Calculus help needed ASAP

Re: Calculus help needed ASAP

y = sqrt(x+2)

All of it. In the real plane, this function only exists for x >= -2.

Let's look at the compound interest problem...

what do we know? the rate of change in your interest is proportional to how much money you have already.

aka, dy/dx = ky where k is some constant, y = your money, x = time

rearrange this simple equation to isolate the variables..

dy/y = kdx

if we integrate this simple equation, we get

ln|y| = kx + c

now we can e both sides, to get:

y = e^(kx+c)

using power distribution rules, this can be rewritten to

y = e^(kx)e^c

but c is some arbitrary constant, so we could replace e^c with a new constant

y = Ce^(kx)

and that's your final equation that describes your principle for compound interest. you probably just memorized P = Ce^(rt) in class, which is the same thing, but now you know how it is derived.

and for your specific problem, the rate is 7.1%, so r = 0.071
t = ? this is the unknown

we don't know how much we started with (C) but we do know that after time t, we have 4 times more...so our principle at time t = 4C

4C = Ce^(rt)

we can cancel out the unknown C, then we just have

4 = e^(0.071t)

using ln rules,

ln(4) = 0.071t

t = ln(4)/0.071

and that's your final answer, which you can simplify with a calculator if you want to be imprecise

ok now onto the inverse problem...

basically the inverse function just means to switch the x and the y, so you can find the inverse by performing this swap and then solving for y

so you originally have y = 20 / (1+3e^(-x))

perform swap:

x = 20 / (1+3e^(-y))

do some multiplying of terms to both sides, and we get:

e^(-y) = (20/x - 1)/3

but a^(-b) = 1/a^(b), so we can do that...

e^y = 3 / (20/x -1)

and remember you can use natural log to cancel out e to a power

y = ln|3/(20/x-1)|

which you would then write in terms of the inverse,

so f^(-1)(x) = ln|3/(20/x-1)|

5. All real numbers less than or equal to 2.

As abs(1+x) has a minumum value of 0 when x = -1.

Therefore f(x) has a maximum value at x = -1 of 2 - 0 = 2. For all other values of x, f(x) < 2