91Ó°ÊÓ

\(\int \sqrt{5 r+1} d r\)

The point \((3,2)\) is on a curve, and at any point \((x, y)\) on the curve the tangent line has a slope equal to \(2 x-3\). Find an equation of the curve.

\(\sqrt{0.042}\)

The equation \(x^{2}=4 a y\) represents a one-parameter family of parabolas. Find an equation of another one-parameter family of curves such that at any point \((x, y)\) there is a curve of each family through it and the tangent lines to the two curves at this point are perpendicular. (HINT: First show that the slope of the tangent line at any point \((x, y)\), not on the \(y\) axis, of the parabola of the given family through that point is \(2 y / x\).)

\(\int \sqrt{3+s}(s+1)^{2} d s\)

An open cylindrical tank is to have an outside coating of thickness \(\frac{1}{8}\) in. If the inner radius is \(6 \mathrm{ft}\) and the altitude is \(10 \mathrm{ft}\), find by differentials the approximate amount of coating material to be used.

A contractor agrees to paint on both sides of 1000 circular signs each of radius \(3 \mathrm{ft}\). Upon receiving the signs, it is discovered that the radius is \(\frac{1}{2}\) in. too large. Use differentials to find the approximate percent increase of paint that will be needed.

The measure of the electrical resistance of a wire is proportional to the measure of its length and inversely proportional to the square of the measure of its diameter. Suppose the resistance of a wire of given length is computed from a measurement of the diameter with a possible \(2 \%\) error. Find the possible percent error in the computed value of the resistance.