Antiderivatives Graphical / Numerical
<< 5.5, 5.FT | 7.0 | 7.1 >>
- Complete today's handout and bring it to our next class.
Graphical consequences of the FT
Second fundamental theorem
if $f$ is a continuous function on an interval, and if $a$ is any number in that interval, then the function $G$ defined on the interval by $$G(x)\equiv\int_a^x f(t)\,dt$$ has derivative $f$; that is, $G'(x)=f(x)$.
Since $G'(x)=f(x)$, what do these mean about $G(x)$?:
- $f(x)=0 \Rightarrow $...?
- $f(x)>0 \Rightarrow $...? This means $G'(x) \gt 0$: At $x$, a line tangent to $G$ would have a positive slope. $G(x)$ is increasing.
- $f(x)<0 \Rightarrow $...?
Since $G''(x)=f'(x)$, what do these mean about $G(x)$?
- $f'(x)>0 \Rightarrow $...?
- $f'(x)<0 \Rightarrow $...?
- $f'(x)=0 \Rightarrow $...?
Today
- Antiderivatives [7.0] (.ppt)
- Handout (.pdf)