If f(x) and f¹(x) are inverse functions of each other and f(x) = 2x+ 5, what is ƒ˜¹ (8) ?

In a previous exercise we formulated a model for learning in the form of the differential equation dp dt = k(m − p) where p(t) m

DaniilM [7]

I assume you mean

$\dfrac{dP}{dt} = k(M-P)$

ANSWER
An expression for P(t) is

$P = M - Me^{-kt}$

EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms

$\begin{aligned} \dfrac{dP}{dt} &= k(M-P) \\ \dfrac{dP}{M-P} &= k\, dt
\end{aligned}$

Integrate both sides of the equation.

$\begin{aligned}
\int \dfrac{dP}{M-P} &= \int k\, dt \\
-\ln|M-P| &= kt + C \\
\ln|M-P| &= -kt - C\end{aligned}$

Note that for the left-hand side, u-substitution gives us

$u = M - P \implies du = -1dP \implies dP = -du$

hence why $\int \frac{dP}{M-P} \ne \ln|M - P|$

Now we use the definition of the logarithm to convert into exponential form.

The definition is

$\ln(a) = b \iff \log_e(a) = b \iff e^b = a$

so applying it here, we get

$\begin{aligned} \ln|M-P| &= -kt - C \\ |M - P| &= e^{-kt - C} \\ 
M - P &= \pm e^{-kt - C} 
 \end{aligned}$

Exponent properties can be used to address the constant C. We use $x^{a} \cdot x^{b} = x^{a+b}$ here:

$\begin{aligned}
 M - P &= \pm e^{-kt - C} \\
M - P &= \pm e^{- C - kt} \\ 
M - P &= \pm e^{- C + (- kt)} \\ 
M - P &= \pm e^{- C} \cdot e^{- kt} \\ 
M - P &= Ke^{- kt} && (\text{\footnotesize Let $K = \pm e^{-kt}$ }) \\ 
M &= Ke^{- kt} + P\\
P &= M - Ke^{- kt}
\end{aligned}$

If we assume that P(0) = 0, then set t = 0 and P = 0

$\begin{aligned} 
0 &= M - Ke^{- k\cdot 0} \\
0 &= M - K \cdot 1 \\
M &= K
 \end{aligned}
$

Substituting into our original equation, we get our final answer of

$P = M - Me^{-kt}$

6 0

3 years ago

Read 2 more answers

Rochelle deposits $5,000 in an IRA. What will be the value of her investment in 25 years if the investment is earning 8% per yea

Iteru [2.4K]

Answer:

$\$36,945.28$

Step-by-step explanation:

we know that

The formula to calculate continuously compounded interest is equal to

$A=P(e)^{rt}$

where

A is the Final Investment Value

P is the Principal amount of money to be invested

r is the rate of interest in decimal

t is Number of Time Periods

e is the mathematical constant number

we have

$t=25\ years\\ P=\$5,000\\ r=8\%=8/100=0.08$

substitute the values in the formula

$A=5,000(e)^{0.08*25}$

$A=5,000(e)^{2}$

$A=\$36,945.28$

5 0

3 years ago

An integer, a, is said to be a type 0 integer if there exists another integer n such that a = 3n. give at least four different i

Elodia [21]

A type 0 integer is simply any whole number that is a multiple of 3. some examples would be : 3,6,9,12...

n=1: 3(1)=3
n=2: 3(2)=6
n=3: 3(3)=9
n=4: 3(4)=12

4 0

3 years ago

Read 2 more answers

Helppp please!!!!!!!!!!!!!!

klio [65]

Answer:

No

Yes

No

Step-by-step explanation:

In this case we are looking at substitution, so any where you find x put three(3) in replace of it.

And remember the answer should be 36,