91Ó°ÊÓ

Solve the given problems. $$\text { Is } \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-1}=\lim _{x \rightarrow 3} \frac{2 x}{1}=6 ? \text { Explain. }$$

Solve the given problems. Show that \(y=x e^{-x}\) satisfies the equation \((d y / d x)+y=e^{-x}\).

Solve the given problems.Find the points where a tangent to the curve of \(y=\tan x\) is parallel to the line \(y=2 x\) if \(0

Show that \(y=e^{-x} \sin x\) satisfies the equation \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+2 y=0\).

Solve the given problems. Evaluate the derivative of \(y=\ln \sqrt{\frac{2 x+1}{3 x+1}},\) where \(x=2.75\)

In Exercises \(37-48,\) solve the given problems. Is \(\lim _{x \rightarrow 3} \frac{x^{3}-3 x^{2}+x-3}{x^{2}-9}=\lim _{x \rightarrow 3} \frac{3 x^{2}-6 x+1}{2 x}=\frac{5}{3} ?\) Explain.

Solve the given problems. Use a calculator to display the graphs of \(y=\tan ^{-1} x\) and \(y=1 /\left(1+x^{2}\right) .\) By roughly estimating slopes of tangent lines of \(y=\tan ^{-1} x,\) note that \(y=1 /\left(1+x^{2}\right)\) gives reasonable values for the derivative of \(y=\tan ^{-1} x\)

Solve the given problems. Find the derivative of the implicit function \(x \cos 2 y+\sin x \cos y=1\)

Solve the given problems. Show that \(y=e^{-x} \sin x\) satisfies the equation \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+2 y=0\)

Solve the given problems by finding the appropriate derivative. A missile is launched and travels along a path that can be represented by \(y=\sqrt{x} .\) A radar tracking station is located \(2.00 \mathrm{km}\) directly behind the launch pad. Placing the launch pad at the origin and the radar station at \((-2.00,0),\) find the largest angle of elevation required of the radar to track the missile.