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Chapter 7: Problem 67

Find the length of the curve \(y^{2}=x^{3}\) from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the \(x\) -axis.

Short Answer

Expert verified
The length of the curve from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the x-axis can be found by using the arc length formula and evaluating the integral over the interval from \(x = 0\) to \(x = 3\). The final answer will require evaluating this integral.

Step by step solution

01

Finding Derivative

The equation \(y^{2} = x^{3}\) can be written as \(y = \sqrt[3]{x^{2}}\). Differentiating both sides with respect to \(x\) using the power rule for differentiation gives us: \(\frac{dy}{dx} = \frac{2x}{3\sqrt[3]{x^{4}}}\).
02

Finding the Tangent

We know that the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. In terms of the derivatives, this means that the tangent of the angle \(\theta\) that the derivative makes with the x-axis at any point is equal to \(\frac{dy}{dx}\). Given this, we set \(\frac{dy}{dx} = \tan(45^{\circ}) = 1\), and solving the equation \(\frac{2x}{3\sqrt[3]{x^{4}}} = 1\), we find that \(x = 3\). Substituting \(x = 3\) back into the equation \(y^{2} = x^{3}\), we get \(y = 3\). So the point where the tangent makes an angle of \(45^{\circ}\) with the x-axis is \((3,3)\).
03

Arc Length Formula

The formula to find the arc length from a point \((a,f(a))\) to \((b,f(b))\) is \(\int_{a}^{b} \sqrt{1+(\frac{dy}{dx})^{2}} dx\). In our case, \(a = 0\) and \(b = 3\). So, substituting the values into the formula, we get: \(\int_{0}^{3} \sqrt{1+(\frac{2x}{3\sqrt[3]{x^{4}}})^{2}} dx\).
04

Solving the Integral

Finally, evaluating this integral (which may require techniques such as substitution or integration by parts) gives us the length of the curve.

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