Unit 5 Flashcards
Fundamental theory of calculus pt. 1
. u
d/du ∫ f(t) dt = f(u) du/dx
a
fundamental theory of calc. pt.2
b
∫ f(x)dx = F(a) - F(b)
a
optimization (revenue & area)
- identify variables
- write a formula
- express as a function of one variable
- find critical points
- check endpoints
- plug back into problem (if needed) to find solution
volume of a sphere
4/3 πr^3
volume of cylinder
π(r^2)h
volume of a rectangle
(l)(w)(h)
surface area of a sphere
4πr^2
surface area of a cylinder
2πr(r+h)
surface area of a rectangular prism
2(w + 2wh + 2lh)
revenue =
(price)(quantity)
- the total amount of money brought in by a company’s operations, measured over a set amount of time
surface area of a cylinder is at a minimum when _____ is equal to diameter
height
position function for a falling object
in feet: s(t) = -16t^2 + Vot + Sot
in meters: s(t) = -4.9t^2 + Vot + Sot
speed of a particle =
|velocity|
concave up, increasing, positive
slope directional fields
pretty straight forward
- show the various solutions of f(x) using its derivative
- could be anywhere, that + C is what locks it in
general exponential model equation
y = Ce^(kt)
compound interest equation
A = Pe^(rt)
- A: end amount
- P: principal (initial balance)
- r: percent rate as a decimal
- t: time in years
constant C to the power of anything =
C
HALF LIFE
half-life equation
y = Ce^(kt)
- C: initial condition (t, C)
k =
ln(0.5)/half-life
Newton’s law of cooling
y - T = Ce^(kt)
- T: ambient temperature (room temp.)
DON’T FORGET ABOUT THE INITIAL CONDITION
