au Naturale
Integrate (10/x) dx
10*ln|x| + c
Find f' for f(x) = 5x
5^x * ln(5)
Find the limit:
lim (x to 3) (x^2 - 2x - 3) / (x - 3)
4
Integrate f(x) = 1/(4+(x-1)^2)
F(x) = .5*arctan((x-1)/2) + C
What is e?
The number n such that int 1/x dx from 1 to n is 1.
Find dy/dx for y = e^(sqrt(x))
y' = .5*x^(-.5)*e^(sqrt(x))
Which investment option has the better yield?
1. $8,000 at 5% compounded continuously for 10 years
2. $6,000 at 7.5% compounded quarterly for 10 years
Investment 1 by $575.67
lim (x to 0) (2 - 2 cos x) / (6x)
0
Find dy/dx for: y = arcsin(t^2)
Chain rule! y' = 2*t/(sqrt(1-t^4))
Given f(x) = 5-2x^3.
Find (f-1)'(7).
f'(x) = -6x^2
f(1) = 7
f'(1) = -6
(f-1)'(7) = 1/f'(1) = -1/6
Find the Derivative of the following function: f(x) = ln(2x/(x+3))
Using the properties of logarithms, rewrite f(x) as:
f(x) = ln(2x) - ln(x+3) = ln(2) + ln(x) - ln(x+3)
Then,
f'(x) = 1/x - 1/(x+3)
Find f' for f(x) = log(x^2-1)
f'(x) = 2x / [ln(10)(x^2-1)]
lim (x to 0+) x^3 cot(x)
0
Integrate 1/(sqrt(4-x^2)) dx from 0 to 1
arcsin(x/2) from 0 to 1 arcsin(.5) - arcsin(0) = pi/6
Integrate cos(t)/(1+sin(t)) dt
u = 1+sin(x) du = cos(x) dx ln|1+sin(t)| + c
Integrate e^(3/x) / (x^2) dx from 1 to 3.
Let u = (3/x) du = (-3/x^2) (-1/3) *e^(3/x) from 1 to 3. (e^3)/3 - e/3
int (from -4 to 4) 3^(x/4) dx
32/(3ln(3))
lim (x to infty) x^(1/x)
1
Integrate 1/(x^2 + 4x + 13) dx
Completing the square for the bottom yields (x+2)^2 -4 + 13 = (x+2)^2 + 9 integrate 1/((x+2)^2 + 9) dx F(x) = (1/3) arctan((x+2)/3) + c
Find the average value of f(x) = tan(x) on [0,pi/4]
-4/pi * ln(sqrt(2)/2)
Use Logarithmic differentiation to find dy/dx for: y=x*(x^2 - 1)^(1/2)
ln(y) = ln(x) + (1/2)*ln(x^2 - 1) (1/y)y' = (1/x) + x/(x^2-1) y' = y[(1/x) + x/(x^2-1)] or y' = y*(2x^2-1)/(x(x^2-1))
int 3^(2x) / (1 + 3^(2x)) dx
ln(1+ 3^(2x)) / (2 ln(3)) +C
lim (x to 0+) (10/x - 3/x^2)
-infinity
Find any relative extreme for the function (identify it as a maximum or minimum): f(x) = arctan(x) - arctan(x-4)
f'(x) = 1/(1+x^2) - 1/(1+(x-4)^2) = 0 1+x^2 = 1 + (x-4)^2 1+x^2 = 1+x^2 - 8x+16 -8x + 16 =0 x=2 By the first derivative test, (2,2.214) is a relative maximum