Problem 22 - Calculus for Biology and Medicine

Question

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=(1-x)\left(2-x^{2}\right)\), at \(x=2\)

Step-by-Step Solution

Answer

The normal line is \( y = -\frac{1}{6}x + \frac{7}{3} \).

Step 1: Understand the Problem

We need to find the equation of the normal line to the function \( y = f(x) = (1-x)(2-x^2) \) at \( x = 2 \). The normal line is perpendicular to the tangent line at this point. To find it, we must first find the derivative of \( f(x) \) using the product rule and evaluate it at \( x = 2 \) to obtain the slope of the tangent line.

Step 2: Apply the Product Rule

The product rule states that \( (uv)' = u'v + uv' \). Let \( u = 1-x \) and \( v = 2-x^2 \). Thus, \( u' = -1 \) and \( v' = -2x \). Applying the product rule:\[ f'(x) = (-1)(2-x^2) + (1-x)(-2x) \]

Step 3: Simplify the Derivative

Simplify the expression from Step 2:\[ f'(x) = -2 + x^2 - 2x + 2x^2 \] Combine like terms:\[ f'(x) = 3x^2 - 2x - 2 \]

Step 4: Calculate Slope of Tangent Line at x=2

Substitute \( x = 2 \) into \( f'(x) \) to find the slope of the tangent line:\[ f'(2) = 3(2)^2 - 2(2) - 2 = 12 - 4 - 2 = 6 \] Thus, the slope of the tangent line at \( x = 2 \) is 6.

Step 5: Find the Slope of the Normal Line

The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of the tangent slope. Since the slope of the tangent line is 6, the slope of the normal line is:\[ m = -\frac{1}{6} \]

Step 6: Find the y-coordinate at x = 2

Evaluate \( f(x) \) at \( x = 2 \) to find the point on the curve:\[ f(2) = (1-2)(2-2^2) = -1(-2) = 2 \] Thus, the point is \((2, 2)\).

Step 7: Write the Equation of the Normal Line

Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( m = -\frac{1}{6} \), \( x_1 = 2 \), and \( y_1 = 2 \):\[ y - 2 = -\frac{1}{6}(x - 2) \] Simplify to slope-intercept form:\[ y = -\frac{1}{6}x + \frac{1}{3} + 2 \] \[ y = -\frac{1}{6}x + \frac{7}{3} \]

Key Concepts

Normal Line

A normal line is an important concept in calculus that is closely linked to tangent lines. While a tangent line touches a curve at just one point and has a slope equal to the derivative of the function at that point, the normal line is perpendicular to this tangent line at the same point. This makes the normal line's slope the negative reciprocal of the tangent line's slope. For example, if the slope of the tangent line is 6, the slope of the normal line would be \[-\frac{1}{6}\].
The normal line helps to analyze the behavior and properties of curves, often used in physics and engineering to study forces and motions. When you find the equation of a normal line, it's often necessary to first find the equation of the tangent line, since these calculations are interdependent.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to represent the equation of a straight line. It’s expressed as \[y = mx + b\], where \(m\) is the slope of the line and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
We use this form to make it easier to graph the line and understand its steepness and position.

• Slope (\(m\)): Indicates the angle and direction of the line's incline.
• Y-intercept (\(b\)): Shows where the line intersects the vertical y-axis.

In the context of the exercise, after finding the slope of the normal line, you use the slope-intercept form to simplify the equation \[y = -\frac{1}{6}x + \frac{7}{3}\]. This makes the properties of the line clear and helps in further analysis.

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point, without crossing it around that locality. The slope of this line at any given point on a curve is equal to the derivative of the function at that point.
In our case, to find the tangent line of the function \(y=f(x)\) at \(x=2\), we first need to find the derivative of the function using the product rule, \((uv)' = u'v + uv'\). By determining the slope of the tangent line, we establish a benchmark from which the normal line can be derived.
Understanding the tangent line is crucial in calculus as it provides insights into the instantaneous rate of change of the function at a given point, which is a fundamental concept in both mathematics and physics.