Chapter 6

\(\int \frac{t d t}{\sqrt{t+3}}\)

The points \((-1,3)\) and \((0,2)\) are on a curve, and at any point \((x, y)\) on the curve \(D_{x}^{2} y=2-4 x .\) Find an equation of the curve.

\(\sqrt{0.042}\)

\(y=\frac{5 x-2}{x^{2}+1} ; x=(2 t-1)^{2}\)

\(\int \frac{2 r d r}{(1-r)^{2 / 3}}\)

An equation of the tangent line to a curve at the point \((1,3)\) is \(y=x+2\). If at any point \((x, y)\) on the curve, \(D_{x}^{2} y=6 x\), find an equation of the curve.

\(\frac{1}{\sqrt[3]{120}}\)

\(y=x^{2}-3 x+1 ; x=\sqrt{t^{2}-t+4}\)

\(\int \sqrt{3-x} x^{2} d x\)

At any point \((x, y)\) on a curve, \(D_{x}^{2} y=1-x^{2}\), and an equation of the tangent line to the curve at the point \((1,1)\) is \(y=2-x\). Find an equation of the curve.

\(\frac{1}{\sqrt[4]{15}}\)

\(y=\sqrt[3]{5 x-1} ; x=\sqrt{2 t+3}\)

\(\int\left(x^{3}+3\right)^{1 / 4} x^{5} d x\)

At any point \((x, y)\) on a curve, \(D_{x}^{3} y=2\), and \((1,3)\) is a point of inflection at which the slope of the inflectional tangent is \(-2 .\) Find an equation of the curve.

The measurement of an edge of a cube is found to be \(15 \mathrm{in}\). with a possible error of \(0.01\) in. Using differentials find the approximate error in computing from this measurement (a) the volume; (b) the area of one of the faces.

\(y=x^{2}-5 x+1 ; x=s^{3}-2 s+1 ; s=\sqrt{t^{2}+1}\)

\(\int \frac{\left(x^{2}+2 x\right) d x}{\sqrt{x^{3}+3 x^{2}+1}}\)

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 \(2 y / x\).)

The altitude of a right-circular cone is twice the radius of the base. The altitude is measured as 12 in., with a possible error of \(0.005\) in. Find the approximate error in the calculated volume of the cone.

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

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.

\(\int \frac{y+3}{(3-y)^{2 / 3}} d y\)

A metal box in the form of a cube is to have an interior volume of 64 in. \(^{3}\). The six sides are to be made of metal \(\frac{1}{4}\) in. thick. If the cost of the metal to be used is 8 cents per cubic inch, use differentials to find the approximate cost of the metal to be used in the manufacture of the box.

\(\int\left(2 t^{2}+1\right)^{1 / 3} t^{3} d t\)

\(\int \frac{\left(r^{1 / 3}+2\right)^{4}}{\sqrt[3]{r^{2}}} d r\)

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.

Evaluate \(\int(2 x+1)^{3} d x\) by two methods: (a) Expand \((2 x+1)^{3}\) by the binomial theorem, and apply Formulas 1,4, and \(5 ;(b)\) make the substitution \(u=2 x+1\). Explain the difference in appearance of the answers obtained in (a) and (b).

If \(t \mathrm{sec}\) is the time for one complete swing of a simple pendulum of length \(l \mathrm{ft}\), then \(4 \pi^{2} l=g t^{2}\), where \(g=32.2\). A clock having a pendulum of length \(1 \mathrm{ft}\) gains \(5 \mathrm{~min}\) each day. Find the approximate amount by which the pendulum should be lengthened in order to correct the inaccuracy.

Evaluate \(\int \sqrt{x-1} x^{2} d x\) by two methods: (a) Make the substitution \(u=x-1 ;(b)\) make the substitution \(v=\sqrt{x}-1 .\)

Let \(f(x)=|x|\) and let \(F\) be defined by $$ F(x)= \begin{cases}-\frac{1}{2} x^{2} & \text { if } x<0 \\ \frac{1}{2} x^{2} & \text { if } x \geq 0\end{cases} $$ Show that \(F\) is an antiderivative of \(f\) on \((-\infty,+\infty)\).