Tangent Line Equation: A Step-by-Step Guide

Let's dive into finding the equation of a tangent line to a curve. This is a classic problem in calculus, and we'll break it down step by step so it's easy to follow. We're given a point P(4, -1) that lies on a curve C defined by y = f(x), where x > 0. The derivative of the function, f'(x), is given by (1/2)x - (6/√x) + 3. Our mission is to find the equation of the tangent line to the curve C at the point P, expressing our answer in the form y = mx + c, where m is the slope and c is the y-intercept.

Understanding the Problem

Before we jump into the calculations, let's make sure we understand what the problem is asking. A tangent line is a straight line that touches the curve at a single point. The slope of this tangent line at that point is given by the derivative of the function at that point. So, our first task is to find the slope m by evaluating f'(x) at x = 4.

Keywords like tangent line equation, derivative, and slope are crucial for understanding this problem. The point (4, -1) gives us the x and y coordinates where the tangent line touches the curve. The equation y = f(x) represents the curve itself, and f'(x) gives us the formula to calculate the slope of the curve at any point x. The form y = mx + c is the standard slope-intercept form of a line, which is what we want to find.

Understanding these concepts is key to tackling similar problems in calculus and analytical geometry.

Step 1: Calculate the Slope (m)

The slope of the tangent line at point P is given by the value of the derivative f'(x) at x = 4. Let's plug in x = 4 into the expression for f'(x):

f'(4) = (1/2)(4) - (6/√4) + 3

Now, simplify the expression:

f'(4) = 2 - (6/2) + 3 f'(4) = 2 - 3 + 3 f'(4) = 2

So, the slope m of the tangent line at point P is 2. This means that for every one unit we move to the right along the tangent line, we move two units up.

This calculation is fundamental to finding the tangent line equation. We are using the derivative, f'(x), as a slope-finding tool. By substituting the x-coordinate of the point of tangency into the derivative, we obtain the exact slope of the curve at that specific point. This slope becomes the m value in our tangent line equation y = mx + c.

Understanding the concept of derivatives allows us to analyze the behavior of functions. In this case, the derivative tells us how the function f(x) is changing at the point x = 4. This information is crucial in various applications, such as optimization problems, curve sketching, and understanding rates of change.

Step 2: Find the y-intercept (c)

Now that we have the slope m = 2, we can plug the coordinates of point P(4, -1) and the slope into the equation y = mx + c to solve for c:

-1 = (2)(4) + c -1 = 8 + c c = -1 - 8 c = -9

So, the y-intercept c of the tangent line is -9. This is the point where the tangent line crosses the y-axis.

The y-intercept, c, is the point where the line intersects the y-axis (when x = 0). Finding the y-intercept is essential for fully defining the tangent line equation. By substituting the known values of x, y, and m into the equation y = mx + c, we isolate c and solve for its value.

This step highlights the importance of understanding the relationship between the slope, y-intercept, and a point on a line. Knowing any two of these three elements allows us to determine the equation of the line. The slope-intercept form (y = mx + c) provides a clear and concise way to represent linear equations, making it easy to visualize and analyze the line's characteristics.

Step 3: Write the Equation of the Tangent Line

Now that we have both the slope m = 2 and the y-intercept c = -9, we can write the equation of the tangent line in the form y = mx + c:

y = 2x - 9

This is the equation of the tangent line to the curve C at the point P(4, -1).

Therefore, the final equation of the tangent line is y = 2x - 9. This equation represents a straight line that touches the curve C at the point (4, -1) and has a slope of 2 and a y-intercept of -9. This equation allows us to calculate the y-coordinate for any given x-coordinate on the tangent line.

Understanding how to find the equation of a tangent line is a fundamental concept in calculus with various applications in fields like physics, engineering, and economics. It provides insights into the behavior of functions and allows us to analyze and model real-world phenomena.

Summary and Conclusion

To summarize, we found the equation of the tangent line to the curve C at the point P(4, -1) by following these steps:

1. Calculate the slope (m): Evaluate the derivative f'(x) at x = 4 to find the slope of the tangent line.
2. Find the y-intercept (c): Plug the coordinates of point P and the slope m into the equation y = mx + c to solve for c.
3. Write the equation of the tangent line: Substitute the values of m and c into the equation y = mx + c.

By following these steps, we determined that the equation of the tangent line is y = 2x - 9.

This problem highlights the power of calculus in analyzing curves and finding tangent lines. The concepts of derivatives and slopes are essential tools for understanding the behavior of functions and solving related problems.

This process can be applied to a variety of curves and points, making it a versatile technique for finding tangent lines. Remember to always start by finding the derivative and evaluating it at the given point to determine the slope.

Calculus is not just about formulas and equations, it’s a powerful tool for understanding change and relationships in the world around us. By mastering these fundamental concepts, you can unlock a deeper understanding of mathematics and its applications.

For more in-depth information on tangent lines and their applications, visit Khan Academy's Calculus Section.