Calc Week 9 Flashcards
arc length
integral from a to b of the square root of x prime squared plus y prime squared with respect to t
finding arc length
find both derivatives
plug
solve integral
concavity
2nd derivative positive = concave up
2nd derivative negative = concave down
d2y/dx2 = d/dt[dy/dx]
/
dx/dt
finding concavity
take first derivative
plug into equation
-will need to take 2nd derivative / x prime
-positive or negative answer
surface area
rotating by x-axis
2pi times the integral from a to b of the function y times the square root of x prime squared plus y prime squared with respect to t
rotating by y-axis
2pi times the integral from a to b of the function x times the square root of x prime squared plus y prime squared with respect to t
finding surface area
find derivatives
plug in equation
solve integral
polar coordinate
have a point
can represent it with (r,theta)
r is the length of the ray by Pythagorean theorem
theta is the angle by rise/run or y/x (tan)
key polar identities
rectangular coordinates
x=rcos0
y=rsin0
polar coordinates
tan0=y/x
r^2=x^2+y^2
finding polar coordinates
use tan0=y/x and r^2=x^2+y^2
plug and solve
finding polar equations
use x=rcos0 and y=rsin0
plug these into the given equation of x and y
sketching polar functions
table
column for theta, then the equation given, then the polar coordinate
plot from center 0,0
finding dy/dx of polar functions
find r prime
use dy/dx formula
plug in theta
find a point for line with x and y identities to find x and y
dy/dx of a polar function
dy/dx=
f’ sin + f’ cos /
f’ cos - f’ sin
area of polar functions
area =
1/2 integral from theta1 to theta2 of r^2 =
1/2 integral from [f]^2
plug and solve integral
approximation of square root of 3 and 2
2 = 1.4
3 = 1.7
