5.4 - The Fundamental Theorem of Calculus

#calc/theorem #calc/concept
The big one :D

The Fundamental Theorem of Calculus relates integrals to derivatives. Here are the two parts of the theorem:

Part 1

If $f$ is continuous on $[a,b]$, then the function ${\displaystyle F(x)={\int }_a^{x}f(t)dt}$ has a derivative at every point $x$ in $[a,b]$, and ${\displaystyle \frac{dF}{dx}=\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)}$, where $f(x)$ or something similar is the function evaluated at the upper limit multiplied by the derivative of the upper limit. This only works if $a$ is a constant.

In human English

The derivative of the integral of a continuous function $f(x)$ is $f(x)$.

Part 2 – AKA the Integral Evaluation Theorem

If $f$ is continuous on $[a,b]$, and $F$ is any antiderivative of $f$ on $[a,b]$, then ${\displaystyle F(b)-F(a)={\int }_a^{b}f(x)dx}$

• This is important for evaluating integrals by hand.

FTC FRQs

• Important to practice, cause they come in a few different forms:
  • In/out problems
  • Geometry problems (that don’t even really require the FTC)
  • Position problems - integrate acceleration to get velocity and integrate velocity to get position
• In general, remember to derive to go “down” a level and integrate to go “up” a level.