Antiderivatives Graphical / Numerical

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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