Final Exam Flashcards
Width
(b-a)/n
length
f(a+ ((b-a)/n)i)
Rolle’s Theorm
Let f be a function that is CONTINUOUS on the CLOSED INTERVAL [a,b] and DIFFERENTIABLE on the OPEN INTERVAL (a,b). If f(a) = f(b), then there is at least one number c in (a,b) such that f’(c)=0.
MVT
AROC=IROC
EVT
If f is continuous on a closed interval [a,b], then f has BOTH a maximum and a minimum on the interval
Critical numbers
F’(c)=0 or if f is not differentiable at c
The Second Derivative Test
f’(c)=0
If f”(c)>0, then f(c) has a relative minimum
If f”(c)<0, then f(c) has a relative maximum
Rate of change between two equations
R(t)= f(t)-g(t)
T=time
Indeterminate forms that require l’hospitals
(0/0), any mix of infinities
L’hospital’s rule
The lim as x approaches c (f(x)/g(x)) is equal to the limit as x approaches c (f’(x)/g’(x))
d/dx (sinx)
cosx
d/dx (cosx)
-sinx
d/dx (e^x)
e^x
d/dx (log b x)
1/(lnb)(x)
derivative of tanx
Sec^2 (x)
Derivative of cotx
-csc^2 x
Derivative of secx
Secxtanx
derivative of cotx
-csc^2x
Derivative of cscx
-cscxcotx
Ln(1)
0
Ln(a^n)
nlna
ln(ab)
ln(a) + ln(b)
ln(a/b)
lna- lnb
d/dx [a^x]
(lna)a^x
d/dx [a^u]
(lna)a^u u’
d/dx (log a u)
1/ (lna)u * u’
Inverse derivative
1/ (f’(f^-1 (x)))
derivative of arcsin
1/√1-x²
Derivative of arccos
-1/√(1 - x²)
Derivative of arctanx
1/(1+x^2)
Derivative of arccotx
-1/(1 + x^2)
Derivative of arcsecu
1 / [|x| √(x2 - 1)]
Derivative of arccscu
-1/(|x|√x²-1)
