4.2 - The Mean Value Theorem and Antiderivatives

#calc/theorem

Increasing and decreasing functions

• A function is increasing on an interval if $f(x_1)<f(x_2)$ for points $x_1$ and $x_2$ in $l$.
  • If the opposite, then the function is decreasing.
    From this we can easily see that a function increases on an interval if the derivative at every point in that interval is positive, and that a function decreases if the opposite is true.

The mean value theorem

If a function $y=f(x)$ is continuous at every point in the closed interval $[a,b]$, then there is at least one point $c$ in that interval such that ${\displaystyle f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}}$.

The theorem essentially just states that the average rate of change in the interval will equal the instantaneous rate of change at some point (or more than one).

Antiderivatives

Antiderivatives are like the reverse of taking a derivative.

• You can do them with some logic: if we see that the derivative of a function has a term of $3x$, then we know that the original function came in the form ${\displaystyle \frac{3}{2}{x}^2+c}$ for some constant $c$.