Problem 60 - Calculus for Biology and Medicine

Question

Find the normal line to $$ f(x)=a x^{2}-3 a x $$ at \(x=2 .\) Assume that \(a\) is a positive constant.

Step-by-Step Solution

Answer

The normal line equation is \( y = -\frac{1}{a}x - 2a + \frac{2}{a} \).

Step 1: Find the Derivative

The first step is to find the derivative of the function \( f(x) = ax^2 - 3ax \). To do this, apply the power rule. The derivative, \( f'(x) \), will be \( 2ax - 3a \).

Step 2: Evaluate the Derivative at x=2

Now, substitute \( x = 2 \) into the derivative to find the slope of the tangent line at this point. \( f'(2) = 2a(2) - 3a = 4a - 3a = a \). This means the slope of the tangent line is \( a \).

Step 3: Calculate the Coordinates of the Point

Calculate \( f(2) \) by substituting \( x = 2 \) into the function: \( f(2) = a(2)^2 - 3a(2) = 4a - 6a = -2a \). The coordinates of the point on the curve are \( (2, -2a) \).

Step 4: Determine the Slope of the Normal Line

The slope of the normal line is the negative reciprocal of the tangent slope. Therefore, if the slope of the tangent is \( a \), the slope of the normal line is \( -1/a \) (since normal line slope \( m_n = -1/m_t \)).

Step 5: Write the Equation of the Normal Line

Using the point-slope form of a line equation, we can write the equation of the normal line: \[ y - (-2a) = \frac{-1}{a}(x - 2) \]. Simplify the equation to: \[ y + 2a = \frac{-1}{a}x + \frac{2}{a} \].

Step 6: Simplify the Normal Line Equation

Bring \( 2a \) to the right side to isolate \( y \): \[ y = \frac{-1}{a}x + \frac{2}{a} - 2a \]. Combine the constant terms: \[ y = \frac{-1}{a}x + \frac{2}{a} - \frac{2a^2}{a} \]. This simplifies to: \[ y = \frac{-1}{a}x - 2a + \frac{2}{a} \].

Key Concepts

Derivative

In calculus, the derivative of a function represents the rate at which the function's value changes with respect to its input. In simpler terms, it's a measure of how a function's output value shifts as the input value changes. Finding the derivative is essentially calculating the slope of the function at any given point. In our exercise, the function is given as \( f(x) = ax^2 - 3ax \). By calculating the derivative, \( f'(x) = 2ax - 3a \), we determine how steep the function is for any value of \( x \). To find the particular slope at \( x = 2 \), substitute this value into the derivative equation, giving \( f'(2) = a \). This result represents the slope of the tangent line at that specific point, which is crucial for later steps.

Tangent Line

The tangent line is a straight line that touches a curve at one and only one point. The slope of the tangent line is equal to the derivative value at that point. This means that its slope indicates the direction in which the function is heading. For example, in our exercise, the function \( f(x) = ax^2 - 3ax \) has a tangent line at \( x = 2 \) with the slope \( a \). This tangent line helps us understand the behavior of the function's curve near the point \( (2, -2a) \) and is used as a reference to find the normal line."

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know a point on the line and the slope. The general formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) are the coordinates of the point. In our situation, after finding the slope of the normal line \( m_n = \frac{-1}{a} \) and knowing the point \( (2, -2a) \), we can plug these into the formula. This gives the equation of the normal line as \( y + 2a = \frac{-1}{a}(x - 2) \). This form is particularly useful for line equations that don't necessarily pass through the origin.

Power Rule

The power rule is a fundamental technique used to find the derivative of functions that contain powers of a variable. If you have a function of the form \( x^n \), the power rule states that its derivative is \( nx^{n-1} \). This rule is instrumental in simplifying the process of differentiation. In our exercise, using the power rule, we differentiated \( ax^2 \) to get \( 2ax \), and \( 3ax \) to get \( 3a \). These derivatives are then combined to get the overall derivative of the function, \( f'(x) = 2ax - 3a \). Remembering and applying the power rule allows us to quickly determine how functions behave and change.