AP Calc AB Formulas
Definition of the derivative
lim h->0 (f(x+h) - f(x))/h
Special Cosine Limit
lim x->0 (1-cosx)/x = 0
Special Sine Limit
lim x->0 sinx/x = 1
[int] 1/x dx =
ln x + c
[int] secx dx =
ln|secx + tanx| + c
[int] cotx dx =
ln|sinx| + c
Volume (disk method)
pi [int](a,b) [(outside/upper)^2 - (inside/lower)^2] dx or dy
d/dx(tanx) =
sec^2(x)
[int] secxtanx dx =
secx + c
d/dx(secx) =
secxtanx
[int] cosx dx =
sinx + c
[int] sec^2 (x) dx =
tanx + c
Second Fundamental Theorem of Calculus
d/dx[int](g(x),a) f(t) dt = f(g(x)) * g'(x)
d/dx(e^x) =
e^x
[int] e^x dx =
e^x + c
[int] a^x dx =
((a^x)/ln a) + c
d/dx(a^x) =
(a^x)(ln a)
d/dx(arccotx) =
-1/(1+x^2)
d/dx(arccosx) =
-1/sqrt(1-x^2)
d/dx(arccscx) =
-1/|x|sqrt(x^2 -1)
[int] sinx dx =
-cosx + c
[int] csc^2 (x) dx =
-cotx + c
d/dx(cotx) =
-csc^2(x)
[int] cscxcotx dx =
-cscx + c
d/dx(cscx) =
-cscxcotx
[int] tanx dx =
-ln|cosx| + c
[int] cscx dx =
-ln|cscx + cotx| + c
d/dx(cosx) =
-sinx
d/dx(arctanx) =
1/(1+x^2)
Average value
1/(b-a) [int] f(x) dx
Derivative of an inverse function (if g is the inverse of f)
1/(f'(g(x)))
d/dx(log base a x) =
1/(x ln a)
d/dx(arcsinx) =
1/sqrt(1-x^2)
d/dx(ln x) =
1/x
d/dx(arcsecx) =
1/|x|sqrt(x^2 -1)
Intermediate Value Theorem
If f is continuous for all x on the closed interval [a,b] and k is any number between f(a) and f(b), then there is a number x=c in (a,b) such that f(c) = k
Mean Value Theorem
If f is differentiable for all values of x on the open interval (a,b) and f is continuous at x=a and x=b, then there is at least one number c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a)
L'Hôpital's Rule
If lim x->a f(x)/g(x) is 0/0, then lim x->a f(x)/g(x) = lim x->a f'(x)/g'(x)
Instantaneous rate of change
Take the derivative
Trapezoidal Rule
[int](a,b) f(x) dx = (1/2)((b-a)/n)[f(x0) + 2f(x1) +...+ 2f(x(n-1)) + f(xn)]
Fundamental Theorem of Calculus
[int](a,b) f(x) dx = F(b)-F(a) where F'(x) = f(x)
[int] 1/x(sqrt(x^2 -1)) dx =
arcsecx + c
[int] 1/sqrt(1-x^2) dx =
arcsinx + c
[int] 1/(1+x^2) dx =
arctanx + c
Average rate of change
change in _/interval
d/dx(sinx) =
cosx