| A | B |
|---|
| Second Derivative Test | f''(c) is greater then zero, have a relative minimum; f''(c) is less than zero, have a relative maximum; f''(c) equals zero, the test fails. |
| derivative of tan u | (sec u) squared times u' |
| derivative of sec u | (sec u tan u) u' |
| derivative of arcsin u | u'/square root of (1-u2) |
| derivative of arctan u | u'/(1+u2) |
| derivative of arcsec u | u'/u times square root of (u2-1) |
| derivative of cos u | - (sin u) u' |
| derivative of cot u | -(csc u)squared times u' |
| derivative of csc u | -(csc u cot u) u' |
| derivative of arccos u | -u'/square root of (1-u2) |
| derivative of arccot u | -u'/(1+u2) |
| derivative of arccsc u | -u'/u times square root of (u2-1) |
| integral of tan u du | -ln (cos u) +C |
| integral of cot u du | ln (sin u) + C |
| integral of sec u du | ln (sec u + tan u) + C |
| integral of csc u du | -ln (csc u + cot u) + C |
| integral of sec2 u du | tan u +C |
| integral of csc2 u du | -cot u + C |
| integral of sec u tan u du | sec u +C |
| integral of csc u cot u du | -csc u +C |
| integral of (du/square root of (a2-u2)) | arcsin (u/a) +C |
| integral of (du/(a2+u2)) | (1/a)arctan (u/a) + C |
| integral of (du/u times the square root of (u2-a2)) | (1/a) arcsec (u/a) +C |
| summation from i=1 to n of i | n (n+1)/2 |
| summation from i=1 to n of i2 | n(n+1)(2n+1)/6 |
| summation from i=1 to n of i3 | n squared times (n+1) squared/4 |
| summation from i=1 to n of c | cn |
| Fundamental Theorem of Calculus | integral from a to b of f(x) dx = F(b) - F(a) |
| Mean Value Theorem | The integral from a to b of f(x) dx = f(c) (b-a) |
| Average Value | 1/(b-a) times the integral from a to b of f(x) dx |
| Second Fundamental Theorem of Calculus | derivative of the integral from a to x of f(t) dt = f(x) |
| Area Between Two Curves | A = the integral from b to a of (f(x)-g(x))dx |
| Disc Method | V = pie times the integral from a to b of [R(x)]2dx or V = pi times the integral from c to d (y-axis) of [R(y)]2 dy |
| Shell Method | V = 2pi times the integral from c to d of p(y)h(y)dy or V = 2pi times the integral from a to b of p(x)h(x)dx |
| Arc Length | s = the integral from a to b of the square root of (1+[f'(x)]2) dx |
