Chapter 6

\(y=x^{3}\)

\(y=\left(3 x^{2}-2 x+1\right)^{3}\)

\(\int 3 x^{4} d x\)

\frac{d y}{d x}=3 x^{2}+2 x-7

\(\int\left(4 x^{3}-3 x^{2}+6 x-1\right) d x\)

\(\frac{d y}{d x}=(3 x+1)^{3}\)

The rate of change of the slope of the total cost curve of a particular company is the constant 2, and the total cost curve contains the points \((2,12)\) and \((3,18)\). Find the total cost function.

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

\(a=800 ; v=20\) when \(s=1\). Find an equation involving \(v\) and s. (HINT: \(\left.a=\frac{d v}{d t}=\frac{d v}{d s} \frac{d s}{d t}=v \frac{d v}{d s}\right)\)

\(\int\left(a x^{2}+b x+c\right) d x\)

The marginal cost function is given by \(3 / \sqrt{2 x+4}\). If the fixed cost is zero, find the total cost function.

\(\frac{d y}{d x}=\frac{3 x \sqrt{1+y^{2}}}{y}\)

A stone is thrown vertically upward from the ground with an initial velocity of \(20 \mathrm{ft} / \mathrm{sec}\). How long will it take the stone to strike the ground, and with what speed will it strike? How long will the stone be going upward, and how high will it go?

\(y=\frac{1}{\sqrt[3]{x}}\)

\(y=(x+2)^{1 / 3}(x-2)^{2 / 3}\)

\(\int \frac{y^{4}+2 y^{2}-1}{\sqrt{y}} d y\)

\(\frac{d y}{d x}=\frac{\sqrt{x}+x}{\sqrt{y}-y}\)

\(\int\left(\sqrt{2 x}-\frac{1}{\sqrt{2 x}}\right) d x\)

A man in a balloon drops his binoculars when it is \(150 \mathrm{ft}\) above the ground and rising at the rate of \(10 \mathrm{ft} / \mathrm{sec}\). How long will it take the binoculars to strike the ground, and what is their speed on impact?

The marginal cost function is given by \(3 x^{2}+8 x+4\), and the fixed cost is \(\$ 6 .\) If \(C(x)\) dollars is the total cost of \(x\) units, find the total cost function, and draw sketches of the total cost curve and the marginal cost curve on the same set of axes.

\(y=x^{2}-3 x ; x=-1 ; \Delta x=0.02\)

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

\(\int \frac{27 t^{3}-1}{\sqrt[3]{t}} d t\)

A stone is thrown vertically upward from the top of a house \(60 \mathrm{ft}\) above the ground with an initial velocity of \(40 \mathrm{ft} / \mathrm{sec}\). At what time will the stone reach its greatest height, and what is its greatest height? How long will it take the stone to pass the top of the house on its way down, and what is its velocity at that instant? How long will it take the stone to strike the ground and with what velocity does it strike the ground?

A ball is thrown vertically upward with an initial velocity of \(40 \mathrm{ft} / \mathrm{sec}\) from a point \(20 \mathrm{ft}\) above the ground. If \(v \mathrm{ft} / \mathrm{sec}\) is the velocity of the ball when it is \(s \mathrm{ft}\) from the starting point, express \(v\) in terms of \(s\). What is the velocity of the ball when it is \(36 \mathrm{ft}\) from the ground and rising?

\(\int x \sqrt[3]{\left(4-x^{2}\right)^{2}} d x\)

\(\frac{d y}{d x}=(x+1)(x+2) ; y=-\frac{3}{2}\) when \(x=-3\)

Suppose that a particular company estimates its growth in income from sales by the formula \(D_{t} S=2(t-1)^{2 / 3}\), where \(S\) millions of dollars is the gross income from sales \(t\) years hence. If the gross income from the current year's sales is \(\$ 8\) million, what should be the expected gross income from sales two years from now?

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

\(\frac{d x}{y}=\frac{4 d y}{x} ; y=-2\) when \(x=4\)

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

\(\frac{d y}{d x}=\frac{x}{4 \sqrt{\left(1+x^{2}\right)^{3}}} ; y=0\) when \(x=1\)

What constant acceleration (negative) will enable a driver to decrease his speed from \(60 \mathrm{mi} / \mathrm{hr}\) to \(20 \mathrm{mi} / \mathrm{hr}\) while traveling a distance of \(300 \mathrm{ft}\) ?

\(\sqrt{37.5}\)

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

\(\frac{d^{2} y}{d x^{2}}=4(1+3 x)^{2} ; y=-1\) and \(y^{\prime}=-2\) when \(x=-1\)

\(\sqrt[3]{7.5}\)

\(\int x^{4} \sqrt{3 x^{5}-5} d x\)

\(\frac{d^{2} y}{d x^{2}}=\sqrt[3]{3 x-1} ; y=2\) and \(y^{\prime}=5\) when \(x=3\)

A ball started upward from the bottom of an inclined plane with an initial velocity of \(6 \mathrm{ft} / \mathrm{sec}\). If there is a downward acceleration of \(4 \mathrm{ft} / \mathrm{sec}^{2}\), how far up the plane will the ball go before rolling down?

\(\sqrt{82}\)

\(3 x^{3}-x^{2} y+2 x y^{2}-y^{3}-3 x^{2}+y^{2}=1\)

\(\int x^{2}\left(4-x^{2}\right)^{3} d x\)

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.

If the brakes on a car can give the car a constant negative acceleration of \(20 \mathrm{ft} / \mathrm{sec}^{2}\), what is the greatest speed it may be going if it is necessary to be able to stop the car within \(80 \mathrm{ft}\) after the brake is applied?

\(\sqrt{82}\)

\(\int\left(x^{2}-4 x+4\right)^{4 / 3} d x\)

The slope of the tangent line at any point \((x, y)\) on a curve is \(3 \sqrt{x}\). If the point \((9,4)\) is on the curve, find an equation of the curve.

\(\sqrt[3]{0.00098}\)

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