Know Cold Flashcards
memorize
limits: vertical asymptote (x=c)
lim f(x) = +/-infinity
x–>c
horizontal asymptote (y=L)
lim f(x) = L
x–> +/-infinity
L’Hospital
f(c)/g(c) = 0/0 or infinity/infinity then lim f(x)/g(x) = lim f’(x)/g’(x)
x–> c
When is a function continuous?
f(c) is defined
lim(-) f(x) = lim(+) f(x)
lim f(c) = f(c)
x–> c
d/dx[C]
0
d/dx[sin(x)]
cos(x)
d/dx[cos(x)]
-sin(x)
d/dx[tan(x)]
sec^2(x)
d/dx[sec(x)]
sec(x)tan(x)
d/dx[e^x]
e^x
d/dx[a^x]
ln(a)a^x
d/dx[ln(x)]
1/x
d/dx[cot(x)]
-csc^2(x)
d/dx[csc(x)]
-csc(x)cot(x)
d/dx[x^(n-1)]
nx^(n-1)
IVT
if f(x) is continuous on the the interval [a,b] and Y is between f(a) and f(b) there exist a c value within [a,b] where f(c) = Y exists
MVT
if f(x) is continuous on [a,b] and differentiable on (a,b) then there exists some C value within (a,b) such that f’(c) = f(b)-f(a)/b-a
EVT
if f(x) is continuous on [a,b] then f(x) has an absolute maximum and absolute minimum on [a,b]
total distance traveled
integral of the absoulte value of v(t)
displacement
integral of v(t)
volume of solids (cross sections) - square
integral S^2
volume of solids (cross sections) - rectangles
integral B * H
volume of solids (cross sections) - isosceles right leg
1/2 integral L^2
volume of solids (cross sections) - isosceles right hypotenuse
1/4 integral H^2
volume of solids (cross sections) - equilateral triangle
sqrt3/4 integral S^2
volume of solids (cross sections) - semicircles
pi/8 integral D^2
volume of solids (discs and washers) - discs
pi integral r^2
volume of solids (discs and washers) - washers
pi integral (R^2 - r^2)
Arc Length
s = integral sqrt(1+[f’(x)]^2)
critical number
f’(x) = 0 or DNE
increasing
f’(x) > 0
decreasing
f’(x) < 0
concave up
f’‘(x) > 0
concave down
f’‘(x) < 0
relative min
f’(x) changes - to +
relative max
f’(x) changes + to -
inflection point
f’‘(x) changes sign or f’(x) has a relative min or max
fraction decomposition
integral 1/(x-a)(x-b) = integral A/(x-a) + B/(x-b)
partial decomposition
intergral udv = uv - integral vdu
euler’s method when change in x and (x(0), y(0)) given
dy/dx = F(x,y)
euler’s method when y(n+1) = y(n) + (change in x)F(x(n), y(n))
x(n+1) = x(n) + change in x
Exponential Model
dy/dx = ky = Ce^kt
Logistic Model
dy/dx = kLy (1-k/L)
dy/dx = ky (L-y)
y = L/(1+Ce^(-kLt))
Logistic –> L = ?
carrying capacity
Logistic –> 2/L = ?
inflection point
