Problem 55 - Calculus for Biology and Medicine

Question

Find the tangent line to $$ f(x)=a x^{2} $$ at \(x=1\). Assume that \(a\) is a positive constant.

Step-by-Step Solution

Answer

The tangent line to \( f(x) = ax^2 \) at \( x = 1 \) is \( y = 2ax - a \).

Step 1: Find the function value

Find the value of the function at the point where we want the tangent line, which is at \( x = 1 \). Compute \( f(1) = a(1)^2 = a \). Thus, the point on the curve is \((1, a)\).

Step 2: Compute the derivative

To find the slope of the tangent line, compute the derivative of the function \( f(x) = ax^2 \). The derivative, using the power rule, is \( f'(x) = 2ax \).

Step 3: Evaluate the derivative at x=1

Evaluate the derivative at \( x = 1 \) to find the slope of the tangent line. Thus, \( f'(1) = 2a(1) = 2a \). The slope of the tangent line at \( x = 1 \) is \( 2a \).

Step 4: Write the equation of the tangent line

Use the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point on the line and \( m \) is the slope. Substitute \( x_1 = 1 \), \( y_1 = a \), and \( m = 2a \) into the equation. This gives: \( y - a = 2a(x - 1) \).

Step 5: Simplify the equation of the tangent line

Simplify the tangent line equation to the slope-intercept form, \( y = mx + c \). Start from \( y - a = 2ax - 2a \). Add \( a \) to both sides: \( y = 2ax - a \). The equation of the tangent line is \( y = 2ax - a \).

Key Concepts

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. Think of it as the slope of the tangent line to the graph of the function at that specific point.
This concept is central to understanding how functions behave and change. For example, when dealing with a quadratic function like \( f(x) = ax^2 \), we can find its derivative to understand how the slope of its graph changes for different values of \( x \).

• The derivative gives us the precise slope of the tangent line at a point, which is crucial for drawing and analyzing graphs.
• From a practical perspective, knowing the derivative of a function allows you to calculate instantaneous rates of change, like speed (if time is the independent variable).

For the function \( f(x) = ax^2 \), the derivative \( f'(x) = 2ax \) tells us how steep the graph is at any value of \( x \). It signifies how much \( y \) changes with a small increase in \( x \), allowing us to predict future values of the function based on its current trend.

Power Rule

The power rule is a fundamental technique in calculus for finding derivatives. It simplifies the process of differentiation for power functions, making it a handy tool for calculus students. When you have a function of the form \( f(x) = x^n \), the power rule states that the derivative is \( f'(x) = nx^{n-1} \).
Let's break it down with an example. Consider \( f(x) = ax^2 \). Here, you apply the power rule:

• The exponent \( n \) is 2.
• Multiply the exponent by the coefficient \( a \), giving \( 2a \).
• Reduce the exponent by 1, resulting in \( x^{2-1} = x \).

Putting it all together, \( f'(x) = 2ax \). This makes finding derivatives effortless for power functions, as it only requires straightforward multiplication and subtraction. The power rule saves time and simplifies complicated derivative calculations!

Point-Slope Form

Working with the point-slope form of a line is an easy way to write the equation of a tangent line when given a point and a slope. The point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Where \((x_1, y_1)\) is the point on the line and \( m \) is the slope. This formula is practical for finding the equation of a line using minimal information.
Let’s see how it applies to finding the tangent line for the function \( f(x) = ax^2 \) at \( x = 1 \).

• We know the point on the curve is \((1, a)\).
• The slope, found from the derivative, is \( 2a \).

Substituting these values into the equation yields \( y - a = 2a(x - 1) \). From there, you can simplify it to find other forms, like the slope-intercept form, which can be beneficial for graphing or further analysis. This method is straightforward and provides a clear structure for solving tangent line problems.