calculus
cos(pai-x)
-cos(x)
cos（x-pai）
-cos(x)
d(csc(x))/dx
-csc(x)cot(x)
d(cot(x))/dx
-csc^2 (x)
d(cos(x))/dx
-sin(x)
sin(-x)
-sin(x)
cos(pai/2+x)
-sin(x)
sin(x-pai)
-sin(x)
differentiability of inverse functions
1. If the domain of a function f is an open interval on which dy/dx>0 or dy/dx<0 then f(x) has an inverse 2. If f is continuous and one-to-one on an interval, then f^-1(x) is continuous on an interval.
L'Hopital's rule
1. both infinitely or both 0
d(f^-1(x))/dx
1/(df(f^-1)/dx)
Extreme-Value theorem
If a function f is continuous on a finite closed interval [a,b], then f has both an absolute max and an absolute min on [a,b]
mean value theorem
Let {\displaystyle f:[a,b]\to \mathbb {R} } {\displaystyle f:[a,b]\to \mathbb {R} } be a continuous function on the closed interval {\displaystyle [a,b]} [a,b] , and differentiable on the open interval {\displaystyle (a,b)} (a,b), where {\displaystyle a<b} a<b . Then there exists some {\displaystyle c} c in {\displaystyle (a,b)} (a,b) such that {\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.} {\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.} The mean value theorem is a generalization of Rolle's theorem, which assumes {\displaystyle f(a)=f(b)} {\displaystyle f(a)=f(b)}, so that the right-hand side above is zero. The mean value theorem is still valid in a slightly more general setting. One only needs to assume that {\displaystyle f:[a,b]\to \mathbb {R} } {\displaystyle f:[a,b]\to \mathbb {R} } is continuous on {\displaystyle [a,b]} [a,b] , and that for every {\displaystyle x} x in {\displaystyle (a,b)} (a,b) the limit {\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}} \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}} exists as a finite number or equals {\displaystyle \infty } \infty or {\displaystyle -\infty } -\infty . If finite, that limit equals {\displaystyle f'(x)} f'(x) . An example where this version of the theorem applies is given by the real-valued cube root function mapping {\displaystyle x\to x^{\frac {1}{3}}} {\displaystyle x\to x^{\frac {1}{3}}} , whose derivative tends to infinity at the origin.
d(cy)/dx
cdy/dx
cos(-x)
cos(x)
d(sin(x))/dx
cos(x)
sin(pai/2-x)
cos(x)
station point VS critical point
critical point include the point that f'(x) is UDF and 0 and two side (if it's close range)
d(f(x)+g(x))/dx
d(f(x))/dx+d(g(x))/dx
sin(pai/2+x)
depend
Find max min
dy/dx=0 get the relative max and min check the two side
Concavity
f", open interval
Increasing and Decreasing
f', close interval
Intermediate value theorem
if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.
lim(x-a)f(x)=b
if value of f(x) can be made as close as b, by making x as infinitely close to a
d(sec(x))/dx
sec(x)tan(x)
d(tan(x))/dx
sec^2 (x)
cos(pai/2-x)
sin(x)
sin(pai-x)
sin(x)
First derivative test
suppose that f is continuous at a and differentiable on some open interval containing a, except possibly at a itself. If there exists a positive number r such that for every x in (a − r, a) we have f′(x) ≥ 0, and for every x in (a, a + r) we have f′(x) ≤ 0, then f has a local maximum at a. If there exists a positive number r such that for every x in (a − r, a) ∪ (a, a + r) we have f′(x) > 0, then f is strictly increasing at a and has neither a local maximum nor a local minimum there. If none of the above conditions hold, then the test fails. (Such a condition is not vacuous; there are functions that satisfy none of the first three conditions such as f(x) = x2⋅sin(1/x).
Second derivative test
suppose that f is twice differentiable at x0 If f'(x)=0 and f"(x)>0, then f has a relative minimum at x If f'(x)=0 and f"(x)<0, then f has a relative maximum at x If f'(x)=0 and f"(x)=0, then dipend
power rule
x^2=2x
