Math 162
Class 2
What is the rage of change of the area function?
\[g(x) = \int_a^x f(t)\;dt\]
The Fundamental Theorem of Calculus Part 1:
If \(f\) is continuous on \([a,b]\) then for every \(c\) in \([a,b]\) the function
\[g(x) = \int_c^x f(t)\;dt\]
is defined on \([a,b]\) and differentiable on \((a,b)\), with
\[g'(x) = f(x)\]
for every \(x\) in \((a,b)\).
What if we accumulate changes?
\[\int_a^b f'(x) \; dx\]
The Fundamental Theorem of Calculus Part 2:
If \(f\) is continuous on \([a,b]\) and has continuous derivative on \(( a,b )\) then
\[\int_a^b f'(x)\;dx = f(b) - f(a)\]
Integrating the rate of change gives us the total change.
Derivative as scaling factor again
Substitution theorem
