91Ó°ÊÓ

To grasp the concept of a tangent line, it's essential to understand derivatives first. A derivative tells us the rate at which a function changes at any given point. Imagine you're trying to find the slope of a curve—kind of like how steep a hill is. The derivative provides this information. It's a powerful tool for mathematicians and scientists because it gives the exact slope of the tangent line at any point on the curve.

In the context of our exercise, we started with the function \( y = 2 \sqrt{x} \). By using calculus, specifically the power rule, we rewrote this as \( y = 2x^{1/2} \) to compute the derivative: \( \frac{dy}{dx} = \frac{1}{\sqrt{x}} \). This resulting derivative formula helps us find the slope of the tangent line whenever we plug in a specific \( x \)-value. In this exercise, by calculating the derivative at \( x = 1 \), we found a slope (or steepness) of 1.

Once we have the slope of the tangent line from the derivative, we can utilize the point-slope form to find the equation of the tangent line itself. The point-slope form is a linear equation formula given by \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope.

For our specific problem, we've already determined that the slope \( m \) is 1, and our given point on the curve is \( (1,2) \). Plugging these into the point-slope form gives us: \( y - 2 = 1(x - 1) \). By simplifying, we arrive at the equation \( y = x + 1 \). This is the equation for the tangent line that touches the curve at the point \( (1,2) \). The point-slope form is a convenient way to translate a slope and a specific point into the familiar structure of a linear equation.

Curve sketching is the art of drawing a graph to represent a function visually. This process helps us understand how the function behaves at different points—that's where the magic of mathematics meets art! For the given function \( y = 2 \sqrt{x} \), we recognize it as a curve that opens upwards starting from the origin.

To sketch this curve, we can plot key points:

• At \( x = 0 \), \( y = 0 \)
• At \( x = 1 \), \( y = 2 \)
• At \( x = 4 \), \( y = 4 \)

These points help us see the shape of the curve.

In our exercise, we also found the tangent line's equation \( y = x + 1 \). It’s a straight line intersecting the curve at the point \( (1,2) \). When both the curve and the line are drawn on the same graph, you can see the point where they touch briefly before parting ways. This visual representation reaffirms the calculations we’ve performed, providing a comprehensive understanding of how the tangent line interacts with the curve.