Use a linear approximation to estimate \(\sqrt{9.2}\). Define the function and base point used.

For \(f(x)=x^3\), find the linear approximation at \(a=2\), then use it to estimate \((1.98)^3\).

Find the linear approximation \(L(x)\) for \(f(x)=\frac{1}{x}\) at \(a=4\), then estimate \(\frac{1}{4.1}\).

Use linear approximation with \(f(x)=\ln x\) at \(a=1\) to estimate \(\ln(1.05)\).

Approximate \(\sin(0.08)\) using a linear approximation at \(a=0\), where angles are measured in radians.

For \(f(x)=\sqrt[3]{x}\), use a linear approximation at \(a=8\) to estimate \(\sqrt[3]{8.3}\).

Find \(dy\) for \(y=x^4\) when \(x=2\) and \(dx=0.03\), and use \(dy\) to approximate the change in \(y\).

The radius \(r\) of a sphere is measured as \(5\) cm with possible error \(dr=0.02\) cm. Use differentials to approximate the resulting error in volume \(V=\frac{4}{3}\pi r^3\).

Use the tangent line to \(f(x)=e^x\) at \(a=0\) to estimate \(e^{-0.04}\).

For \(f(x)=\frac{1}{\sqrt{x}}\), use a linear approximation at \(a=4\) to estimate \(\frac{1}{\sqrt{3.9}}\).