Tangent Line: F(x) = X² + 2x At (1, 3) Explained!

Introduction

Hey guys! Today, we're diving into a classic calculus problem: finding the equation of the tangent line to a curve at a specific point. We'll be working with the function f(x) = x² + 2x and the point (1, 3). This is a fundamental concept in calculus, and mastering it will help you understand rates of change, derivatives, and a whole lot more. So, let's break it down step by step!

At its core, finding the tangent line involves using the derivative of a function. Think of the derivative as the function that gives you the slope of the original function at any given point. The tangent line is the straight line that "just touches" the curve at that specific point, sharing the same slope as the curve at that location. The point-slope form of a line, which is y - y₁ = m(x - x₁), is going to be our best friend here. We already have a point (1, 3), so we just need to find the slope, m. That's where the derivative comes in! Remember, the derivative will give us the slope of the tangent line at that exact point. There are different ways to find the derivative. We can use the power rule, which is a quick and efficient method for polynomial functions like ours. We might also talk about using the limit definition of the derivative, which is a more fundamental approach, but for this problem, the power rule will get us there faster. Once we have the slope, we plug it into the point-slope form along with our point (1, 3), and then simplify to get the equation of the tangent line in slope-intercept form (y = mx + b), which is often the easiest way to visualize and work with a line. So, buckle up, because we're about to embark on a calculus adventure!

1. Finding the Derivative

Okay, first things first: we need to find the derivative of our function, f(x) = x² + 2x. Remember, the derivative will give us the slope of the tangent line at any point on the curve. To find the derivative, we're going to use the power rule. The power rule is a handy shortcut that states if you have a term of the form axⁿ, its derivative is naxⁿ⁻¹. Basically, you multiply by the exponent and then reduce the exponent by 1. Let's apply this to our function.

Our function, f(x) = x² + 2x, has two terms: x² and 2x. Let's tackle them one at a time. For the first term, x², we have a = 1 (since it's 1*x²) and n = 2. Applying the power rule, we get 2 * 1 * x²⁻¹ = 2x¹. So, the derivative of x² is 2x. Now, let's move on to the second term, 2x. We can think of this as 2x¹, so a = 2 and n = 1. Applying the power rule again, we get 1 * 2 * x¹⁻¹ = 2x⁰. Remember anything raised to the power of 0 is 1, so this simplifies to 2 * 1 = 2. Therefore, the derivative of 2x is 2. Now, we just add the derivatives of the individual terms together. So, the derivative of f(x) = x² + 2x, which we often denote as f'(x), is f'(x) = 2x + 2. This is a crucial step! We've now got a new function, f'(x), that tells us the slope of the original function at any x-value. Think of it as a slope-finding machine! We're not quite done yet; we need to find the specific slope at the point (1, 3), but we've made excellent progress.

2. Calculating the Slope at (1, 3)

Great job, guys! We've found the derivative, f'(x) = 2x + 2. Now, we need to figure out the slope of the tangent line at the specific point (1, 3). Remember, the x-coordinate of our point is 1. To find the slope at this point, we simply plug x = 1 into our derivative function, f'(x). So, we're calculating f'(1).

Let's do the substitution: f'(1) = 2 * (1) + 2. This simplifies to f'(1) = 2 + 2, which equals 4. So, the slope of the tangent line at the point (1, 3) is 4. This is a super important piece of information! We now know the steepness of our tangent line at the point where it touches the curve. Think of this slope as the direction the tangent line is heading at that specific instant. A positive slope like 4 means the line is going upwards as we move from left to right. A larger slope value means it's going upwards more steeply. We're one step closer to finding the full equation of the tangent line. We have the slope (m = 4) and a point (1, 3). Now, we just need to put it all together using the point-slope form of a line!

3. Finding the Equation of the Tangent Line

Alright, we're on the home stretch! We've got the slope of the tangent line, m = 4, and the point where it touches the curve, (1, 3). Now, we're going to use the point-slope form of a line to find the equation of the tangent line. Remember, the point-slope form is: y - y₁ = m(x - x₁), where (x₁, y₁) is our point and m is the slope.

Let's plug in our values. Our point (1, 3) means x₁ = 1 and y₁ = 3. And we know our slope, m, is 4. Substituting these values into the point-slope form, we get: y - 3 = 4(x - 1). Now, we just need to simplify this equation to get it into a more familiar form, like slope-intercept form (y = mx + b). First, let's distribute the 4 on the right side of the equation: y - 3 = 4x - 4. Next, we want to isolate y, so let's add 3 to both sides of the equation: y = 4x - 4 + 3. Finally, simplify the constants: y = 4x - 1. And there you have it! The equation of the tangent line to f(x) = x² + 2x at the point (1, 3) is y = 4x - 1. This equation tells us everything we need to know about the tangent line. We know it has a slope of 4 (which we already calculated), and we know its y-intercept is -1. If you were to graph this line and the original function, you'd see the line just kissing the curve at the point (1, 3).

Conclusion

Awesome work, everyone! We've successfully found the equation of the tangent line to f(x) = x² + 2x at the point (1, 3). We went through all the crucial steps: finding the derivative using the power rule, calculating the slope at the specific point, and then using the point-slope form to derive the equation of the tangent line. This process is a cornerstone of calculus, and understanding it opens the door to more advanced concepts.

Finding tangent lines is super useful in many real-world applications. Think about it: it allows us to approximate the behavior of a function near a specific point. For example, in physics, it can help us determine instantaneous velocity. In economics, it can help us analyze marginal cost or revenue. The possibilities are endless! The key takeaway here is understanding the relationship between a function, its derivative, and the tangent line. The derivative gives us the slope, which is the key to finding the tangent line's equation. So, practice these steps, and you'll become a tangent line master in no time! Remember, calculus is all about understanding change, and tangent lines are a powerful tool for analyzing that change. Keep exploring, keep practicing, and keep having fun with math!