Problem 82 - Essential Calculus

Question

Use the parametric equations \(x=t^{2} \sqrt{3} \quad\) and \(\quad y=3 t-\frac{1}{3} t^{3}\) to answer the following. (a) Use a graphing utility to graph the curve on the interval \(-3 \leq t \leq 3\) (b) Find \(d y / d x\) and \(d^{2} y / d x^{2}\). (c) Find the equation of the tangent line at the point \(\left(\sqrt{3}, \frac{8}{3}\right)\). (d) Find the length of the curve. (e) Find the surface area generated by revolving the curve about the \(x\) -axis.

Step-by-Step Solution

Answer

The tangent is \(y = 8/3\), the lengths and surface area calculation will require numerical integrations, which should be calculated using a calculator.

Step 1: Find \(dy/dx\)

We know that \(dy/dx = (dy/dt) / (dx/dt)\). First find \(dx/dt\)= \(2t \sqrt{3}\), \(dy/dt = 3 - t^2\), so \(dy/dx = (3 - t^2) / (2t \sqrt{3})\).

Step 2: Find \(d^{2} y / d x^{2} \)

To find \(d^{2} y / d x^{2}\), we differentiate \(dy/dx\) with respect to \(t\) and then divide it by \(dx/dt\). The derivative of \(dy/dx\) with respect to \(t\) gives \(-2t / (2t \sqrt{3}) - (6t - 2t^2) / (4t^2 \sqrt{3})\). After dividing by \(dx/dt\) we get \(d^{2} y / d x^{2} = (-1 / \sqrt{3} - 3/T) / (2 \sqrt{3})\).

Step 3: Tangent line at point

The equation of a tangent line is given by \(y - y1 = m(x - x1)\), where \(m\) is the slope of the tangent at that point and \(x1, y1\) are coordinates of the given point. For the given point \((\sqrt{3}, \frac{8}{3})\), \(m\) for this will be the value of \(dy/dx\) at \(t = 1\). After substituting \(t = 1\) in \(dy/dx\), we get \(m = 0\), so the equation of the tangent line is \(y = 8/3\).

Step 4: Length of the curve

The length of the section of a curve from \(a\) to \(b\) given the parametric equations is given by the formula \(\int_{a}^{b} \sqrt{1+ (dy/dx)^2} dt\). After substituting the \(dy/dx\) in the formula and simplifying, the integral gives us the length of the curve feature.

Step 5: Surface area

The surface area of a volume rotated about the x-axis is given by \(\int_{a}^{b} 2\pi y \sqrt{1+ (dy/dx)^2} dt \). Substitute \(dy/dx\) in the equation and 'y' from the parametric equation to find the surface area. Luna currently doesn't support the integration of this equation, we would highly advice you to use a calculator for the integration.