The Theorems of Integral Calculus
Below are statements of theorems from integral calculus with accompanying visualizations.
The Fundamental Theorem of Calculus
If [math]f[/math] is continuous on [math][a,b][/math], then:[br][list=1][*]If [math]g\left(x\right)=\int_a^xf\left(t\right)dt[/math], then [math]g'\left(x\right)=f\left(x\right)[/math]. Another way of saying this is [math]\frac{d}{dx}\int_a^xf(t)dt=f(x)[/math].[/*][*][math]\int_a^bf(x)dx=F(b)-F(a)[/math], where [math]F[/math] is any antiderivative of [math]f[/math] (that is, [math]F'=f[/math]).[/*][/list]
The Mean Value Theorem for Integrals
If [math]f[/math] is continuous on [math][a,b][/math], then there is a number [math]c[/math] in [math](a,b)[/math] such that [math]f(c)=\frac{1}{b-a}\int_a^bf(x)dx[/math].
Mean Value Theorem for Integrals
Explain what the Mean Value Theorem for Integrals says using plain English (i.e., without using any mathematical notation).
Use the applet above to determine what you can say about the value of [math]c[/math] guaranteed by the Mean Value Theorem for Integrals if the function [math]f[/math] is linear.
Edit the applet below to find (another example of) a function f and an interval [math][a,b][/math] to show that your finding from the previous problem is not true in general. Experiment to see how close you can get [math]c[/math] to [math]a[/math].