Tangent Line Slope Of -1 For F(x) = 1/x: Find The X-Values

by Andrew McMorgan 59 views

Hey math enthusiasts! Today, we're diving into a cool calculus problem that involves finding the tangent line slope of a function. Specifically, we'll be looking at the function f(x) = 1/x and figuring out at which x-values the slope of the tangent line is equal to -1. This is a classic problem that combines the concepts of derivatives and slopes, so let's break it down step by step.

Understanding the Problem

Before we jump into the math, let's make sure we understand what the question is asking. We're given the function f(x) = 1/x. This function represents a hyperbola, and if you were to graph it, you'd see a curve in the first and third quadrants. The tangent line at any point on this curve is a straight line that touches the curve at that point and has the same slope as the curve at that instant. The slope of a line, as you might remember, tells us how steep the line is. A slope of -1 means the line is decreasing as you move from left to right, and it's doing so at a 45-degree angle.

So, our mission is to find the x-values where the tangent line slope to the curve f(x) = 1/x is exactly -1. To do this, we'll need to use the concept of derivatives. The derivative of a function gives us a new function that tells us the slope of the tangent line at any point. Basically, it's a slope-calculating machine for our original function. Once we find the derivative, we'll set it equal to -1 and solve for x. These x-values will be our answer.

Step 1: Find the Derivative of f(x) = 1/x

The first step in solving this problem is to find the derivative of the function f(x) = 1/x. Remember that the derivative, often denoted as f'(x), gives us the slope of the tangent line at any point x on the curve. To find the derivative, we can rewrite f(x) = 1/x as f(x) = x⁻¹. This makes it easier to apply the power rule of differentiation.

The power rule states that if we have a function of the form f(x) = xⁿ, then its derivative is f'(x) = nxⁿ⁻¹. Applying this rule to our function f(x) = x⁻¹, we get:

f'(x) = (-1)x⁻¹⁻¹ = -x⁻² = -1/x²

So, the derivative of f(x) = 1/x is f'(x) = -1/x². This new function, f'(x), is super important because it tells us the tangent line slope at any given x-value on the original function.

Step 2: Set the Derivative Equal to -1

Now that we have the derivative, f'(x) = -1/x², we need to find the x-values where this derivative is equal to -1. Remember, we're looking for the points where the tangent line slope is -1. So, we'll set f'(x) equal to -1 and solve for x:

-1/x² = -1

This equation is the key to finding our answer. It's saying, "Hey, at what x-values does the slope of the tangent line, represented by f'(x), equal -1?" Solving this equation will give us the specific x-values we're looking for.

Step 3: Solve for x

Let's solve the equation -1/x² = -1 for x. To do this, we can follow these steps:

  1. Multiply both sides by x² to get rid of the fraction: (-1/x²) * x² = -1 * x², which simplifies to -1 = -x².
  2. Divide both sides by -1: -1 / -1 = -x² / -1, which simplifies to 1 = x².
  3. Take the square root of both sides: √1 = √(x²). Remember, when we take the square root, we need to consider both the positive and negative roots.
  4. This gives us two possible solutions: x = 1 and x = -1.

So, we've found two x-values where the tangent line slope of f(x) = 1/x is -1. These values are x = 1 and x = -1.

Step 4: Verify the Solutions (Optional)

To be absolutely sure we've got the right answer, it's always a good idea to verify our solutions. We can do this by plugging the x-values we found back into the derivative, f'(x) = -1/x², and making sure we get -1.

  • For x = 1: f'(1) = -1/(1)² = -1/1 = -1. This checks out!
  • For x = -1: f'(-1) = -1/(-1)² = -1/1 = -1. This also checks out!

Both x = 1 and x = -1 give us a tangent line slope of -1, so we're confident in our answer.

The Answer

Alright, guys, we've cracked the code! The x-values at which the slope of the tangent line to the function f(x) = 1/x is equal to -1 are x = 1 and x = -1. This means that at the points (1, 1) and (-1, -1) on the graph of f(x) = 1/x, the tangent lines will have a slope of -1.

Visualizing the Solution

It's often helpful to visualize what's going on in a calculus problem. If you were to graph the function f(x) = 1/x and draw tangent lines at the points x = 1 and x = -1, you'd see that these tangent lines indeed have a slope of -1. They slope downwards from left to right, and for every one unit you move to the right, you move one unit down. This visual confirmation can help solidify your understanding of the problem.

Why This Matters

This type of problem is a fundamental concept in calculus. Understanding how to find tangent line slopes is crucial for many applications, including optimization problems (finding maximums and minimums), related rates problems (how things change in relation to each other), and understanding the behavior of functions in general. The ability to work with derivatives is a powerful tool in mathematics, physics, engineering, and many other fields.

Key Takeaways

Let's recap the key steps we took to solve this problem:

  1. Understanding the problem: We clarified what the question was asking and made sure we understood the concepts of tangent lines and slopes.
  2. Finding the derivative: We used the power rule to find the derivative of f(x) = 1/x, which gave us the function f'(x) = -1/x².
  3. Setting the derivative equal to -1: We set f'(x) equal to -1 to find the x-values where the tangent line slope is -1.
  4. Solving for x: We solved the equation -1/x² = -1 and found two solutions: x = 1 and x = -1.
  5. Verifying the solutions: We plugged our solutions back into the derivative to make sure they were correct.
  6. Visualizing the solution: We discussed how graphing the function and tangent lines can help understanding.

Practice Makes Perfect

To really master this concept, it's essential to practice similar problems. Try finding the tangent line slopes for other functions, or try finding the points where the tangent line has a different slope. The more you practice, the more comfortable you'll become with derivatives and tangent lines.

Conclusion

So, there you have it! We've successfully found the x-values where the tangent line slope of f(x) = 1/x is equal to -1. This problem highlights the power of calculus and how derivatives can be used to analyze the behavior of functions. Keep exploring, keep practicing, and keep pushing your mathematical boundaries. You've got this!

If you guys found this helpful, be sure to check out more math problems and explanations. Happy calculating!