Solve Derivatives Questions with Step-by-Step Help

[seiferseph]

1) I'm not really sure how to get this, i know the answers but i can't get them
for the first range, f(x) is obviously 0.15x, but i can't get the next ones. The second range answer is 0.28x - 3295.50. can someone tell me how to get this answer?
http://i2.photobucket.com/albums/y15/seiferseph/2.jpg

2) I'm not quite sure how to find the limit when t aproaches infinity. i don't know the rules of exponents w/ infinity, the answer is 6.5, how do i get it?
http://i2.photobucket.com/albums/y15/seiferseph/3.jpg

3) how do i solve b? some hints would be perfect, thanks!
http://i2.photobucket.com/albums/y15/seiferseph/1.jpg

thanks again!

 

[Fermat]

For 1), if you read the table, the end bit, where it says that the 0.28% applies to the amount over x, the person's income.

Tax = 3,802.50 + 0.28(X- 25,350)
Tax = 0.28x + 3,802.50 - 0.28*25,350
Tax = 0.28x - 3,2950.50
===================

 

[Fermat]

For 2),

$$e^{-kt} = \frac{1}{e^{kt}}$$

where k is a positive number.

As t increases, how does $$e^{kt}$$ vary ?

 

[Disar]

Reply

for2):

lim(t->infinity) [6.5 -2 exp(-.035*t)
= lim(t->infinity) (6.5 - 2/exp(.035*t))

the limit of a constant is the constant, correct?
Then,
lim(t->infinity) 6.5 = 6.5

-2* lim(t->infinity) 1/exp(0.035*t)
What is the exp(infinity), look at the graph of the exponetial function, it should be infinity.

What is 1/(a large#)? It should be a small number.

And infinitly large number inverted is an infinitely small number; ie zero

So,
-2* lim(t->infinity) 1/exp(0.035*t) = -2*1/(infinity) = -2*0 = 0

Putting the pieces together,
lim(t->infinity) [6.5 -2 exp(-.035*t)
= 6.5 - 2*lim(t->infinity) 1/exp(0.035*t)
= 6.5 -2*0 = 6.5

 

[Fermat]

For 3),
You have the differential eqn,

$$\frac{dH}{dt} = 175 - 0.35H$$

this becomes,

$$\frac{dH}{0.35H - 175} = -dt$$

giving,

$$\int \frac{dH}{0.35H - 175} = - \int dt$$

Can you solve this now ?

 

[seiferseph]

For 1), if you read the table, the end bit, where it says that the 0.28% applies to the amount over x, the person's income.
Tax = 3,802.50 + 0.28(X- 25,350)
Tax = 0.28x + 3,802.50 - 0.28*25,350
Tax = 0.28x - 3,2950.50
===================

oh i see now, i didn't get how to get the minus sign, i didn't understand that it was the amount over 25350, so x - 25350. thanks!

For 2),
$$e^{-kt} = \frac{1}{e^{kt}}$$
where k is a positive number.
As t increases, how does $$e^{kt}$$ vary ?

for2):
lim(t->infinity) [6.5 -2 exp(-.035*t)
= lim(t->infinity) (6.5 - 2/exp(.035*t))
the limit of a constant is the constant, correct?
Then,
lim(t->infinity) 6.5 = 6.5
-2* lim(t->infinity) 1/exp(0.035*t)
What is the exp(infinity), look at the graph of the exponetial function, it should be infinity.
What is 1/(a large#)? It should be a small number.
And infinitly large number inverted is an infinitely small number; ie zero
So,
-2* lim(t->infinity) 1/exp(0.035*t) = -2*1/(infinity) = -2*0 = 0
Putting the pieces together,
lim(t->infinity) [6.5 -2 exp(-.035*t)
= 6.5 - 2*lim(t->infinity) 1/exp(0.035*t)
= 6.5 -2*0 = 6.5

oh i see, thanks!, so its basically just 6.5 - 2/infnity which obviously simplifies to 6.5, ugh i feel so stupid.

For 3),
You have the differential eqn,
$$\frac{dH}{dt} = 175 - 0.35H$$
this becomes,
$$\frac{dH}{0.35H - 175} = -dt$$
giving,
$$\int \frac{dH}{0.35H - 175} = - \int dt$$
Can you solve this now ?

i'm not sure how to solve it, it is the same form as Newton's law of cooling right? but I'm not sure how to sovle it

 

[Fermat]

The DE is the same form as Newton's law of cooling.

As regards solving it, have you integrated,

$$\int \frac{dH}{0.35H - 175}$$

yet ?

What did you get ?

 

[seiferseph]

i see, i got it, thanks for all the help!