quick math to-knows

trigonomic derivatives and antiderivatives, fundamental theorems, exponentials, logs

Fundamental Theorem of Calculus 1; If a function is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval, then ∫^b_a f(x) dx = …

F(x)|^b_a = F(b) - F(a)

Fundamental Theorem of Calculus 2; d/dx ∫^h(x)_g(x) f’(t) dt = …

f’(h(x))* h’(x) - f’(g(x))* g’(x)

Integrals are also:

Area under the curve

Order of f(x) evolution

∫f’(x) = f(x) (+C)

What do you always need when solving a indefinite integral

+C

derivative of cos =

-sin

derivative of sin =

cos

derivative of csc =

csc*cot

derivative of sec =

sec*tan

derivative of tan =

sec²

derivative of cot =

-csc²

MVT for Integration; If f is continuous on [a, b] then there exists a “c” on the interval, found by:

∫^b_a f(x) dx = (b-a) f(c)

This is the exact value under a curve

AVT for Integration; (Where f(c) given in MVT is the AV of f) If f is integrable on the interval [a, b] then the AV of f on the interval is:

f(c) = 1/(b-a)* ∫^b_a f(x) dx

AVT is NOT the same as

average rate of change. Instead, it is the slope of secant line

Derivative of a natural log function

d/dx (ln u) AND d/dx (ln |u|) = u’/u

Log rule for integration

∫1/u du = ln |u| + C

∫cos(x) dx

sin x

∫sin(x) dx

-cos(x)

∫sec²(x) dx

tan(x)

∫sec(x)*tan(x) dx

sec(x)

∫csc²(x) dx

-cot(x)

∫csc(x)*cot(x) dx

-csc(x)

∫e^x dx

e^x + C

Derivative for exponential bases other than e; d/dx [a^u] =

ln(a)*a^u*u’

Derivative for log bases other than e; d/dx log_au =

1/ln(a)*1/u*u’

Integrating Bases Other Than e; ∫a^u du =

1/ln(a)*a^u + C

Derivative of e; d/dx e^u =

e^u*u’

Derivative quotient rule format

[(High-D Low) - (Low-D High)]/Low*Low