Problem 27 - Thomas Calculus

Question

Moving along a parabola A particle moves along the parabola \(y=x^{2}\) in the first quadrant in such a way that its \(x\) -coordinate (measured in meters) increases at a steady 10 \(\mathrm{m} /\) sec. How fast is the angle of inclination \(\theta\) of the line joining the particle to the origin changing when \(x=3 \mathrm{m} ?\)

Step-by-Step Solution

Answer

The angle \(\theta\) changes at \(\frac{1}{3}\) radians per second when \(x=3\).

Step 1: Define the problem and given variables

We need to find the rate of change of the angle \(\theta\) at which a line from the origin to a point \((x, y)\) on the parabola makes with the positive \(x\)-axis, as \(x\) increases at a rate of 10 m/s. The angle \(\theta\) is expressed as \(\theta = \arctan\left(\frac{y}{x}\right)\). We will calculate \(\frac{d\theta}{dt}\) when \(x = 3\).

Step 2: Express y in terms of x

Since the particle is on the parabola \(y = x^2\), \(y = x^2\). This is important because we will use this relationship to find the rate of change of \(y\).

Step 3: Differentiate the angle \(\theta\)

Given \(\theta = \arctan\left(\frac{y}{x}\right)\), use the chain rule to differentiate with respect to \(t\): \[\frac{d\theta}{dt} = \frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{d}{dt}\left(\frac{y}{x}\right)\]. Simplify the first term to get \[\frac{x^2}{x^2+y^2}\].

Step 4: Differentiate y and apply quotient rule

Differentiate \(y = x^2\) to get \(\frac{dy}{dt} = 2x \frac{dx}{dt}\).For \(\frac{d}{dt}\left(\frac{y}{x}\right)\): Use the quotient rule: \(\frac{d}{dt}\left(\frac{y}{x}\right) = \frac{x \frac{dy}{dt} - y \frac{dx}{dt}}{x^2}\) Substitute \(y = x^2\) and \(\frac{dy}{dt} = 2x \frac{dx}{dt}\):\[\frac{d}{dt}\left(\frac{y}{x}\right) = \frac{x(2x \frac{dx}{dt}) - x^2 \frac{dx}{dt}}{x^2} = \frac{x(2x)\frac{dx}{dt} - x^2 \frac{dx}{dt}}{x^2} = \frac{x \frac{dx}{dt}}{x^2} = \frac{\frac{dx}{dt}}{x}\].

Step 5: Calculate \(\frac{d\theta}{dt}\) when \(x = 3\)

Replace \(\frac{dy}{dt}\) and the simplified expression for \(\frac{d}{dt}\left(\frac{y}{x}\right)\) in the derivative formula:\(\frac{d\theta}{dt} = \frac{x^2}{x^2+y^2} \cdot \frac{\frac{dx}{dt}}{x}\).Set \(x = 3\), \(\frac{dx}{dt} = 10\), and \(y = 9\). Substitute these into the formula:\[\frac{d\theta}{dt} = \frac{3^2}{3^2+9^2} \cdot \frac{10}{3} = \frac{9}{9+81} \cdot \frac{10}{3} = \frac{9}{90} \cdot \frac{10}{3} = \frac{1}{10} \cdot \frac{10}{3} = \frac{1}{3} \text{ radians per second}.\]

Step 6: Conclusion

The rate at which the angle \(\theta\) is changing, at \(x = 3\), is \(\frac{1}{3}\) radians per second.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, and it involves finding the rate at which a function is changing at any given point. It can be thought of as a measure of how a quantity varies with another. In our exercise, differentiation helps us determine how quickly the inclination angle \(\theta\), formed by a line from the origin to the point on the parabola, changes as the particle moves along the curve.

Key points to remember about differentiation:

• It provides a way to calculate instantaneous rates of change, such as velocity or acceleration.
• It is essential for finding derivatives, which are functions that give these rates at any point.
• The derivative of a function \(f(x)\) with respect to \(x\) is denoted by \(f'(x)\) or \(\frac{df}{dx}\).

In this problem, we are differentiating the angle \(\theta\) with respect to time \(t\) to find how fast the angle changes. This involves applying the chain rule on the function \(\theta = \arctan\left(\frac{y}{x}\right)\). Differentiation thus becomes a tool to compute the inter-relationship between varying quantities as the particle moves along the path in the first quadrant of the parabola.

Chain Rule

The Chain Rule is a crucial technique in differentiation, especially when dealing with composite functions. It allows us to differentiate functions made up of multiple layers, or nested functions, with respect to a single variable.

In simple terms, the Chain Rule states that if you have a composite function, say \(y = f(g(x))\), then the derivative of \(y\) with respect to \(x\) is \(f'(g(x)) \, g'(x)\). Think of it like a conveyor belt—each stage must be differentiated in sequence.

• Consider a function, \(\theta = \arctan\left(\frac{y}{x}\right)\), where both \(y\) and \(x\) change with time.
• To differentiate this equation, you first find the derivative of the outer function \(\arctan(z)\), which is \(\frac{1}{1+z^2}\), and apply it to the inner function \(z = \frac{y}{x}\).
• Next, you differentiate the inner function \(\frac{y}{x}\), leveraging the quotient rule to find \(\frac{d}{dt}\left(\frac{y}{x}\right)\).

In the case of our problem involving movement along a parabola, the Chain Rule helps combine the changes in both \(x\) and \(y\) to determine how they collectively impact \(\theta\) as the particle follows its path.

Parabola

A parabola is a symmetrical, curved shape that is a fundamental form in algebra and calculus. Its mathematical description is known as a quadratic function, typically expressed as \(y = ax^2 + bx + c\). Parabolas are distinctive for their U-shaped curve and have important properties in physics, engineering, and various fields of mathematics.

• The vertex is the highest or lowest point on the parabola, depending on its orientation.
• The parabola is symmetric with respect to its axis, which is a vertical line passing through the vertex.
• In this exercise, the equation \(y = x^2\) describes a simple parabola opening upwards, positioned in the first quadrant.

Our focus is on the path traced by a particle moving along the parabola \(y = x^2\). As the particle moves, it increases its \(x\)-coordinate at a steady rate of 10 m/s, which affects the \(y\)-coordinate.
Describing the movement and position on the parabola is essential since these coordinates dictate how the line from the origin to the moving point affects the angle \(\theta\). Therefore, understanding the characteristics of a parabola becomes crucial in solving related rate problems involving curves and their geometric properties.