calc ch 10

dot product equations

1. abs v1 times abs v2 times cos theta

2. x1x2 + y1y2

when do you use dot product

getting tangential acceleration and projection

what do unit vectors represent

pure direction

MVT for integrals

if f is continuous on the closed interval [a,b] then there exists at least one value, x = c, in the closed interval [a,b] suchh that f(c ) = the integral from a to b of f(x)dx over b-a or f(c ) times b-a equals the integral from a to b of f(x)dx

mass equation for radially changing density in 2D

the integral from a to b of 2 pi R dr times rho of r because the the area is so small its basically a ring, which would be 2 pi r times dr

axial vs radial

axial means like a rectangular prism type of cross section away from the axis or plane

radial means obviously radial in all directions

addition and subtraction of vectors

• tip to tail to tip to tail = addition

• two tails in same place, distance between tips = subtraction

• tip to tail to tip to tail but one of them is facing opposite way = subtraction

what is P

the length of a projection, use dot product to find it

equation for P

a vector times v vector over |v vector| because projecting a onto v takes away all direction so that the projection you’re left with has to be acceleration

how to know if vectors are perpendicular

if dot product = 0

equation for tangential accel

a sub t = P times v vector over |v vector| because P is found by taking away v and a’s difference in direction, such that P literally equals the numerical acceleration. to get real tangential acceleration, you have to take this magnitude and multiply It by v’s unit vector so that it is given direction once again and is a vector again. P represents the magnitude or scalar value of A, and A sub t represents (tangential) acceleration as a vector

equation for normal accel

accel vector - tangential accel

what is another equation for P

|a vector| times cos theta. this makes sense because its hypotenuse times adjacent over hypotenuse, which equals adjacent

derivative of speed function and P

should be the same thing

how would you find total population or mass given a chart of x or t versus density

density times volume breaks up so that it’s average of rho times average of x or r times whatever needed to get that to equal area, times dx or dr.

so for circle its 2 pi times average r times average rho times dr. 2 pi average r gets you the average circumference, so you have to multiply by dr to get the area. and then by average rho to get mass of that ring.

another way is to do pi times R² minus r² times average rho. no dr here because area of annulus doesn’t need it. for this method, you don’t need to take the average radius because you’re already incorporating both radii.