91Ó°ÊÓ

The equation of motion of a particle is \(s=t^{3}-3 t,\) where \(s\) is in meters and t is in seconds. Find (a) the velocity and acceleration as functions of t, (b) the acceleration after \(2 \mathrm{s},\) and (c) the acceleration when the velocity is \(0 .\)

\(45-54\) Find the derivative of the function. Simplify where possible. $$y=\tan ^{-1}\left(x-\sqrt{1+x^{2}}\right)$$

If \(\mathrm{f}\) is a differentiable function, find an expression for the derivative of each of the following functions. $$ \text { (a) } y=x^{2} f(x) \quad \text { (b) } y=\frac{f(x)}{x^{2}}$$ $$ \text { (c) } y=\frac{x^{2}}{f(x)} \quad \text { (d) } y=\frac{1+x f(x)}{\sqrt{x}}$$

Find \(y^{\prime}\) if \(x^{y}=y^{x}\)

A flexible cable always hangs in the shape of a catenary \(y=c+a \cosh (x / a),\) where \(c\) and a are constants and a \(>0\) (see Figure 4 and Exercise 52 ). Graph several members of the family of functions \(y=a \cosh (x / a) .\) How does the graph change as a varies?

\(47 - 50\) Find the first and second derivatives of the function. $$y = e ^ { e ^ { x } }$$

The equation of motion of a particle is \(\mathrm{s}=2 \mathrm{t}^{3}-7 \mathrm{t}^{2}+4 \mathrm{t}+1,\) where \(\mathrm{s}\) is in meters and \(\mathrm{t}\) is in seconds. (a) Find the velocity and acceleration as functions of t. (b) Find the acceleration after 1 \(\mathrm{S}\) . (c) Graph the position, velocity, and acceleration functions on the same screen.

How many tangent lines to the curve \(y=x /(x+1)\) pass trough the point \((1,2) ?\) At which points do these tangent lines touch the curve?

Find a formula for \(f^{(n)}(x)\) if \(f(x)=\ln (x-1)\)

Find the points on the curve \(y=2 x^{3}+3 x^{2}-12 x+1\) where the tangent is horizontal.