Formulas Flashcards
area underneath a line
integral (a to b) F(x) dx
exact total area between to functions
integral (xhigh - xlow) dy
volume of a 3D shape
(slice area) (thickness or height)
area of a circle
pie r^2
volume of a ring
(ring area)(thickness)
ring area = (pier^2out - pier^2in)
mass when density is constant
[lineal density (Kg/m)][length(m)]
total mass-circular
E (density)(2pie*r * delta r)
center of mass
(Eximi)/(Emi) = moment/total mass of all objects
xi = m
mi = Kg
y = integral ( y * mass of slice)/ integral (mass of slices)
volume of a circle
pier^2*h
Newtons law of heating and cooling
dH/dt = -a(H -Tenv)
drug dosing
dM (mg in body)/dt (time) = rate in - rate out
logistic growth
dp/dt = KP(L-P)
distance between 2 points in R^3
sqrt( (x-a)^2 + (y-b)^2 + (z-c)^2))
formula for a circle centered on the origin
x^2 + y^2 = r^2
linear two variable function
f(x,y) = mx + ny + c
partial derivative of f with respect to x (if no formula is given and only a chart)
(Fend-Fstart)/ (Xend-Xstart)
point-slope form
z = c +m(x-a) + n(y-b)
tangent plane formula
z = f(a,b) + fx(a,b)(x-a) + fy(a-b)(y-b)
- can easily be adjusted for a three or more variable function
z = f(a,b,c) + fx(a,b,c)(x-a) + fy(a-b,c)(y-b) + fz(a,b,c)(z-c)
|error bound|
= |fx(a, b)(max delta x)| + |fy(a, b)(max delta y)
length of a vector
sqrt( v1^2 + v2^2)
special triangles
(1,1, sqrt2), (1, sqrt3, 2)
normalization of a non-unit vector
(1/||V||)V = (1/(sqrt(V1^2 + V2^2))) V
directional derivative
Duf(a, b) = fx(a, b) *U1 + fy(a, b) * U2
or
grad f * U
slope of the surface in the direction of U
grad f
= < fx, fy>
direction of maximum increase/slope
chain rule
dz/dt = (pz/pz * dx/dt) + (pz/py *dy/dt)
p = partial derivative
is + fxx concave up or down
up
volume of a cylinder
Pie(r)^2 h
where do you find saddle points
where two contours cross and in circle clusters
second derivative test
D = fxx(a,b) * fyy(a, b) - (fxy(a, b))^2
D> 0 and fxx > 0 = minimum
D > 0 and fxx < 0 = maximum
D < 0 = saddle point
D = 0 then the test is inconclusive
surface area of a closed rectangular box (x, y, z)
= 2xz + 2yx + 2xy
if lambda is > 0 then grad f and grad g…
have the same direction and are parallel
if lambda is < 0 then grad f and grad g…
have opposite directions but are still parallel
exponential growth and decay
dp/dt = KP —> P(t) = Poe^kt
if the line is concave up Euler’s method will be an …
underestimate
integration by parts
integral u (dv) = uv - integral v(du)
1. choose part of the integral to be u and the remaining to be dv
2. differentiate u to get du
3. integrate dv to get v
4. replace (integral u dv) with (uv - integral v(du))
5. hope/check that the new integral is easier to evaluate
area of a triangle
1/2(b*h)
sin^2(x) + cos^2(x) =
1
derivative of tan(x)
1/ (cos(x))^2
derivative of arctan(x)
1/(x^2 +1)
derivative of arcsin(x)
1/sqrt(1-x^2)
where is the radius on a cone, what will be the radius at x?
1/3 from the vertex
r = x/3
what does proportional mean when making an equation
multiplication
what does inversely mean when creating an equation?
1/x, times the reciprocal, times the inverse of the function…
product of
multiplication
difference between
subtraction
acceleration
second derivative
if the line is concave up Eulers method will be?
under estimate
density
mass/volume
how do you know if two vectors are perpendicular
angle between them is 90 and the dot product is 0
partial f of x and f of y in vector terms
The partial derivative fx is the rate of change of f in the direction of the unit
vector ~i (towards larger x values)
* The partial derivative fy is the rate of change of f in the direction of the unit
vector ~j (towards larger y values)
what are the three key properties of the gradient
- direction of the grad vector is the direction of maximum increase of function f at point (a, b)
- the grad vector is orthogonal to the level curve at (a, b)
- magnitude of the grad vector is the maximum rate of change of the function f.
equation for D in the second derivative test
D = Fxx * Fyy - (Fxy)^2
explain the lambda method
grad f = lambda grad g. find you partial derivatives, set the x and y’s equal to each other (equations one and 2) and then rewrite the constraint. then solve for lambda by dividing out x but x could equal 0 so make sure you evaluate x at 0 by plugging it into equation 3 and solving for y (this will give you critical points). once you have lambda plug it into equation 2 and solve for y, then plug the new y value into equation 3 to get the critical points. once you have all of your points make the D value table (second derivatives) and solve for D at each point.
