Exercises

01

Fundamental Theorem of Calculus

Compute each of the following derivatives:

• (a) $\frac{d}{dx}{\int }_2^x\sin (1/t)dt$
• (b) $\frac{d}{dx}{\int }_x^3{e}^{-t^2}dt$
• (c) $\frac{d}{dx}{\int }_2^{x^2}\sin (1/t)dt$

02

Area under a curve

Find the area under the curve $y=\cos x$ for $-\pi /2\le x\le \pi /2$.

03

Particle distance travelled from acceleration

A particle is moving on the number line with position $x(t)$, velocity $v(t)$, and acceleration $a(t)$. So $v(t)=x^{\prime }(t)$ and $a(t)=v^{\prime }(t)$.

Suppose that $a(t)=3t+4$ and $v(0)=2$. How far does the particle travel during the time interval starting at $t=5$ and ending at $t=10$?