91Ó°ÊÓ

Solve the given problems by finding the appropriate derivatives. Using the product rule, find the point(s) on the curve of \(y=\left(2 x^{2}-1\right)(1-4 x)\) for which the tangent line is \(y=4 x-1\).

S represents the displacement, and t represents the time for objects moving with rectilinear motion, according to the given functions. Find the instantaneous velocity for the given times. $$s=120+80 t-16 t^{2} ; t=2.5$$

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 \(t.\) $$s=26 t-4.9 t^{2}, t=3.0 \mathrm{s}$$

\(\text {Solve the given problems.}\) At what point on the curve of \(y=2 x^{2}-16 x\) is there a tangent line that is horizontal?

\(\text {Solve the given problems.}\) 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. 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 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 \(t.\) $$s=3(1+2 t)^{4}, t=0.500$$

S represents the displacement, and t represents the time for objects moving with rectilinear motion, according to the given functions. Find the instantaneous velocity for the given times. $$s=0.5 t^{4}-1.5 t^{2}+2.5 ; t=3$$

Evaluate the derivatives of the given functions for the given values of \(x\). Check your results, using the derivative evaluation feature of a calculator. $$y=x^{2} \sqrt[3]{3 x+2}, x=2$$

Evaluate the indicated limits algebraically as in Examples \(10-14\). Change the form of the function where necessary. $$\lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{3-x}$$