22. Antiderivatives, Areas and the FTC

Homework

For each function, find its general antiderivative. Be sure to check by differentiating.

\(f(x)=4x^3-3e^{3x}+2\cos(2x)\)
\(g(x)=2x^2\sin(4x^3+7)\)
\(p(x)=x^2(x^4+4)^2\)
\(f(x)=\dfrac{\cos x}{\sin^2x}\)
\(p(x)=\dfrac{4}{x^3}+\dfrac{3}{x^2}+\dfrac{2}{x}\)
\(q(x)=\dfrac{6}{1+4x^2}\)
\(r(t)=\dfrac{6}{\sqrt{1-9t^2}} \)
\(f(x) = \dfrac{6}{x\sqrt{4x^2-1}}\) for \(x \ge 1\)
\(u(x)=e^{2x}+\dfrac{4}{e^{3x}}+\dfrac{1}{2x}\)

If a car's velocity at time \(t\) is given by \(v(t)=3t^2+\sin t\), and its position at \(t=0\) is \(x(0)=4\), what is its position at \(t=\pi\)?
A rocket's acceleration is: \[ a(t)=\sin\left(\dfrac{\pi t}{30}\right) \] where time is in seconds and distance is in kilometers. Its initial altitude and velocity are: \[ y(0)=0.01 \quad \text{and} \quad v(0)=0 \] Find its altitude after \(15\) seconds.
On the Jupiter's moon Io, the acceleration of gravity is: \(g_\text{io}=1.81\,\dfrac{\text{m}}{\text{sec}^2}\).
If a ball is dropped from \(10\,\text{m}\), how long will it take to reach the ground?
NOTE: You must find the necessary antiderivatives, not just use a formula from physics.
On the Jupiter's moon Io, the acceleration of gravity is: \(g_\text{io}=1.81\,\dfrac{\text{m}}{\text{sec}^2}\).
If a ball is thrown at \(30\,\dfrac{\text{m}}{\text{sec}}\) at an angle of \(30^\circ\) above horizontal, from a height of \(2\,\text{m}\), find its maximum height and its range.
NOTE: You must find the necessary antiderivatives, not just use a formula from physics.
Consider the differential equation: \(\dfrac{dy}{dx}=6x^2+6e^{2x}\) and the initial condition \(y(0)=9\).
1. Find the general solution of the differential equation.
2. Solve the initial value problem.
A bucket is filling with rain at the rate \(R(t)=\dfrac{2}{(t+1)^3}\,\dfrac{\text{gal}}{\text{hr}}\). Find the volume of water in the bucket at time \(t=4\,\text{hr}\) if it starts with \(5\,\text{gal}\) at time \(t=0\).