Basic Derivatives
The derivative of x to the n
d/dx (x^n)
WHAT IS...
n times x to the n minus 1
n(x^n-1)
If the function f(x) is continuous on [a,b], for all k between f(a) and f(b), there exists at least one number x=c in the open interval (a,b) such that f(c)=k
WHAT IS...
Intermediate Value Theorem
1 + x + ((x^2)/2!) + ((x^3)/3!) +...+ ((x^n)/n!) +...
WHAT IS...
e^x
diverges if the limit as n approaches infinity of a sub n does not = 0
WHAT IS...
nth term
2sinxcosx
WHAT IS...
sin(2x)
The derivative of cos(x)
d/dx (cos(x))
WHAT IS...
-sin(x)
If the function f(x) is continuous on [a,b], then there exists an absolute max and min on that interval.
WHAT IS...
Extreme Value Theorem
1 - (x^2)/(2!) + (x^4)/(4!) -...+ (x^2n)(-1^n)/(2n)! +...
WHAT IS...
cos(x)
converges if the limit as n approaches infinity of the absolute value of a sub (n+1) over a sub n is less than 1
(if the limit = 1, you know nothing so check the endpoints using another test)
WHAT IS...
Ratio Test
cos^2(x)-sin^2(x)
(= 1-2sin^2(x))
WHAT IS...
cos(2x)
The derivative of ln(u)
d/dx (ln(u))
WHAT IS...
1/u du/dx
If the function f(x) is continuous on [a,b], AND differentiable on the interval (a,b), AND f(a)=f(b), then there is at least one number x=c in (a,b) such that f'(c)=0
WHAT IS...
Rolle's Theorem
x - (x^3)/(3!) + (x^5)/(5!) -...+ (x^2n+1)(-1^n)/(2n+1)! +...
WHAT IS...
sin(x)
converges if the limit as n approaches infinity of the nth root of a sub n is less than 1
WHAT IS...
Root Test
(1+cos(2x))/2
WHAT IS...
cos^2(x)
The derivative of a^u(x)
d/dx (a^u(x))
WHAT IS...
(a^u(x))(ln(a)) du/dx
d/dx integral from a(x) to b(x) of f(t)dt =
f(b(x))b'(x)-f(a(x))a'(x)
WHAT IS...
2nd Fundamental Theorem of Calculus
1 + x + x^2 + x^3 +...+ x^n +...
WHAT IS...
1/(1-x)
The series from n=0 to infinity of ar^n converges if
|r|<1
WHAT IS...
Geometric Series Test
sin^2(x) + cos^2(x)
WHAT IS...
1
The derivative of arccsc(x)
WHAT IS...
-1/(|x|((x^2)-1)^(1/2))
If the function f(x) is continuous on [a,b], and differential on the interval (a,b), then there exists at least one number x=c on (a,b) such that
x=c on (a,b) such that
f(c)(b-a) = integral from a to b of f(x) dx
f(c) = (integral from a to b of f(x) dx)/(b-a)
This value f(c) is the "average value" of the function on the interval [a,b]
WHAT IS...
MVT of Integrals (Average Value)
(x-1) - ((x-1)^2)/2 + ((x+1)^3)/3 -...+
((x-1)^n)(-1^(n+1))/n
WHAT IS...
ln(x)
If the limit as n approaches infinity of a sub n over b sub n exists and does not equal zero, what one series does (c or d), so does the other.
WHAT IS...
Limit Comparison Test
1/cos(x)
WHAT IS...
sec(x)