Formulas

Question 1

area underneath a line

Answer 1

integral (a to b) F(x) dx

Question 2

exact total area between to functions

Answer 2

integral (xhigh - xlow) dy

Question 3

volume of a 3D shape

Answer 3

(slice area) (thickness or height)

Question 4

area of a circle

Answer 4

pie r^2

Question 5

volume of a ring

Answer 5

(ring area)(thickness)
ring area = (pier^2out - pier^2in)

Question 6

mass when density is constant

Answer 6

[lineal density (Kg/m)][length(m)]

Question 7

total mass-circular

Answer 7

E (density)(2pie*r * delta r)

Question 8

center of mass

Answer 8

(Eximi)/(Emi) = moment/total mass of all objects
xi = m
mi = Kg

y = integral ( y * mass of slice)/ integral (mass of slices)

Question 9

volume of a circle

Answer 9

pier^2*h

Question 10

Newtons law of heating and cooling

Answer 10

dH/dt = -a(H -Tenv)

Question 11

drug dosing

Answer 11

dM (mg in body)/dt (time) = rate in - rate out

Question 12

logistic growth

Answer 12

dp/dt = KP(L-P)

Question 13

distance between 2 points in R^3

Answer 13

sqrt( (x-a)^2 + (y-b)^2 + (z-c)^2))

Question 14

formula for a circle centered on the origin

Answer 14

x^2 + y^2 = r^2

Question 15

linear two variable function

Answer 15

f(x,y) = mx + ny + c

Question 16

partial derivative of f with respect to x (if no formula is given and only a chart)

Answer 16

(Fend-Fstart)/ (Xend-Xstart)

Question 17

point-slope form

Answer 17

z = c +m(x-a) + n(y-b)

Question 18

tangent plane formula

Answer 18

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)

Question 19

|error bound|

Answer 19

= |fx(a, b)(max delta x)| + |fy(a, b)(max delta y)

Question 20

length of a vector

Answer 20

sqrt( v1^2 + v2^2)

Question 21

special triangles

Answer 21

(1,1, sqrt2), (1, sqrt3, 2)

Question 22

normalization of a non-unit vector

Answer 22

(1/||V||)V = (1/(sqrt(V1^2 + V2^2))) V

Question 23

directional derivative

Answer 23

Duf(a, b) = fx(a, b) *U1 + fy(a, b) * U2
or
grad f * U

slope of the surface in the direction of U

Question 24

grad f

Answer 24

= < fx, fy>
direction of maximum increase/slope

Question 25

chain rule

Answer 25

dz/dt = (pz/pz * dx/dt) + (pz/py *dy/dt) p = partial derivative

Question 26

is + fxx concave up or down

Answer 26

up

Question 27

volume of a cylinder

Answer 27

Pie(r)^2 h

Question 27

where do you find saddle points

Answer 27

where two contours cross and in circle clusters

Question 28

second derivative test

Answer 28

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

Question 29

surface area of a closed rectangular box (x, y, z)

Answer 29

= 2xz + 2yx + 2xy

Question 30

if lambda is > 0 then grad f and grad g...

Answer 30

have the same direction and are parallel

Question 31

if lambda is < 0 then grad f and grad g...

Answer 31

have opposite directions but are still parallel

Question 32

exponential growth and decay

Answer 32

dp/dt = KP ---> P(t) = Poe^kt

Question 33

if the line is concave up Euler's method will be an ...

Answer 33

underestimate

Question 34

integration by parts

Answer 34

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

Question 35

area of a triangle

Answer 35

1/2(b*h)

Question 36

sin^2(x) + cos^2(x) =

Answer 36

1

Question 37

derivative of tan(x)

Answer 37

1/ (cos(x))^2

Question 38

derivative of arctan(x)

Answer 38

1/(x^2 +1)

Question 39

derivative of arcsin(x)

Answer 39

1/sqrt(1-x^2)

Question 40

where is the radius on a cone, what will be the radius at x?

Answer 40

1/3 from the vertex r = x/3

Question 41

what does proportional mean when making an equation

Answer 41

multiplication

Question 42

what does inversely mean when creating an equation?

Answer 42

1/x, times the reciprocal, times the inverse of the function...

Question 43

product of

Answer 43

multiplication

Question 44

difference between

Answer 44

subtraction

Question 45

acceleration

Answer 45

second derivative

Question 46

if the line is concave up Eulers method will be?

Answer 46

under estimate

Question 47

density

Answer 47

mass/volume

Question 48

how do you know if two vectors are perpendicular

Answer 48

angle between them is 90 and the dot product is 0

Question 49

partial f of x and f of y in vector terms

Answer 49

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)

Question 50

what are the three key properties of the gradient

Answer 50

1. direction of the grad vector is the direction of maximum increase of function f at point (a, b) 2. the grad vector is orthogonal to the level curve at (a, b) 3. magnitude of the grad vector is the maximum rate of change of the function f.

Question 51

equation for D in the second derivative test

Answer 51

D = Fxx * Fyy - (Fxy)^2

Question 52

explain the lambda method

Answer 52

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.