Derivartives
f(x) = secx + cscx then, f'(x) =
f'(x) = secxtanx + cscxcotx
lim x -> 0 3x - 3sinx/ x2
0
∫18x2 sec2x (3x3) dx
2tan(3x3) + C
This Theorem States that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].
Mean Value Theorem
Average Value
∫1/b-a f(x) dx
y = ln (6x3 - 2x2), then f'(x) =
f'(x) = (9x + 2)/(3x2 - x)
lim x ->0 sin2x/2sinx
2
∫ln3x/x dx
ln4x/4 + C
This theorem states that if a function f(x) is continuous over an interval [a, b], then the function takes on every value between f(a) and f(b).
Intermediate Value Theorem
Area of a Triangle
1/2bh
f(x) = 6x2/ x- 2
f'(x) = 24x - 6x2/ (2-x)2
lim h-> 0 5(½ + h)4 - 5(½)4/ h
5⁄2
-11∫ 4/ 1+x2 dx
2π
This Theorem states that if a function f has an antiderivative F, then the definite integral of f from a to b is equal to F(b)-F(a)
Fundamental Theorem of Calculus
∫ax
ax/ln(a) + C
f(x) = tan(x)sec(x)
f'(x) = sec3(x)+sec(x)tan2(x)
lim x->∞ 3xe-3x
0
∫(x2+2x)cos(x3 + 3x2) dx
⅓ sin (x3+ 3x2) + C
If a function f(x) is continuous on a closed interval [ a, b], then f(x) has both a maximum and minimum value on [ a, b].
Extreme Value Theorem
if dy/dx = Ky then
y = Cekx
Find the second derivative of x2y = 2
f"(x) = 6y/x2
lim x-> π⁄2 1 - sinx/1+ cos2x
¼
∫x2 sin(3x3+2) dx
- cos(3x3+2)/9 + C
if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them.
Squeeze Theorem
Special Trig Limits
limθ->0 cosθ-1/θ=0
limθ->0 sinθ/θ=1