Problem 29 - Calculus for Biology and Medicine

Question

Find the equation of the tangent line to the curve \(y=\sqrt{x}\) at the point \((4,2)\).

Step-by-Step Solution

Answer

The equation of the tangent is \(y = \frac{1}{4}x + 1\).

Step 1: Determine the Derivative

To find the equation of the tangent line, we need the slope of the tangent at the given point. This is found using the derivative. The function is given as \(y = \sqrt{x}\), which is also \(y = x^{1/2}\). The derivative using the power rule is \(y' = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}\).

Step 2: Evaluate the Derivative at the Point

We need the slope of the tangent line at \(x = 4\). Substitute \(x = 4\) into the derivative: \(y' = \frac{1}{2\sqrt{4}} = \frac{1}{2 \times 2} = \frac{1}{4}\). So the slope of the tangent line at \((4, 2)\) is \(\frac{1}{4}\).

Step 3: Write the Equation of the Tangent Line

We can use the point-slope form of the equation for a line, which is \(y - y_1 = m(x - x_1)\). Here, \(m = \frac{1}{4}\), \(x_1 = 4\), and \(y_1 = 2\). Substitute these into the equation: \(y - 2 = \frac{1}{4}(x - 4)\).

Step 4: Simplify the Equation

Now, simplify the equation: \(y - 2 = \frac{1}{4}x - 1\). Adding 2 to both sides gives the equation of the tangent line: \(y = \frac{1}{4}x + 1\).

Key Concepts

Understanding Derivatives

A derivative represents the rate at which a function changes at any given point, essentially capturing the idea of a slope in the context of calculus. When you're looking at a curve, the derivative at a specific point tells you the slope of the tangent line to the curve at that point. In simpler terms, if you draw a tangent line that just touches the curve at that point, the gradient or steepness of that line is the derivative. For the function given in the exercise, which is expressed as \( y = \sqrt{x} \), we can rewrite it using exponents as \( y = x^{1/2} \) for ease in differentiation. Calculating the derivative gives us a function that we can use to find slopes at any given point on the original curve.

Power Rule for Derivatives

The power rule makes finding derivatives much simpler, especially when dealing with functions like polynomials and radicals. According to the power rule, if you have a function \( y = x^n \), its derivative is \( y' = nx^{n-1} \). This rule helps in breaking down the process of differentiation into quick and easy steps. In our example, with the function \( y = x^{1/2} \), applying the power rule gives us a derivative of \( y' = \frac{1}{2}x^{-1/2} \). This is a straightforward process that can be used for various functions, simplifying what might otherwise be a more complicated process.

Using the Point-Slope Form

The point-slope form of a line equation is a fantastic tool to quickly find the equation of a line when you know the slope and a point on the line. It's expressed as \( y - y_1 = m(x - x_1) \), where \( m \) stands for the slope, and \( (x_1, y_1) \) is a point on the line. This formula comes in handy when you've calculated the slope, as we did by finding the derivative at a point. With the slope \( m = \frac{1}{4} \) and point \( (4, 2) \), plugging these into the formula gives us \( y - 2 = \frac{1}{4}(x - 4) \). This step bridges our understanding of the curve's behavior at a point to the tangible equation of its tangent line.

Slope of the Tangent Line

The slope of a tangent line at a particular point on a curve provides insight into the curve's behavior right at that point. It's essential for understanding how the curve is moving—whether it's ascending, descending, or remaining constant. Calculating the slope at a point means evaluating the derivative at that point's x-value. In our case, substituting \( x = 4 \) into the derivative \( y' = \frac{1}{2\sqrt{x}} \) allows us to find that the slope at the point \( (4, 2) \) is \( \frac{1}{4} \). This means that at the point \( (4, 2) \), the line just gently rises, indicating a shallow incline there on the graph of \( y = \sqrt{x} \).