The Second Fundamental Theorem of Calculus

Evaluate each of the following derivatives and definite integrals. Clearly cite whether you use the First or Second FTC in so doing.

1. $\displaystyle \frac{d}{dx} \left[ \int_4^x e^{t^2} \, dt \right]$

2. $\displaystyle \int_{-2}^x \frac{d}{dt} \left[ \frac{t^4}{1+t^4} \right] \, dt$

3. $\displaystyle \frac{d}{dx} \left[ \int_{x}^1 \cos(t^3) \, dt \right]$

4. $\displaystyle \int_{3}^x \frac{d}{dt} \left[ \ln(1+t^2) \right] \, dt$

5. $\displaystyle \frac{d}{dx} \left[ \int_4^{x^3} \sin(t^2) \, dt \right]$

The tide removes sand from the beach at a small ocean park at a rate modeled by the function

A pumping station adds sand to the beach at rate modeled by the function

Both $R(t)$ and $S(t)$ are measured in cubic yards of sand per hour, $t$ is measured in hours, and the valid times are $0 \le t \le 6$. At time $t = 0$, the beach holds 2500 cubic yards of sand.

1. What definite integral measures how much sand the tide will remove durin…