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.

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- Jul 25th 2011, 02:06 AMasaverDefinite 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, 02:45 AMSironRe: Coming up with an exponential equation
Why are you so sure the exponential function has base $\displaystyle e$? We know the exponential function is increasing so the base >1.

- Jul 25th 2011, 04:51 AMSorobanRe: 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 $\displaystyle y$-intercept $\displaystyle (0,3)$

. . you believe the function has the form: $\displaystyle f(x) \,=\,3e^x$

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

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

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

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

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

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

We have three equations in three variables $\displaystyle \{a,b,c\}.$

. . Solve the system.