Finals Flashcards
Derivative f(x) / g(x)
(g(x)f’(x) - f(x)g’(x))/ g(x)^2
Derivative f(x)g(x)
f(x)g’(x) + g(x)f’(x)
Derivative f(g(x))
f’(g(x)) * g’(x)
Derivative cos(x)
-sin(x)
Derivative sin(x)
cos(x)
Derivative ln(u)
1/u
Derivative tan(x)
sec^2(x)
Derivative cot(x)
-csc^2(x)
Derivative csc(x)
-csc(x)cot(x)
Derivative sec(x)
sec(x)tan(x)
Derivative e^u
U’e^u
Derivative logau
u’ /ulna
Derivative a^u
u’a^u * lna
Integral u^n
u^(n+1) / n+1 +c
Integral 1/u
ln(u) + c
Integral e^u
e^u + c
Integral sinu
-cosu + c
Integral cosu
sinu + c
Integral tanu
-ln(cosu) + c
Integral cotu
ln(sinu) + c
Derivative arctanu
u’/1+u^2
Integral cscu
-ln(cscu + cotu) + c
Integral secu
ln(secu + tanu) +c
Derivative arccosu
-u’/√(1-u^2)
Derivative arcsinu
u’/ √(1-u^2)
Derivative arccotu
-u’/1+u^2
Inverse Steps
- Set function = to x-value and solve
- Plug that answer into f’(x)
- Take reciprocal
ln(1) =
0
ln(ab)
ln(a) + ln(b)
ln(a/b)
ln(a) - ln(b)
Fundamental Theorem of Calc
Integral a to b f(x) = f(b) - f(a)
MVT for integrals
1/(b - a) Integral a to b of f(x)
Average Value for integrals
1(b - a) Integral a to b of f(x) solve for c
2nd Fundamental Theorem of Calc
f(b) * b’ - f(a) * a’
Use 2nd ftoc when
taking the derivative of an integral
Trapezoidal Sum Estimation
.5 * w * (1a + 2b + 2c +1d)
Riemann Sum Estimation
w * all values except rightmost or leftmost value
Midpoint Rectangle Estimation
w * points in between left and right added together
Use finding areas of shapes when
you’re solving an integral when you don’t know the equation
square cross section
integral s^2
equilateral triangle cross section
√3/4 integral s^2
isosceles right triangle cross section
1/2 integral s^2
semicircle cross section
1/8 integral s^2
Rectangle cross section
k integral s^2
Washer method (for gaps)
pi integral R(x)^2 - r(x)^2
Disk method (no gap)
pi integral (R(x))^2
Exponential growth equation
f(x) = a(1 + r)^2
