Finding F'(x) And F(3) With Given Conditions A Calculus Solution

Hey guys! Let's dive into this calculus problem where we need to find the first derivative, f'(x), and the value of the function at a specific point, f(3). We're given the second derivative, f''(x), and some initial conditions. It might sound a bit intimidating, but we'll break it down step by step and see how integration and these initial conditions help us solve it. So, grab your thinking caps, and let's get started!

Determining f'(x)

Okay, so the first key step here is to find f'(x). We know that f''(x) = 6x + 1. Remember, the derivative is just the rate of change, and integration is the reverse process of differentiation. So, to get f'(x) from f''(x), we need to integrate! This means we need to find the antiderivative of 6x + 1. Integration is like unwrapping a present – we're trying to figure out what function, when differentiated, would give us 6x + 1. Think of it as going backward on the derivative process.

The power rule of integration is our friend here. It tells us that the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. This C is super important because when we differentiate a constant, it disappears, so when we integrate, we need to remember there might have been a constant term there originally. It's like finding a missing piece of the puzzle!

Let's apply this to our problem. The integral of 6x is 6 * (x^(1+1))/(1+1) = 3x^2. The integral of 1 (which can be thought of as x^0) is x^(0+1)/(0+1) = x. So, integrating 6x + 1 gives us 3x^2 + x + C. Therefore, we have f'(x) = 3x^2 + x + C. We're getting closer, but we still need to figure out what that C is. This is where the initial condition comes in handy!

We're given that f'(-1) = -3. This is a crucial piece of information. It tells us that when x is -1, the slope of the function (that's what the derivative represents) is -3. We can plug these values into our expression for f'(x) to solve for C. It’s like having a secret code that unlocks the value of C!

So, substituting x = -1 into f'(x) = 3x^2 + x + C gives us -3 = 3(-1)^2 + (-1) + C. Simplifying this, we get -3 = 3 - 1 + C, which means -3 = 2 + C. Solving for C, we find that C = -5. Now we've found our missing constant! It’s like finding the last piece of a jigsaw puzzle, and the picture is finally complete!

Therefore, the first derivative is f'(x) = 3x^2 + x - 5. We've nailed the first part of the problem! We now have an expression for the slope of the original function at any point x. That’s pretty powerful stuff!

Calculating f(x)

Now that we've found f'(x), the next step is to find f(x). Guess what? We need to integrate again! This time, we'll be integrating f'(x) = 3x^2 + x - 5 to get back to the original function, f(x). It's like reversing the derivative process a second time, peeling back another layer to reveal the original function. We’re basically doing the same process as before, but this time on a different function.

Again, we'll use the power rule of integration. Let's integrate term by term. The integral of 3x^2 is 3 * (x^(2+1))/(2+1) = x^3. The integral of x is (x^(1+1))/(1+1) = (1/2)x^2. The integral of -5 (which is -5x^0) is -5 * (x^(0+1))/(0+1) = -5x. So, integrating 3x^2 + x - 5 gives us x^3 + (1/2)x^2 - 5x + D. Notice that we're using a different constant, D, because it might not be the same as the C we found earlier. It's important to keep track of these constants, as they can affect the final answer.

So, we have f(x) = x^3 + (1/2)x^2 - 5x + D. Just like before, we need to find the value of this constant D. And how do we do that? With the initial condition we were given! This is where f(-1) = 2 comes into play. This tells us that when x is -1, the value of the function is 2. We can plug these values into our expression for f(x) to solve for D. It's like using a second secret code to unlock the final constant!

Substituting x = -1 into f(x) = x^3 + (1/2)x^2 - 5x + D gives us 2 = (-1)^3 + (1/2)(-1)^2 - 5(-1) + D. Simplifying, we get 2 = -1 + (1/2) + 5 + D, which means 2 = (9/2) + D. Solving for D, we find that D = 2 - (9/2) = -5/2. Fantastic! We've found our constant D. It’s the last piece of the puzzle needed to fully describe our original function.

Therefore, the original function is f(x) = x^3 + (1/2)x^2 - 5x - 5/2. We now have a complete expression for f(x). We started with the second derivative and, through the power of integration and initial conditions, we've found the original function. That’s a pretty cool journey, if you ask me!

Evaluating f(3)

Alright, we're in the home stretch now! The final part of the problem asks us to find f(3). This means we need to evaluate our function f(x) at x = 3. It's like plugging in a specific value to see what the function does at that point. We’ve built the whole machine, and now we get to see it in action!

We have f(x) = x^3 + (1/2)x^2 - 5x - 5/2. To find f(3), we simply substitute x = 3 into this expression. So, f(3) = (3)^3 + (1/2)(3)^2 - 5(3) - 5/2. Now, we just need to do the arithmetic.

First, (3)^3 = 27. Then, (1/2)(3)^2 = (1/2)(9) = 9/2. Next, 5(3) = 15. And we already have -5/2. So, f(3) = 27 + 9/2 - 15 - 5/2. Let's combine the whole numbers and the fractions separately.

We have 27 - 15 = 12. And 9/2 - 5/2 = 4/2 = 2. So, f(3) = 12 + 2 = 14. We’ve done it! The value of the function at x = 3 is 14. We’ve successfully navigated through derivatives, integrals, and constants to arrive at our final answer!

Summary

So, to recap, we started with the second derivative f''(x) = 6x + 1 and the initial conditions f'(-1) = -3 and f(-1) = 2. Through integration and using the initial conditions, we found the first derivative to be f'(x) = 3x^2 + x - 5 and the original function to be f(x) = x^3 + (1/2)x^2 - 5x - 5/2. Finally, we evaluated f(3) and found it to be 14. What a journey! This problem beautifully illustrates the relationship between derivatives and integrals, and how initial conditions help us pin down the specific function we're looking for.

Calculus might seem like a mountain to climb, but by breaking it down into smaller steps and understanding the underlying concepts, we can conquer even the trickiest problems. Keep practicing, keep exploring, and you'll be amazed at what you can achieve! And remember, even mathematicians sometimes need to work through the steps carefully to avoid mistakes – it’s all part of the process. Keep up the great work, guys!