Question

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. $$ y=\frac{1}{x^{2}} $$

Short answer

The graph of the function \(y = \frac{1}{x^{2}}\) does not have any points where the tangent line is horizontal.

Step-by-step solution

Compute the Derivative

The derivative of \(y = \frac{1}{x^{2}}\) can be determined using the power rule for differentiation, which states: If \(y = x^n\) (where \(n ≠ 0\)), then \(y' = n*x^{(n-1)}\). In this case, rewrite \(\frac{1}{x^{2}}\) as \(x^{-2}\), so the derivative \(y' = -2*x^{(-2 - 1)} = -2*x^{-3} = -\frac{2}{x^{3}}\).

Set the Derivative Equal to Zero

In order to find the points where the tangent line is horizontal, set the derivative equal to zero and solve for \(x\). Solving \(-\frac{2}{x^{3}} = 0\) for \(x\) is not possible because for any real number \(x\), \(-\frac{2}{x^{3}}\) never equals 0.

Conclusion

Since there is no real solution to the equation \(-\frac{2}{x^{3}} = 0\), the graph of the function \(y = \frac{1}{x^{2}}\) does not have any points where the tangent line is horizontal.

Key concepts

Derivative

Understanding the concept of a derivative is essential in calculus. The derivative measures how a function's output value changes as its input value changes. In simpler terms, it tells us the rate at which one quantity changes with respect to another. For example, in physics, the derivative of position with respect to time represents velocity, indicating how position changes over time.

Mathematically, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. The slope of the tangent line represents the instantaneous rate of change of the function. If you were to draw a tangent line at any point on a curve, the slope of that line gives you the derivative value at that specific point. Derivatives are foundational to solving problems involving rates of change in various scientific and engineering fields.

Power Rule for Differentiation

The power rule for differentiation is a fundamental rule for finding the derivative of functions that can be written as a power of x. The rule states that if you have a function of the form \(y = x^n\), where \(n\) is any real number, the derivative of \(y\) with respect to \(x\) is \(y' = n \cdot x^{(n-1)}\). This rule simplifies the differentiation process significantly, especially for polynomial functions.

For instance, if you are given \(y = x^4\), using the power rule, the derivative would be \(y' = 4 \cdot x^{(4-1)} = 4x^3\). This works for any exponent, including negative and fractional powers. Remember, the power rule is a quick and powerful tool to differentiate a wide range of functions, as long as they can be expressed as a power of x.

Tangent Line in Calculus

The tangent line to a curve at a given point is a straight line that just 'touches' the curve at that point. It represents the direction in which the curve is heading at the point of tangency. More formally, it's the line with the slope equal to the derivative of the function at that particular point.

When a function has a horizontal tangent line at a certain point, it signifies that the derivative at that point is zero since the slope of a horizontal line is zero. This can be an indication of a local maximum or minimum if the function's slope changes from positive to negative (or vice versa) at that point, or a point of inflection if the slope does not change. Calculating horizontal tangents is a common task and involves finding where the derivative of the function equals zero. However, as demonstrated with the exercise, if the derivative cannot equal zero, the function has no horizontal tangent lines.