Coming up with an exponential equation

• Jul 25th 2011, 03:06 AM
asaver
Definite Integrals
I need to come up with an exponential equation which fulfils the following criteria;
-is an increasing function
-is concave up
-pass through the points (0,3) and (4,11)
-has a definite integral of 22 between x=0 and x=4.
• Jul 25th 2011, 03:45 AM
Siron
Re: Coming up with an exponential equation
Why are you so sure the exponential function has base $e$? We know the exponential function is increasing so the base >1.
• Jul 25th 2011, 05:51 AM
Soroban
Re: Coming up with an exponential equation
Hello, asaver!

Quote:

I need an exponential equation which fulfils the following criteria:
. . [1] is an increasing function
. . [2] is concave up
. . [3] pass through the points (0,3) and (4,11)
. . [4] has a definite integral of 22 between x=0 and x=4.

Because of the $y$-intercept $(0,3)$
. . you believe the function has the form: $f(x) \,=\,3e^x$

But it could have the form: . $f(x) \,=\,e^x + 2$

You had best begin with the general exponential function: . $f(x) \:=\:ae^{bx} + c$

[1] and [2] tells us that $a > 0$

[3] says $f(0) = 3$ and $f(4) = 1.$

. . We have: . $\begin{Bmatrix} a + c &=& 3 \\ ae^{4b} + c &=& 11 \end{Bmatrix}$

[4] gives us: . $\int^4_0\left(ae^{bx} + c\right)\,dx \:=\:22$

We have three equations in three variables $\{a,b,c\}.$
. . Solve the system.