a to b of f'(x)dx = f(b)-f(a)

Second Fundamental Theorem of Calculus

deriv of integral from a to x of f(t)dt = f(x)

Average value of f(x) on [a,b]

(1/(b-a))*(integral from a to b of f(x)dx)

Volume around a horizontal axis by discs

V=pi(integral from a to b of [r(x)]^2dx

Volume around a horizontal axis by washers

V=pi(integral from a to b of [R(x)]^2-[r(x)]^2

V= ∫(from a to b) A(x)dx

Volume by cross sections perpendicular to the x-axis

velocity

The rate at which distance is changing with time.

Speed

D / T

acceleration

a=v¹= dx/dt

Displacement

Fluid being pushed back out of the way by an object and talking its place.

Total Distance (from x=a to x=b)

∫(from a to b) |v(t)dt| or |∫(from a to c) v(t)dt| + |∫(from c to b) v(t)dt|

Integration by parts

If u = f(x) and v = g(x), and if f'(x) and g'(x) are continuous, then

Length of Arc for functions

s= ∫(from a to b) √1+[f'(x)]^2 dx

dP/dt= kP(L-P)

Logistic Growth Formula

P= L/[1+Ce^(-Lkt)]

Solution of Logistics Differential Equation

LaGrange Formula for Error Bound

|R(n)|≤ | [M/(n+1)!]* (b-c)^(n+1) |

Parametrics

dy/dx= (dy/dt)/ (dx/dt)

Position Vector

(x(t), y(t))

Velocity Vector

(x'(t), y'(t))

Acceleration Vector

(x"(t), y"(t))

|v(t)|= √[(dx/dt)^2 + (dy/dt)^2]

Speed (Magnitude of Velocity Vector)

Distance traveled from t=a to t=b (arc length)

s= ∫(from a to b) √[(dx/dt)^2 + (dy/dt)^2] dt

x= rcosθ, y= rsinθ

Relationship of Polar and Rectangular

Slope of a Polar Curve

dy/dx= (rcosθ+ r'sinθ)/ (-rsinθ + r'cosθ) or

Area of a Polar Curve

A= (1/2) ∫ (from alpha to beta) r^2 dθ

Arc Length of a Polar Curve

s= ∫(from alpha to beta) √[r^2 + (dr/dθ)^2] dθ

nth term test for divergence

lim (as n approaches infinity) of a{n} ≠ 0, then the series diverges

geometric series

sum of the terms of an arithmetic sequence

P-Series

Sum where 1/(n^p) converges if and only if p > 1

Alternating Series

∑(from n=1 to infinity) (-1)^n-1 a{n}. Converges if 0<a{n+1}<a{n} and lim (as n approaches infinity) a{n}= 0

Integral Test

take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent

Ratio Test

limit as n→∞ of an+1/an

Direct Comparison

Series converges if 0<a{n}<b{n} and ∑(from n=1 to infinity) b{n} converges; Series diverges if 0<b{n}<a{n} and

limit comparison

Series converges if lim (as n approaches infinity) a{n}/ b{n}= L>0 and

Power series for e^x

1+ x + x^2/2! + x^3/3! + x^4/4! +...=∑(from n=1 to infinity) x^n/n!

Power Series for sin x

x - x^3/3! + x^5/5! - x^7/7! + x^9/9! -...=