Problem 64 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The point with a horizontal tangent is (0, 2).

Step 1: Understand the Problem

We need to find the points on the curve of the function \( f(x) = 2 - x^2 \) where the tangent to the curve is horizontal. This means the slope of the tangent is zero.

Step 2: Find the Derivative

To find where the tangent is horizontal, we first need to calculate the derivative of the function, \( f'(x) \), which represents the slope of the tangent line at any point \( x \). For \( f(x) = 2 - x^2 \), use the power rule to differentiate.\[ f'(x) = \frac{d}{dx}(2-x^2) = 0 - 2x = -2x \]

Step 3: Set the Derivative Equal to Zero

Since a horizontal tangent has a slope of zero, we set the derivative \( f'(x) \) equal to zero and solve for \( x \).\[ -2x = 0 \]

Step 4: Solve for x

Solve the equation \( -2x = 0 \) to find the x-coordinate where the tangent is horizontal.\[ x = 0 \]

Step 5: Find the y-coordinate

To find the y-coordinate, substitute \( x = 0 \) back into the original function \( f(x) \).\[ f(0) = 2 - (0)^2 = 2 \]

Step 6: State the Coordinates

The point on the graph where the tangent is horizontal is \((0, 2)\).

Key Concepts

Derivatives

In calculus, a derivative represents a fundamental concept which allows us to determine the rate at which a function changes. Finding the derivative, denoted as \( f'(x) \), of a given function \( f(x) \) helps identify how the value of \( f \) changes as \( x \) changes. This is crucial in understanding the behavior of functions.
Derivatives are important because they give us the slope of the tangent line to the function at any given point. This slope is essentially the instantaneous rate of change — showing how fast or slow something occurs.

• To find the derivative of a simple polynomial function, like \( f(x) = 2 - x^2 \), we use different rules of differentiation including the power rule.
• Differentiation transforms the original function into a new function, providing insights on the original function's rate of change.

Having a firm grasp on derivatives allows one to solve complex problems related to rates of change, motion, growth, and decay. It's a vital tool in real-world applications across science and engineering.

Tangent lines

A tangent line is a straight line that just "touches" a curve at a particular point, without crossing over. It represents the instantaneous direction of the curve at that specific point.
In the context of calculus, tangent lines are crucial for understanding the slope or rate of change of the curve at a given point.

• For the function \( f(x) = 2 - x^2 \), the tangent line at any point \( x \) is calculated using the derivative \( f'(x) \).
• The tangent line at the point \((x_0, f(x_0))\) can be given by the equation \( y = f'(x_0) \cdot (x - x_0) + f(x_0) \).

Tangent lines tell us a lot about the local behavior of the graph. They show if the graph is increasing, decreasing, or constant at specific points and provide linear approximations of the function.

Power rule

The power rule is a basic technique used to find the derivative of functions of the form \( x^n \), where \( n \) is any real number. It provides a quick and effective way to compute derivatives, especially useful for polynomial functions.
The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) = n \cdot x^{n-1} \).

• In our exercise, the function \( f(x) = 2 - x^2 \) incorporates the power rule for \( x^2 \), resulting in the simple derivative \( f'(x) = -2x \).
• Using the power rule helps in swiftly solving derivative-related problems, simplifying the differentiation process.

The power rule is fundamental in calculus, simplifying differentiation of polynomial functions and forming the foundation for more complex rules and methods.

Slope of a tangent

The slope of a tangent line is a key concept in calculus, representing how steep a line is at the point where it touches the curve. It is obtained from the derivative of the function at that specific point.
For the function \( f(x) = 2 - x^2 \), finding \( f'(x) \) gives us the slope of the tangent line at any \( x \).

• A horizontal tangent line has a slope of zero, indicating no increase or decrease at the point where the tangent touches the curve.
• In the exercise, we found \( f'(x) = -2x \), and setting this equal to zero helps locate where the slope is zero—hence identifying the horizontal tangent.

Understanding the slope of a tangent line provides insights into the function's rate of change and is necessary for analyzing and graphing the behavior of functions.