91Ó°ÊÓ

Solve the given problems. At what point on the curve of \(y=9-2 x^{2}\) is there a tangent line that is parallel to the line \(12 x-2 y+7=0 ?\)

Solve the given problems by finding the appropriate derivatives. If \(f(x)\) is a differentiable function, find an expression for the derivative of \(y=f(x) / x^{2}\).

Find the acceleration of an object for which the displacement \(s\) (in \(\mathrm{m}\) ) is given as a function of the time \(t\) (in s) for the given value of \(t\). $$s=3(1+2 t)^{4}, t=0.500 \mathrm{s}$$

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{x \rightarrow 1 / 3} \frac{9 x-3}{3 x^{2}+5 x-2}$$

In Exercises \(35-40,\) solve the given problems. At what point on the curve of \(y=9-2 x^{2}\) is there a tangent line that is parallel to the line \(12 x-2 y+7=0 ?\)

Find the indicated instantaneous rates of change. The bottom of a soft-drink can is being designed as an inverted spherical segment, the volume of which is \(V=\frac{1}{6} \pi h^{3}+2.00 \pi h\) where \(h\) is the depth (in \(\mathrm{cm}\) ) of the segment. Find the instantaneous rate of change of \(V\) with respect to \(h\) for \(h=0.60 \mathrm{cm}\)

Find the acceleration of an object for which the displacement \(s\) (in \(\mathrm{m}\) ) is given as a function of the time \(t\) (in s) for the given value of \(t\). $$s=\frac{16}{0.5 t^{2}+1}, t=2 \mathrm{s}$$

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{x \rightarrow 1} \frac{(2 x-1)^{2}-1}{2 x-2}$$

Find the indicated instantaneous rates of change. The total solar radiation \(H\) (in \(\mathrm{W} / \mathrm{m}^{2}\) ) on a particular surface during an average clear day is given by \(H=\frac{5000}{t^{2}+10},\) where \(t\) \((-6 \leq t \leq 6)\) is the number of hours from noon (6 A.M. is equivalent to \(t=-6 \mathrm{h}\) ). Find the instantaneous rate of change of \(H\) with respect to \(t\) at 3 P.M.

Solve the given problems by finding the appropriate derivatives. Find the derivative of \(y=\frac{x^{2}-1}{x-1}\) by (a) the quotient rule, and (b) by first simplifying the function.