Calc BC Test

lim definition of derivative

lim h→0 f(x+h)-f(x) divided by h

u substitution

integarla f(g(x))dx let u=g(x)

integration by parts

integaral u dv = uv-integral v du

d/dx tan x

sec²x

d/dx secx

secxtanx

how to find displacement

integral a to b v(t)dt

how to find total distance traveled

integral a to b |v(t)|dt

how to find speed using velocity

speed=|velocity|

e^x mclaurin series

1 +x+x²/2! + x³/3! … x^n/n!

sinx maclaruin series

sinx = x - x³/3! + x^5/5! …(-1)^n * x^(2n+1) divided by (2n+1)!

cosx mclurin series

cosx = 1 - x²/2! + x^4/4! … (-1)^n * x^(2n)/2n!

maclaurin seriesseires

taylor siries with a=0

taylor seires

f(x)= f(a)+f’(a)(x-a) + f;;(a)/2! (x-a)² … f^(n) (a) divided by n! * (x-a)^n

disc method formula

V=pi integral from a to b r² dx

washer method formula

pi integral a to b (R² - r²) dx

volume of a shell formula

v=2pi integral from a to b rh dx

volume of a cross section formula

V = integral from a to b A dx

M

carrying capacity

Logistic formula

M divided by 1+Ce^-kt

population growth

dp/dt = k/m *p(m-p)

Euler’s method chart

(x,y), dy/dx, change in x, change in y = dy/dx * change in x, (x,y)

average rate of chnage

f(b)-f(a) divided by b-a

instantaneous rate of change

f’(x)

average value of a function

integral from a to b f(x) dx divided by b-a

intermediate value theorm

a function f(x) that is continuous on [a, b] takes on every y-value between f(a) and f(b)

extreme value theorm

if f(x) is continuous on [a, b], then f(x) must have both an absolute minimum and absolute maximum on the interval [a,b]

arc length formula for cartesian

integral from a to b √(1 + (dy/dx)^2) dx

arc length formula for parametric

integral from a to b √((dx/dt)² + (dy/dt)²) dt

formula for speed of a parametric

square root of ((dx/dt)² + (dy/dt)²)

total distance travelled formula parametric

integral from a to b square root of ((dx/dt)² + (dy/dt)²) dt

polar area formula

½ integral from theta 1 to theta 2 r² dtheta

parametric 1st derivative

dy/dx = (dy/dt) divided by (dx/dt)

parametric 2nd derivative

d²y/dx² = (d/dt(dy/dx)) divided by (dx/dt)

nth term test

diverges if lim as n approaches infinity of a_{n does not equal 0}

if g=f^-1 (x) then g’(x) = ?

1 divided by f’(g(x))

alternating series error

error <= | a_n+1 |

langrange error

error <= M divided by (n+1)! times (x-a)^n+1

M

largest value for next term

slope of the tangent line

y-y1=m(x-x1)

decomposing into partial fraction

1 divided by ((cx+d)(hx+k)) = A divided by (cx+d) + B divided by (hx+k)

ratio test

1. 1. Calculate the limit:

   For a series with terms a_n, compute the limit: L = lim (n→∞) |a<sub>n+1</sub> / a<sub>n</sub>|. 

2. 2. Interpret the limit:

   • If L < 1: The series converges (absolutely). 

   • If L > 1 or L = ∞: The series diverges. 

   • If L = 1: The test is inconclusive, and you need to use another test to determine convergence or divergence. 

alternating series test

∑(-1)^n*a_n, where a_n > 0 for all n. The series converges if the terms (a_n) are decreasing and approach zero as n approaches infinity

p-series test

∑ 1/n^p p>1 converges, p<= 1 diverges

geometric series test

∑ar^n |r| < 1 converges, |r| >= 1 diverges, S=a divided by (1-r)

limit comparison test

1. 2. Calculate the Limit:

   Compute the limit of the ratio an/bn as n approaches infinity. Let's call this limit c. 

2. 3. Interpret the Limit: 

   • If c is a positive finite number (i.e., 0 < c < ∞), then both series either converge or both diverge. 

   • If c is 0 or infinite, the Limit Comparison Test is inconclusive