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Calculus 1 Reference
Derivatives
|
Function |
Derivative |
| xn |
nxn-1 |
| ax |
axln a |
| ex |
ex |
| ln x |
1/x |
| sin x |
cos x |
| cos x |
- sin x |
| tan x |
sec2 x |
| cot x |
- csc2 x |
| sec x |
sec x tan x |
| csc x |
- csc x cot x |
| y |
y' |
Integrals
| Function |
Integral |
| xn |
xn+1/(n+1) + c |
| ax |
ax/ln a + c |
| ex |
ex + c |
| 1/x |
ln |x| + c |
| sin x |
- cos x + c |
| cos x |
sin x + c |
Geometry
| Area of a circle |
∏r2 |
| Circumference of a circle |
2∏r |
| Surface area of a sphere |
I think 4∏r2 |
| Volume of a sphere |
(4/3)∏r3 |
| Volume of a cylinder |
∏r2h |
| Volume of a cone |
(1/3)∏r2h |
Trigonometry
|
0 |
∏/6 |
∏/4 |
∏/3 |
∏/2 |
∏ |
2∏ |
| sin |
0 |
1/2 |
1/√2 |
√3/2 |
1 |
0 |
0 |
| cos |
1 |
√3/2 |
1/√2 |
1/2 |
0 |
- 1 |
1 |
| tan |
0 |
1/√3 |
1 |
√3 |
infinity |
0 |
0 |
| sin2 x + cos2 x = 1 |
sin 2x = 2sin x cos x |
| sec2 x = tan2 x + 1 |
sin2 x = (1/2)(1 - cos 2x) |
| csc2 x = cot2 x + 1 |
cos2 x = (1/2)(1 + cos 2x) |
| cos (-x) = cos (x) |
sin (-x) = -sin (x) |
Law of Cosines:
For a triangle with angles A, B and C, and sides opposite their respective
angles of length a, b, and c, you can use the following formula for any angle:
a2 = b2 + c2 - [2 (b) (c) cos A]
To convert from logs to natural logs:
Logb a = (ln a)/(ln b)
Calculus 2 Reference
More Derivatives
| Function |
Derivative |
| arctan x |
1/(1 + x2) |
| arcsin x |
1/√(1 - x2) |
More Integrals
| Function |
Integral |
| sec x |
ln |sec x + tan x| + c |
| csc x |
- ln |csc x + cot x| + c |
| 1/(1 + x2) |
arctan x + c |
| 1/√(1 - x2) |
arcsin x + c |
| sink x cos x |
Notice how each one of these is next to its
derivative, before it's solved. |
sink+1 x/(k+1) + c |
| cosk x sin x |
- cosk+1 x/(k+1) + c |
| tank x sec2 x |
tank+1 x/(k+1) + c |
| seck x (sec x tan x) |
seck+1 x/(k+1) + c |
In Integrals, the following trig substitutions are allowed:
| √(a2 - x2) = a cos € when x =
a sin € |
| √(x2 - a2) = a tan € when x =
a sec € |
| √(a2 + x2) = a sec € when x =
a tan € |
Areas and Volumes with Integrals
| Cartesian (Standard) Coordinates |
| Area between curves |
∫ f(x) - g(x) dx |
| Length of a curve |
∫ √[1 + f'(x)2]
dx |
| Volume around x axis (disc/donut) |
∏
* ∫ f(x)2 - g(x)2
dx |
| Volume around y axis (shells) |
2∏
* ∫ x[f(x) - g(x)] dx |
| Polar Coordinates |
| Area inside curves |
(1/2) * ∫ r2 d€ |
| Length of a curve |
∫ √[r2 + (r')2] d€ |
Polar Coordinates
| sin € = y/r |
tan € = y/x |
| cos € = x/r |
r2 = x2 + y2 |
Simpson's Rule
(n is always even)
[(b-a)/3n][f(x0) + 4f(x1) + 2f(x2)
+ 4f(x3) + ... + f(xn)]
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