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Trigonometric formulas
Differentiation formulas
Integration formulas
y = D + A sin B ( x − C ) A is amplitude B is the affect on the period (stretch or shrink)
C is vertical shift (left/right) and D is horizontal shift (up/down)
Limits:
sin x
=1
x −> 0
x
lim
sin x
=0
x −>∞
x
lim
1 − cos x
=0
x −> 0
x
lim
Exponential Growth and Decay
y = Ce kt
Rate of Change of a variable y is proportional to the value of y
dy
= ky or
dx
y ' = ky
Formulas and theorems
1
...
f(a) exists
ii
...
2
...
A function y = f(x) is even if f(-x) = f(x) for every x in the function's domain
...
2
...
Every
odd function is symmetric about the origin
...
Horizontal and vertical asymptotes
1
...
2
...
4
...
To find the maximum and minimum values of a function y = f(x), locate
1
...
the end points, if any, on the domain of f(x)
...
Let f be differentiable for a < x < b and continuous for a ≤ x ≤ b
...
If f'(x) > 0 for every x in (a,b), then f is increasing on [a,b]
...
If f'(x) < 0 for every x in (a,b), then f is decreasing on [a,b]
...
Suppose that f''(x) exists on the interval (a,b)
...
If f''(x) > 0 in (a,b), then f is concave upward in (a,b)
...
If f''(x) < 0 in (a,b), then f is concave downward in (a,b)
...
These are the only candidates where f(x) may have a
point of inflection
...
8
...
in (a,b) such that
9
...
The converse is
false, i
...
continuity does not imply differentiability
...
L'Hôpital's rule
If
11
...
Area between curves
If f and g are continuous functions such that f(x) ≥ g(x) on [a,b], then the area between
...
Inverse functions
a
...
b
...
c
...
d
...
13
...
The exponential function y = ex is the inverse function of y = ln x
...
The domain is the set of all real numbers, −∞ < x < ∞
...
The range is the set of all positive numbers, y > 0
...
15
...
e
...
The domain of y = ln x is the set of all positive numbers, x > 0
...
The range of y = ln x is the set of all real numbers, −∞ < y < ∞
...
y = ln x is continuous and increasing everywhere on its domain
...
ln(ab) = ln a + ln b
...
ln(a / b) = ln a − ln b
...
ln ar = r ln a
...
...
Let f be nonnegative and continuous on [a,b], and let R be the region bounded
above by y = f(x), below by the x-axis, and the sides by the lines x = a and x =
b
...
When this region R is revolved about the x-axis, it generates a solid (having
...
When R is revolved about the y-axis, it generates a solid whose volume
17
...
...
If a particle moving along a straight line has a positive function x(t), then its
instantaneous velocity v(t) = x'(t) and its acceleration a(t) = v'(t)
...
v(t) = ∫ a(t)dt and x(t) = ∫ v(t)dt
...