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Chapter 2: Q38E (page 128)

\(y = a{x^{\rm{3}}}\), \({x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\).

Short Answer

Expert verified

The family of curves \(y = a{x^{\rm{3}}}\) and \({x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\) are orthogonal trajectories of each other and their sketch is,

Step by step solution

01

Given Information

The given family of curves are \(y = a{x^{\rm{3}}}\)and \({x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\).

02

Definition of Derivative

The derivative of a real-valued function measures the sensitivity of the function's value (output value) to changes in its argument in mathematics (input value).

03

Slopes of family of curves

Differentiate and find the slope of curves.

Differentiate the curve \(y = a{x^{\rm{3}}}\) with respect to \(x\).

\(\begin{array}{c}y = a{x^{\rm{3}}}\\{{y'}_{\rm{1}}} = {\rm{3a}}{x^{\rm{2}}}\\{{y'}_{\rm{1}}} = {\rm{3}}\left( {\frac{y}{{{x^{\rm{3}}}}}} \right){x^{\rm{2}}}\\{{y'}_{\rm{1}}} = \frac{{{\rm{3y}}}}{x}\end{array}\)

Differentiate the curve \({x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\) with respect to \(x\).

\(\begin{array}{c}{x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\\{\rm{2}}x + {\rm{6}}y{{y'}_{\rm{2}}} = {\rm{0}}\\{{y'}_{\rm{2}}} = - \frac{x}{{{\rm{3}}y}}\end{array}\)

04

Orthogonality of family of curves

The product of slopes of two orthogonal slopes is always equals to \( - {\rm{1}}\).

Find the product of slopes of the curves.

\(\begin{array}{c}{{y'}_{\rm{1}}}{{y'}_{\rm{2}}} = \left( {\frac{{{\rm{3}}y}}{x}} \right)\left( { - \frac{x}{{{\rm{3}}y}}} \right)\\ = - {\rm{1}}\end{array}\)

So, the two curves are orthogonal to each other.

05

Sketch of curves

Sketch the graph of the given family of curves on the same axes and show their orthogonal trajectories.

Therefore, the family of curves \(y = a{x^{\rm{3}}}\) and \({x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\) are orthogonal trajectories of each other and their sketch is,

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Most popular questions from this chapter

(a) The van der Waals equation for \({\rm{n}}\) moles of a gas is \(\left( {P + \frac{{{n^{\rm{2}}}a}}{{{V^{\rm{2}}}}}} \right)\left( {V - nb} \right) = nRT\) where \(P\)is the pressure,\(V\) is the volume, and\(T\) is the temperature of the gas. The constant\(R\) is the universal gas constant and\(a\)and\(b\)are positive constants that are characteristic of a particular gas. If \(T\) remains constant, use implicit differentiation to find\(\frac{{dV}}{{dP}}\).

(b) Find the rate of change of volume with respect to pressure of \({\rm{1}}\) mole of carbon dioxide at a volume of \(V = {\rm{10}}L\) and a pressure of \(P = {\rm{2}}{\rm{.5atm}}\). Use \({\rm{a}} = {\rm{3}}{\rm{.592}}{{\rm{L}}^{\rm{2}}}{\rm{ - atm}}/{\rm{mol}}{{\rm{e}}^{\rm{2}}}\)and \(b = {\rm{0}}{\rm{.04267}}L/mole\).

43-45 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f?

44. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{{\bf{2}}^x}}&{{\bf{if}}\,\,\,x \le {\bf{1}}}\\{{\bf{3}} - x}&{{\bf{if}}\,\,\,{\bf{1}} < x \le {\bf{4}}}\\{\sqrt x }&{{\bf{if}}\,\,\,x > {\bf{4}}}\end{array}} \right.\)

Show by implicit differentiation that the tangent to the ellipse \(\frac{{{x^{\rm{2}}}}}{{{a^{\rm{2}}}}} + \frac{{{y^{\rm{2}}}}}{{{b^{\rm{2}}}}} = {\rm{1}}\) at the point \(\left( {{x_{\rm{0}}},{y_{\rm{0}}}} \right)\)is \(\frac{{{x_{\rm{0}}}x}}{{{a^{\rm{2}}}}} + \frac{{{y_{\rm{0}}}y}}{{{b^{\rm{2}}}}} = {\rm{1}}\).

19-32 Prove the statement using the \(\varepsilon \), \(\delta \) definition of a limit.

20. \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{5}}} \left( {\frac{{\bf{3}}}{{\bf{2}}}x - \frac{{\bf{1}}}{{\bf{2}}}} \right) = {\bf{7}}\)

For the function f whose graph is given, state the value of each quantity if it exists. If it does not exist, explain why.

(a)\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{1}}} f\left( x \right)\)

(b)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{3}}^ - }} f\left( x \right)\)

(c) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{3}}^ + }} f\left( x \right)\)

(d) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{3}}} f\left( x \right)\)

(e) \(f\left( {\bf{3}} \right)\)
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