Problem 83 - Calculus Volume 3

Question

For the curve \(x=4 t, y=3 t-2,\) find the slope and concavity of the curve at \(t=3\).

Step-by-Step Solution

Answer

The slope at \( t = 3 \) is \( \frac{3}{4} \), and the curve is linear (no concavity).

Step 1: Find the Derivative with Respect to t

To find the slope, we need to determine \( \frac{dy}{dx} \). Begin by finding \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):\[ \frac{dx}{dt} = \frac{d}{dt}(4t) = 4 \] \[ \frac{dy}{dt} = \frac{d}{dt}(3t - 2) = 3 \] The slope \( \frac{dy}{dx} \) is then \( \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3}{4} \).

Step 2: Evaluate the Slope at t=3

Since the slope \( \frac{dy}{dx} = \frac{3}{4} \) is constant, it remains \( \frac{3}{4} \) at any value of \( t \), including \( t = 3 \).

Step 3: Determine the Second Derivative

The second derivative \( \frac{d^2y}{dx^2} \) determines concavity. Start with \( \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{3}{4}\right) = 0 \) since \( \frac{3}{4} \) is a constant. Then, \( \frac{d^2y}{dx^2} = \frac{d\left(\frac{dy}{dx}\right)/dt}{dx/dt} = \frac{0}{4} = 0 \).

Step 4: Interpret the Results

A second derivative of 0 means that the curve has no concavity, i.e., it is linear at \( t = 3 \) and indeed everywhere, as the second derivative is 0 for any \( t \).

Key Concepts

Slope of a Curve

The slope of a curve defined by parametric equations like \(x = 4t\) and \(y = 3t - 2\) can be thought of as the rate at which the curve rises or falls. For a parametric curve, this involves finding the ratio of derivatives of y with respect to t and x with respect to t.

In simple terms, the slope \(\frac{dy}{dx}\) tells us how much y changes for a small change in x. To calculate it, we conduct the following steps:

• Compute \(\frac{dx}{dt}\), the derivative of x in terms of t. For \(x = 4t\), \(\frac{dx}{dt} = 4\).
• Compute \(\frac{dy}{dt}\), the derivative of y in terms of t. For \(y = 3t - 2\), \(\frac{dy}{dt} = 3\).
• The slope \(\frac{dy}{dx}\) is then calculated as \(\frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3}{4}\).

The value \(\frac{3}{4}\) indicates a steady positive slope wherever t is along the curve, demonstrating that the line is neither very steep nor very flat.

At \(t=3\), the slope remains \(\frac{3}{4}\), since the ratio doesn't change.

Concavity

Concavity, in relation to curves, describes whether a curve opens upwards or downwards and indicates the curvature's direction. In this context, we're looking for the second derivative \(\frac{d^2y}{dx^2}\). This second derivative tells us about the acceleration of the curve, or how the slope itself is changing.

In parametric equations, finding the second derivative involves:

• First calculating the derivative of our slope, \(\frac{d}{dt}(\frac{dy}{dx})\). Since \(\frac{dy}{dx} = \frac{3}{4}\) remains constant, \(\frac{d}{dt}(\frac{dy}{dx}) = 0\).
• Then, dividing by \(\frac{dx}{dt}\), so \(\frac{d^2y}{dx^2} = \frac{0}{4} = 0\).

A second derivative of 0 signifies no curvature, meaning the line is straight, portraying neither concavity upward nor downward.

This understanding of concavity will help you recognize the nature of different curves: a positive second derivative means upward concavity, while a negative one means downward.

Differentiation

Differentiation is a crucial technique in calculus, used to find the rate of change of functions. When dealing with parametric equations, it assists in finding important characteristics of the curve they describe, such as slope and concavity.

With parametric functions like \(x = 4t\) and \(y = 3t - 2\), differentiation follows a few ordered steps:

• Differentiate both x and y with respect to t to find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).
• Use these derivatives to find \(\frac{dy}{dx}\), the slope of the curve, by taking the ratio \(\frac{dy/dt}{dx/dt}\).
• To explore further details of the curve, differentiate \(\frac{dy}{dx}\) with respect to t to find \(\frac{d}{dt}(\frac{dy}{dx})\).
• Finally, derive the second derivative \(\frac{d^2y}{dx^2}\) to explore the curve's concavity.

Differentiation not only uncovers the geometry of the curve but also provides insight into its dynamic properties, such as how steep or curved the path might be, aiding in problem-solving across different contexts.