6.4 The Fundamental Theorem of Calculus and Accumulation Functions

3 practice questions

Notes

For an accumulation function $\displaystyle g(x)=\int_a^x f(t)\,dt$:

$$g'(x)=f(x)$$

With a variable upper limit $u(x)$: $\displaystyle \frac{d}{dx}\int_a^{u(x)}f(t)\,dt=f(u(x))\,u'(x)$.

$\displaystyle \frac{d}{dx}\int_2^x \cos(t^2)\,dt=$

$\displaystyle \frac{d}{dx}\int_0^{x^3}\sqrt{1+t}\,dt=$

If $\displaystyle g(x)=\int_1^x f(t)\,dt$, then $g(1)=$