Problem 16 - Precalculus: Mathematics for Calculus

Question

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\frac{1}{x^{2}}, \quad \text { at }(-1,1)$$

Step-by-Step Solution

Answer

The tangent line equation is \( y = 2x + 3 \).

Step 1: Understand the Problem

We need to find the equation of the tangent line to the curve \( y = \frac{1}{x^2} \) at the point \((-1, 1)\). The general equation for a tangent line at a point \((x_0, y_0)\) is \( y - y_0 = m(x - x_0) \), where \( m \) is the slope at \( x_0 \).

Step 2: Find the Derivative

To find the slope of the tangent line, we need to compute the derivative of the function \( y = \frac{1}{x^2} \). Let's denote the function as \( f(x) = x^{-2} \). The derivative \( f'(x) \) is calculated using the power rule: \[ f'(x) = -2x^{-3} = -\frac{2}{x^3}. \]

Step 3: Evaluate the Derivative at the Given Point

Substitute \( x = -1 \) into the derivative to find the slope \( m \) at the point:\[ f'(-1) = -\frac{2}{(-1)^3} = 2. \]Thus, the slope of the tangent line at \((-1, 1)\) is 2.

Step 4: Write the Equation of the Tangent Line

Use the point-slope form of a line \( y - y_0 = m(x - x_0) \) with \( m = 2 \), \( x_0 = -1 \), and \( y_0 = 1 \):\[ y - 1 = 2(x + 1). \]

Step 5: Simplify the Equation

Expand and simplify the equation derived in Step 4:\[ y - 1 = 2x + 2 \] \[ y = 2x + 3. \]This is the equation of the tangent line at the point \((-1, 1)\).

Step 6: Graph the Curve and the Tangent Line

Graph the function \( y = \frac{1}{x^2} \) and the tangent line \( y = 2x + 3 \) on the same set of axes. At point \((-1, 1)\), the line should just touch the curve, representing the tangent line.

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus that help us understand how a function changes at any given point. Specifically, the derivative of a function at a point gives us the slope of the tangent line to the function's curve at that point. This concept is crucial because it allows us to measure how a small change in the input of a function affects its output.

• A derivative is denoted as \( f'(x) \) for some function \( f(x) \).
• It can be thought of as the rate of change or the slope of the function at a specific point.
• Calculating derivatives involves rules, such as the power rule, product rule, and chain rule.

In our exercise, we calculated the derivative of the function \( y = \frac{1}{x^2} \) to determine the slope of the tangent at a specific point. Understanding derivatives helps us predict and analyze behaviors of functions, which is indispensable in fields like physics, engineering, and economics.

Slope of a Curve and Tangent Lines

The slope of a curve at a particular point gives insight into how steeply the curve is rising or falling. Calculating this slope is achieved using derivatives. The tangent line to a curve at a point is a straight line that touches the curve only at that particular point, without crossing it immediately.

• The slope of the tangent line is equal to the derivative of the function at that point.
• If the slope is positive, the function is increasing at that point.
• If it's negative, the function is decreasing at that point.
• A slope of zero indicates a flat or stationary point on the curve.

In our example, the slope of the tangent line at the point \((-1, 1)\) was calculated by finding the derivative of \( y = \frac{1}{x^2} \) and then substituting \( x = -1 \). This told us how the function was changing at that exact point and allowed us to write the equation for the tangent line.

The Power Rule for Derivatives

The power rule is a quick and efficient method for finding the derivative of a function that is in the form of \( x^n \), where \( n \) is any real number. This rule simplifies the differentiation process significantly and is one of the first rules students learn in calculus.
The power rule states that if \( f(x) = x^n \), then its derivative \( f'(x) = nx^{n-1} \). It is especially useful for polynomial functions.

• For example, considering \( f(x) = x^{-2} \): using the power rule, the derivative is \( f'(x) = -2x^{-3} \).
• This rule can be applied to both positive and negative powers of \( x \).
• The power rule simplifies the process of finding slopes for more complex functions.

In our exercise, the power rule was used to calculate the derivative of \( y = x^{-2} \), making it easier to find the slope of the tangent line.

Graph of a Function and Tangent Lines

Graphing a function provides a visual representation that can make understanding complex equations much easier. It allows us to see the behavior of the function, including how it slopes, where it increases or decreases, and where the tangent line might touch the curve.

• A graph can show various characteristics like intercepts, asymptotes, and concavity.
• The tangent line graphically represents the instantaneous direction of a function at a given point.
• It plays a critical role in calculus problems, helping visualize changes in a function.

In the provided exercise, the function \( y = \frac{1}{x^2} \) was graphed alongside its tangent at the point \((-1, 1)\), expressed as the line \( y = 2x + 3 \). This visual tool illustrates how the tangent line only touches the curve at one specific point, providing insight into the concept of derivations and slopes.