Question

Finding a Tangent Line Find the equation of the tangent line to the curve defined by the equations $$ x(t)=t^{2}-3, \quad y(t)=2 t-1, \quad-3 \leq t \leq 4 \text { when } t=2 $$

Short answer

The equation of the tangent line is \(y = \frac{1}{2}x + \frac{5}{2}\).

Step-by-step solution

Find the point on the curve

To find the point on the curve when \(t = 2\), substitute \(t = 2\) into the parametric equations: \[ x(2) = 2^2 - 3 = 1 \] \[ y(2) = 2(2) - 1 = 3 \] Therefore, the point on the curve is \((1, 3)\).

Calculate the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)

Calculate the derivative of \(x(t)\) with respect to \(t\): \[ \frac{dx}{dt} = \frac{d}{dt}(t^2 - 3) = 2t \]Calculate the derivative of \(y(t)\) with respect to \(t\):\[ \frac{dy}{dt} = \frac{d}{dt}(2t - 1) = 2 \]

Find the slope of the tangent line \(\frac{dy}{dx}\)

To find the slope \(\frac{dy}{dx}\), use the relationship:\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]Substitute the derivatives from Step 2:\[ \frac{dy}{dx} = \frac{2}{2t} \]When \(t = 2\), the slope \(m\) becomes:\[ m = \frac{2}{2(2)} = \frac{2}{4} = \frac{1}{2} \]

Write the equation of the tangent line

The equation of a line is given by the formula \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line and \(m\) is the slope.Using the point \((1, 3)\) and slope \(\frac{1}{2}\) from previous steps:\[ y - 3 = \frac{1}{2}(x - 1) \]Simplify to find:\[ y = \frac{1}{2}x + \frac{5}{2} \]

Key concepts

Parametric Equations

Parametric equations allow us to describe a curve by expressing coordinates as functions of a parameter, usually denoted as \(t\). This method is particularly useful when dealing with curves that are not functions in the traditional sense (where every \(x\) maps to exactly one \(y\)).
For instance, the parametric equations \(x(t) = t^2 - 3\) and \(y(t) = 2t - 1\) describe a curve in the plane. Here, each value of \(t\) results in a corresponding position on the curve, specifically giving the \(x\) and \(y\) coordinates.

• As \(t\) varies within the range \(-3 \leq t \leq 4\), the entire path of the curve is traced out.
• These equations offer flexibility, allowing us to explore parts of the curve by selecting specific \(t\) values.

In this problem, we're interested in the curve's behavior specifically when \(t=2\). By substituting \(t=2\) into the parametric equations, we obtain a particular point on the curve. Understanding this pair of equations is fundamental in transitioning from a parametric to a Cartesian framework.

Derivatives

Derivatives in the context of parametric equations are crucial for analyzing how each coordinate changes with respect to the parameter \(t\). They help us to understand the dynamics of the curve.
For a parametric curve defined by \(x(t)\) and \(y(t)\), we calculate the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\):

• \(\frac{dx}{dt} = \frac{d}{dt}(t^2 - 3) = 2t\)
• \(\frac{dy}{dt} = \frac{d}{dt}(2t - 1) = 2\)

These derivatives reveal how each of the coordinates changes as \(t\) changes. Such rates of change are foundational to understanding the slope of the curve at any point. Calculating these derivatives for a specific \(t\) helps derive the slope of the tangent to the curve.

Slope of a Curve

The slope of a curve at a specific point is a measure of its steepness and can be found using the derivatives obtained from parametric equations. When dealing with parametric equations, the slope of the tangent line to the curve is determined using the formula:
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]This formula highlights that the slope of the tangent line depends on the ratio of the rate of change of \(y\) to the rate of change of \(x\).

• For this curve, substituting \(t=2\) into the derivatives, we find \(\frac{dy}{dx} = \frac{2}{2 \cdot 2} = \frac{1}{2}\).
• This slope \(\frac{1}{2}\) indicates that for every 2 units moved horizontally, the curve rises by 1 unit vertically at the point \((1, 3)\).

The slope is instrumental in calculating the full equation of the tangent line, thereby giving us a linear approximation of the curve at the given point.

Equation of a Line

The equation of a line provides a complete mathematical description of the line's behavior, namely its slope and its passage through a specific point. When you have a point \((x_1, y_1)\) and a known slope \(m\), the equation of the line can be written as:
\[ y - y_1 = m(x - x_1) \]
Substituting the point \((1, 3)\) and the slope \(\frac{1}{2}\) obtained earlier, the equation of the tangent line becomes:
\[ y - 3 = \frac{1}{2}(x - 1) \]

• Simplifying this equation yields: \(y = \frac{1}{2}x + \frac{5}{2}\).
• This is the tangent to the curve at the point where \(t = 2\).

Understanding how to form this equation not only confirms the slope's utility but also provides insight into how the line intersects the curve at a specific point.