# Thread:

1. ## 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.

2. ## 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.

3. ## Re: Coming up with an exponential equation

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 $y$-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.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8940156102180481, "perplexity_flag": "middle"}