91Ó°ÊÓ

Find \(d y / d x\). $$ y=\sin ^{-1} x+\cos ^{-1} x $$

Coffee is poured at a uniform rate of \(20 \mathrm{cm}^{3} / \mathrm{s}\) into a cup whose inside is shaped like a truncated cone (see the accompanying figure). If the upper and lower radii of the cup are \(4 \mathrm{cm}\) and \(2 \mathrm{cm}\) and the height of the cup is \(6 \mathrm{cm},\) how fast will the coffee level be rising when the coffee is halfway up? [Hint: Extend the cup downward to form a cone.]

Find \(d y / d x\). $$ y=x^{2}\left(\sin ^{-1} x\right)^{3} $$

Find a formula for the area \(A(w)\) of the triangle bounded by the tangent line to the graph of \(y=\ln x^{2}\) at \(P\left(w, \ln w^{2}\right),\) the horizontal line through \(P,\) and the \(y\) -axis.

(a) Find the error in the following calculation: $$\lim _{x \rightarrow 2} \frac{e^{3 x^{2}-12 x+12}}{x^{4}-16}=\lim _{x \rightarrow 2} \frac{(6 x-12) e^{3 x^{2}-12 x+12}}{4 x^{3}}=0$$ (b) Find the correct limit.

Verify that \(y=\ln (x+e)\) satisfies \(d y / d x=e^{-y},\) with \(y=1\) when \(x=0 .\)

Make a conjecture about the limit by graphing the function involved with a graphing utility; then check your conjecture using L'Hôpital's rule. $$ \lim _{x \rightarrow+\infty} \frac{\ln (\ln x)}{\sqrt{x}} $$

Find \(d y / d x\). $$ y=\sec ^{-1} x+\csc ^{-1} x $$

Find \(d y / d x\). $$ y=\csc ^{-1}\left(e^{x}\right) $$

Verify that \(y=-\ln \left(e^{2}-x\right)\) satisfies \(d y / d x=e^{y},\) with \(y=-2\) when \(x=0\)