Problem 63 - Calculus for Biology and Medicine

Question

In Problems 63-70, find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=x^{2} $$

Step-by-Step Solution

Answer

The point with a horizontal tangent is (0, 0).

Step 1: Understand the Problem

We need to find points on the graph of \( y = f(x) = x^2 \) where the tangent line is horizontal. A horizontal tangent occurs when the derivative \( f'(x) \) is equal to zero.

Step 2: Differentiate the Function

Find the derivative of the function \( f(x) = x^2 \). Use the power rule for differentiation, which gives \( f'(x) = 2x \).

Step 3: Set the Derivative Equal to Zero

To find where the tangent is horizontal, set the derivative equal to zero: \( 2x = 0 \).

Step 4: Solve for \( x \)

Solve the equation \( 2x = 0 \) to find \( x \). Divide both sides by 2, which gives \( x = 0 \).

Step 5: Find the Corresponding \( y \)-Coordinate

Substitute \( x = 0 \) back into the original function to find the \( y \)-coordinate. Therefore, \( f(0) = 0^2 = 0 \).

Step 6: State the Point with Horizontal Tangent

Combine the \( x \) and \( y \) coordinate to express the point. The point on the graph with a horizontal tangent is \( (0, 0) \).

Key Concepts

Horizontal Tangents

In calculus, a horizontal tangent is a fascinating topic for students to explore. Imagine you're riding a rollercoaster. There might be a moment when you are at the top of a rise, and for a brief period, the path is flat. Similarly, in mathematics, this is where the graph of a function is momentarily not increasing or decreasing. This flat spot is due to what's called a horizontal tangent.

Horizontal tangents occur where the slope of the tangent line to the graph is zero. The slope is described by the derivative of the function. When the derivative is set to zero, we find these horizontal tangent points.

• It's a handy tool for identifying points of local maxima, minima, or saddle points.
• Useful in optimization problems and in analyzing the shape of graphs.

Understanding where and why horizontal tangents exist can help you predict the behavior of functions better.

Derivative

The concept of a derivative in calculus is extremely crucial. A derivative is essentially a tool that measures how a function changes as its input changes. You can think of it as the function's rate of change or the slope of the function at a given point.

Derivatives allow us to:

• Find slopes of tangent lines to curves.
• Analyze increasing or decreasing functions.
• Determine the concavity of functions.

The derivative is denoted by either \( f'(x) \) or \( \frac{dy}{dx} \), representing the rate of change of the function \( y \) with respect to \( x \). In our exercise, to find the horizontal tangent of \( f(x) = x^2 \), the derivative \( f'(x) = 2x \) pointed us to where the slope (2x) is zero.

Power Rule for Differentiation

The power rule for differentiation is one of the simplest and most commonly used rules in calculus. It makes finding derivatives straightforward for power functions. If you have a function that can be expressed in the form \( x^n \), where \( n \) is any real number, the power rule states:

\[ \frac{d}{dx}x^n = nx^{n-1} \]

This means you multiply by the exponent and then decrease it by one.

• For example, the derivative of \( f(x) = x^2 \) is obtained by applying the power rule, resulting in \( f'(x) = 2x^{2-1} = 2x \).

Applying the power rule is quick and helps when dealing with polynomial functions, effectively simplifying the differentiation process.

Graph of a Function

Graphs of functions offer a visual representation of equations and are fundamental in understanding their behavior. They provide insights into continuity, peaks, valleys, and symmetries.

A graph's features include:

• Intercepts: Points where the graph crosses the axes.
• Slopes: How steep or flat the graph is at exactly any point, which can be found using derivatives.
• Curvature: Indicates whether the graph is curving upwards or downwards.

For the function \( f(x) = x^2 \), graphing helps illustrate a parabolic upward shape, symmetric about the y-axis. At \( x = 0 \), where we found the horizontal tangent, the graph smoothly flattens before curving back up.