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.
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.
Hello, asaver!
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.