91Ó°ÊÓ

Integrating Inverse Functions Assume that the function \(f\) has an inverse. Use integration by parts directly to show that $$\int f^{-1}(x) d x=x f^{-1}(x)-\int x\left(\frac{d}{d x} f^{-1}(x)\right) d x$$

Resistance Proportional to Velocity It is reasonable to assume that the air resistance encountered by a moving object, such as a car coasting to a stop, is proportional to the object's velocity. The resisting force on an object of mass \(m\) moving with velocity \(v\) is thus \(-k v\) for some positive constant \(k\). (a) Use the law Force = Mass \(\times\) Acceleration to show that the velocity of an object slowed by air resistance (and no other forces) satisfies the differential equation $$m \frac{d y}{d t}=-k v$$ (b) Solve the differential equation to show that \(v=v_{0} e^{-(k / m) t}\) where \(v_{0}\) is the velocity of the object at time \(t=0 . (c) If \)k$ is the same for two objects of different masses, which one will slow to half its starting velocity in the shortest time? Justify your answer.

Percentage Error Let \(y=f(x)\) be solution to the initial value problem \(d y / d x=2 x-1\) such that \(f(2)=3 .\) Find the per- centage error if Euler's Method with \(\Delta x=-0.1\) is used to ap- proximate \(f(1.6) .\)

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity \(v(t)\) is modeled by the initial value problem $$\begin{array}{ll}{\text { Differential equation: }} & {m \frac{d v}{d t}=m g-k v^{2}} \\ {\text { Initial condition: }} & {v(0)=0}\end{array} $$ where \(t\) represents time in seconds, \(g\) is the acceleration due to gravity, and \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that variation in the air's density will not affect the outcome.) (a) Show that the function $$v(t)=\sqrt{\frac{m g}{k}} \frac{e^{a t}-e^{-a t}}{e^{a t}+e^{-a t}}$$ where \(a=\sqrt{g k / m},\) is a solution of the initial value problem. (b) Find the body's limiting velocity, \(\lim _{t \rightarrow \infty} v(t)\) (c) For a 160 -lb skydiver \((m g=160),\) and with time in seconds and distance in feet, a typical value for \(k\) is \(0.005 .\) What is the diver's limiting velocity in feet per second? in miles per hour?

In Exercises 67 and \(68,\) make a substitution \(u=\cdots(\) an expression in \(x), \quad d u=\cdots .\) Then (a) integrate with respect to \(u\) from \(u(a)\) to \(u(b)\) . (b) find an antiderivative with respect to \(u,\) replace \(u\) by the expression in \(x,\) then evaluate from \(a\) to \(b\) . $$\int_{\pi / 6}^{\pi / 3}(1-\cos 3 x) \sin 3 x d x$$