Formula Quiz
d/dx[cotx] =
-csc^2x
d/dx[cscx] =
-cscxcotx
d/dx[cosx] =
-sinx
Definition of a continuity : F is continuous at x = c if and only if
1) F (c) is defined 2)lim as x → C F(x) exists 3) lim as x → C F(X) = F (c)
Let f be a function such that the second derivative of f exists on an open interval containing c. (Second derivatives test)
1) if f'(c) = 0 and f''(c)>0 then (c,f(c)) is a relative minimum of f 2) if f'(c) = 0 and f''(c) < 0, then (c, f(c)) is a relative maximum of f
csc^2x
1+cot^2x
sec^2x
1+tan^2x
(F^-1)' (a) =
1/f'(f^-1(a))
d/dx[lnx] =
1/x
d/dx[loga(x)] =
1/xln(a)
Sin(2x) =
2sinxcosx
Alternate form definition of a derivative f'(c) =
F'(c) = F(x) - F (C)/x-c as x →c
Mean Value Theorem
If a function f is... 1) continuous on [a,b] 2) and differentiable on (a,b), 3) then there exists a number c on (a,b) such that... 1) avg rate of change = instant rate at c 2) secant line (parallel) - tangant line 3) f'(c)=(f(b)-f(a))/(b-a)
Let c be a critical number of a function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except at possibly x = c then f(c) can be classified as follows...(1st derivative test)
If f '(x) changes from a negative to a positive at c, then f(c) is a relative minimum of f. If f' (x) changes from a positive to a negative at c, then f(c) is a relative maximum of f
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
Definition of a critical number
Let f be defined at c. If f'(c)=0, or if f' is undefined at c, then c is a critical number of f.
d/dx[a^x] =
a^x ln(a)
Cos(2x) = (there are three)
cos^2x-sin^2x or 1 - 2sin^2x or 2cos^2x -1
d/dx[sinx] =
cosx
Chain rule
d/dx f(g(x)) = f'(g(x)) g'(x)
Power Rule
d/dx x^n = nx^n-1
Derivative of a constant rule
d/dx[c] = 0
Constant Multiplier Rule
d/dx[cx^n] = (cn)(x)^n-1
Sums and Differences
d/dx[f(x) + g(x)] = [f'(x) + g(x)] and d/dx[f(x) - g(x)] = [f'(x) - g(x)]
Product Rule
d/dx[f(x) g(x)] = f'(x)g(x)+f(x)g'(x)
Quotiant Rule
d/dx[f(x)/ g(x)] = [f'(x)g(x)- (x)g'(x)]/g(x)^2
definition of e:
e = lim n->infinity (1+1/n)^n
d/dx[e^x] =
e^x
Average rate of change
f(b)-f(a)/b-a
Rolle's Theorem
if f(x) is.... 1) continuous on [a,b] and 2) differentiable on (a,b), and 3) if f(a)=f(b), then there is at least one point c on (a,b) where... 1) instant rate of change = 0 2) tangant is horizontal 3) f'(c) = 0
definition of a derivative f'a =
limit of [ f(a + h) - f(a) / h ] as h→0
Definition of a derivative f'(x) =
limit of [ f(x + h) - f(x) / h ] as h→0
d/dx[tanx] =
sec^2x
d/dx[secx] =
secxtanx
Pythagorean identity
sin²θ + cos²θ = 1
Definition of absolute value
|x| = x if x ≥ 0 and -x if x < 0
