91Ó°ÊÓ

Find \(d y / d x\). $$ y=\sqrt[4]{1+\cos \left(x^{2}+2 x\right)} $$

Use the definition of the derivative to show that \(D_{x}\left(\sin x^{2}\right)=2 x \cos x^{2}\)

Use a graphing calculator or a CAS to do Problems \(31-34\). Draw the graph of \(y=f(x)=x^{3}-2 x^{2}+1\). Then find the slope of the tangent line at (a) -1 (b) 0 (c) 1 (d) 3.2

It can be shown that if \(\left|d^{2} y / d x^{2}\right| \leq M\) on a closed interval with \(c\) and \(c+\Delta x\) as end points, then $$ |\Delta y-d y| \leq \frac{1}{2} M(\Delta x)^{2} $$ Find, using differentials, the change in \(y=3 x^{2}-2 x+11\) when \(x\) increases from 2 to 2.001 and then give a bound for the error that you have made by using differentials.

Two objects move along a coordinate line. At the end of \(t\) seconds their directed distances from the origin, in feet, are given by \(s_{1}=4 t-3 t^{2}\) and \(s_{2}=t^{2}-2 t,\) respectively. (a) When do they have the same velocity? (b) When do they have the same speed? (c) When do they have the same position?

Evaluate the indicated derivative. \(F^{\prime}(1)\) if \(F(t)=\sin \left(t^{2}+3 t+1\right)\)

The given limit is a derivative, but of what function and at what point? \(\lim _{x \rightarrow 3} \frac{x^{3}+x-30}{x-3}\)

Find \(D_{x} y\) using the rules of this section. $$y=\left(5 x^{2}-7\right)\left(3 x^{2}-2 x+1\right)$$

The given limit is a derivative, but of what function and at what point? \(\lim _{p \rightarrow x} \frac{p^{3}-x^{3}}{p-x}\)

Find \(D_{x} y\) using the rules of this section. $$y=\left(3 x^{2}+2 x\right)\left(x^{4}-3 x+1\right)$$