Finding F(x) Given F'(x) And F(0) A Calculus Guide

Hey everyone! Today, we're diving into a classic calculus problem: finding a function f(x) when we know its derivative, f'(x), and an initial value, f(0). It's like detective work with functions, and it's super useful in many real-world applications. Let's break it down step by step so you guys can master this skill.

Understanding the Problem

So, what exactly are we trying to do? We're given that the derivative of our mystery function f(x) is f'(x) = 5x² + 9x - 6. Remember, the derivative tells us the slope of the function at any point. We also know that f(0) = 2. This is our initial condition – it tells us the value of the function at x = 0. Our mission, should we choose to accept it (and we do!), is to find the actual function f(x).

Think of it like this: we have the blueprint for the slope of a curve, and we have one specific point that the curve passes through. Can we reconstruct the entire curve? The answer, my friends, is a resounding yes! We'll use the power of antiderivatives, also known as indefinite integrals, to reverse the process of differentiation.

Antiderivatives: The Key to Reversing Differentiation

The antiderivative, in simple terms, is the opposite of a derivative. If we differentiate a function and get f'(x), then the antiderivative of f'(x) will give us back f(x) (almost!). The catch is that when we find an antiderivative, we always have to add a constant of integration, usually denoted as C. This is because the derivative of a constant is always zero, so when we reverse the process, we lose that constant information.

To put it mathematically, if F'(x) = f(x), then the indefinite integral of f(x) is:

∫f(x) dx = F(x) + C

Where:

• ∫ is the integral symbol
• f(x) is the integrand (the function we're integrating)
• F(x) is the antiderivative
• C is the constant of integration

Applying the Power Rule for Integration

To find the antiderivative of our given f'(x) = 5x² + 9x - 6, we'll use the power rule for integration. This rule is a cornerstone of calculus, and it's essential for solving problems like this. The power rule states:

∫xⁿ dx = (xⁿ⁺¹) / (n + 1) + C, where n ≠ -1

In plain English, to integrate a power of x, we increase the exponent by 1 and then divide by the new exponent. And, of course, we always add our constant of integration, C.

Let's see how this works with our problem. We have three terms in f'(x): 5x², 9x, and -6. We'll apply the power rule to each term individually:

• For 5x²: We increase the exponent 2 by 1 to get 3, and then divide by 3. This gives us (5x³) / 3.
• For 9x: Remember that x is the same as x¹. We increase the exponent 1 by 1 to get 2, and then divide by 2. This gives us (9x²) / 2.
• For -6: This is the same as -6x⁰. We increase the exponent 0 by 1 to get 1, and then divide by 1. This gives us -6x, which is just -6x.

Don't forget our constant of integration, C! Now we can assemble the antiderivative:

∫(5x² + 9x - 6) dx = (5x³) / 3 + (9x²) / 2 - 6x + C

The General Solution: A Family of Functions

So, we've found the antiderivative, but what does this C really mean? It means that we haven't found a single function, but rather a family of functions. Each different value of C gives us a different function that has the same derivative, f'(x). They're all vertical shifts of each other.

Think of it like this: imagine a set of parallel curves. They all have the same shape (the same derivative), but they're positioned at different heights on the graph (different values of C). The constant of integration is what allows us to move these curves up and down.

That's why we call this the general solution. It's a general form of the function, but it doesn't pinpoint the exact function we're looking for. To do that, we need more information – and that's where our initial condition comes in!

Using the Initial Condition to Find the Particular Solution

Remember that we were given the initial condition f(0) = 2. This tells us that when x = 0, the function f(x) has a value of 2. This is the key to finding the specific value of C that gives us the particular solution we're after.

Our general solution is:

f(x) = (5x³) / 3 + (9x²) / 2 - 6x + C

We can substitute x = 0 and f(0) = 2 into this equation:

2 = (5(0)³)/3 + (9(0)²)/2 - 6(0) + C

This simplifies to:

2 = 0 + 0 - 0 + C

So, C = 2! Now we know the specific value of the constant of integration that satisfies our initial condition.

The Particular Solution: Our Unique Function

Now that we've found C, we can plug it back into our general solution to get the particular solution. This is the one and only function that satisfies both the derivative condition f'(x) = 5x² + 9x - 6 and the initial condition f(0) = 2.

Substituting C = 2 into our general solution, we get:

f(x) = (5x³) / 3 + (9x²) / 2 - 6x + 2

This is our answer! We've successfully found the function f(x).

Putting It All Together: A Step-by-Step Recap

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

1. Understand the problem: We were given the derivative f'(x) and an initial condition f(0), and we needed to find the function f(x).
2. Find the antiderivative: We used the power rule for integration to find the general antiderivative of f'(x), remembering to add the constant of integration, C.
3. Use the initial condition: We substituted the values from the initial condition into the general solution to solve for C.
4. Find the particular solution: We plugged the value of C back into the general solution to get the specific function f(x).

Why This Matters: Real-World Applications

Finding a function from its derivative might seem like a purely mathematical exercise, but it has tons of real-world applications. Here are just a few examples:

• Physics: If you know the acceleration of an object (the derivative of its velocity), you can find its velocity and position functions by integration. This is crucial for understanding motion, projectile trajectories, and much more.
• Engineering: Engineers use integration to calculate things like the stress and strain on a structure, the flow of fluids, and the accumulation of heat.
• Economics: Economists use integration to model things like the growth of a population, the accumulation of capital, and the consumer surplus.
• Biology: Biologists use integration to model the growth of populations, the spread of diseases, and the decay of radioactive substances.

Practice Makes Perfect: Try These Problems!

Now that you've seen how to solve this type of problem, it's time to put your skills to the test! Here are a few practice problems you can try:

1. Find f(x) such that f'(x) = 3x² - 4x + 1 and f(1) = 5.
2. Find g(x) such that g'(x) = 2cos(x) + sin(x) and g(0) = 1.
3. Find h(x) such that h'(x) = eˣ - 2x and h(0) = 0.

Work through these problems, and you'll become a master of finding functions from their derivatives. Remember, the key is to understand the power rule for integration, the concept of the constant of integration, and how to use initial conditions to find the particular solution.

Conclusion: You've Got This!

So, there you have it! We've successfully navigated the world of finding functions from their derivatives and initial values. It's a fundamental concept in calculus, and it's a powerful tool for solving real-world problems. Keep practicing, and you'll be amazed at what you can accomplish. You guys are awesome, and I know you can nail this! Happy integrating!