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Differentiate equation!!! Really need help!

HELP!
I have this question, and I've got about 20 different answers to it but I'm completely lost.

T= a+b e^-kt

T is the temperature at time t, and a, b and k are constants with values a = 20 °C, b = 80 °C and k = 2.6 × 10^-4 s^-1.

Differentiate the equation with respect to t and thus give values (with appropriate units and to two significant figures) for the rate of change of temperature with time at times of 1.0 × 103 s and 5.0 × 103 s.

Can anybody please help me with the steps in doing this please??

Reply 1

Zacken

Original post by thegallowsgirl

Constants, go away, obviously.

\frac{d}{dt}(e^{-kt}) = -k e^{-kt}

Reply 2

thegallowsgirl

So far I have

dT/dt = (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4)

Reply 3

thegallowsgirl

Original post by Zacken

Constants, go away, obviously.

\frac{d}{dt}(e^{-kt}) = -k e^{-kt}

okay, so if I am understanding correctly (which I may not be sorry) then:
-k e^-kt would be:
(-2.6 x 10^-4) x (-2.6 x 10^4) x (t = 1 x 10^3) ?

Reply 4

DPD99

Internet search came up with http://mymathforum.com/algebra/330327-differentiation-problem.html

It may be some help.

Reply 5

Zacken

Original post by thegallowsgirl

So far I have

dT/dt = (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4)

Where'd the small t in the power go?

Reply 6

PossiblyNotGod

Is this T= (a+b) x (e^-kt)?
or T= a+b + (e^-kt)?
or T= a+(b*e^-kt)?

(edited 7 years ago)

Reply 7

thegallowsgirl

Original post by Zacken

Where'd the small t in the power go?

Sorry, that was a typing error.

dT/dt = (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4) x t

= (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4) x (1 x 10^3)

With constants a & b removed:

= (-2.6 x 10^-4)e ^(2.6 x 10^-4) x (1 x 10^3)

Quick question - e is normally 2.718, but I'm unsure as to whether I need to include that in this equation?

Reply 8

thegallowsgirl

Original post by XOR_

Is this T= (a+b) x (e^-kt)?
or T= (a+b) + (e^-kt)?
or T= a+(b*e^-kt)?

It's written T = a+b e^-kt so I presume it's T= (a+b) x (e^-kt)? At least that's what I've been doing.

Reply 9

Zacken

Original post by thegallowsgirl

It's written T = a+b e^-kt so I presume it's T= (a+b) x (e^-kt)? At least that's what I've been doing.

I would presume it's a + [be^(-kt)].

Reply 10

1420787

Original post by thegallowsgirl

It's written T = a+b e^-kt so I presume it's T= (a+b) x (e^-kt)? At least that's what I've been doing.

I presume it would be as written, without the brackets.

The questions looks set up to calculate temperature change of someting in a room of temperature 20 degrees. As time goes on the temperature will tend to 20 degrees, as is natural.

Reply 11

thegallowsgirl

Original post by Zacken

I would presume it's a + [be^(-kt)].

That could be where I'm going wrong.
If I did it that way, it would be:

20 + [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000) ?

Reply 12

Zacken

Original post by thegallowsgirl

That could be where I'm going wrong.
If I did it that way, it would be:

20 + [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000) ?

No, when you differentiate it, the

a

disappears.

Reply 13

thegallowsgirl

Original post by Zacken

No, when you differentiate it, the

a

disappears.

Right!
So it would be [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000)] without the 20 in front.
So calculating it would result in 0.005408 = 5.4 x 10^-3
Is that in the right area or is there a step that I've missed out? Sorry for asking so many questions, I'm trying to understand the steps and how they result in the answer.

Reply 14

Zacken

Original post by thegallowsgirl

It would be (stuff in front seems fine) but the

e^{-2.6\times 10^{-4} \times 1000}

.

That is, e raised to the power of the the number in the exponent, not e * number.

Reply 15

thegallowsgirl

Thank you so much! I was thinking I'd missed something out.
[80 x (-2.6 x 10^-4)e^ (-2.6 x 10^-4)x (1000)] I'm trying to figure out how I can input it into a calculator now. Thank you for your help!!

Reply 16

Zacken

Original post by thegallowsgirl

Thank you so much! I was thinking I'd missed something out.
[80 x (-2.6 x 10^-4)e^ (-2.6 x 10^-4)x (1000)] I'm trying to figure out how I can input it into a calculator now. Thank you for your help!!