Important Info Flashcards
Intermediate Value Theorem (derivative)
A function y=f(x) that is continuous on the closed interval [a,b] takes on every value between f(a) and f(b).
Mean Value Theorem (derivative)
If y=f(x) is continuous on [a,b] & differentiable on (a,b), then there is at least one number c between a and b.
where
f’(c)= f(b) - f(a)
———-
b - a
Extreme Value Theorem (derivative)
If f(x) is continuous on the closed interval [a,b], then f(x) has both an absolute max value f(c) and an absolute min value f(c) at some numbers c and d in [a,b].
Mean Value Theorem (integral)
If f is continuous on [a,b], then there exists a number c in [a,b] such that :
integral from a to b of f(x)
divided by b - a
= f(c)
Fundamental Theorem of Calculus
derivative of integral
plug in b and multiply by derivative
(subtract) plug in a and multiply by derivative
Continuity Rules
f is continuous at x=a if:
- lim as x approaches a exists
- f(a) exists
- lim as x approaches a = f(a)
Types of Discontinuities
- point (removable)
- jump (non removable)
- infinite (non removable)
Differentiable Rules
f is differentiable on [a,b] if: 1. f is continuous on [a,b] 2. f has a derivative at every point on [a,b] Not differentiable when: *Corner, cusp, vertical tangent
average velocity AKA…
slope of secant line = change in position/change in time
instantaneous rate of change AKA
slope of tangent line=derivative of position= s’(t) = v(t)
speed
lvelocityl
maximum height of projectile
when v(t) = 0
projectile hits ground when
s(t) = 0
particle speeds up when…
velocity and acceleration has the same sign
particle slows down when…
velocity and acceleration have different signs
limit definition of the derivative
f’(x) = lim as h approaches 0
f(x+h) - f(x) / h
