Difference Quotient: F(a), F(a+h) For F(x) = 7x^2 + 3

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Hey guys! Today, we're diving into a classic calculus concept: the difference quotient. This might sound intimidating, but trust me, it's super useful, especially when you start exploring derivatives. We're going to break down how to find f(a), f(a+h), and the difference quotient (f(a+h) - f(a))/h for the function f(x) = 7x² + 3. So, buckle up, and let's get started!

What is the Difference Quotient?

Before we jump into the specifics, let's quickly define what the difference quotient actually is. The difference quotient is a fundamental concept in calculus that represents the average rate of change of a function over a small interval. Mathematically, it's expressed as:

(f(a + h) - f(a)) / h

Where:

  • f(x) is the function we're working with.
  • a is a specific point in the function's domain.
  • h is a small change in x (a small interval).

Essentially, the difference quotient is the slope of the secant line through the points (a, f(a)) and (a + h, f(a + h)) on the graph of f(x). As h approaches zero, this secant line approaches the tangent line, and the difference quotient approaches the derivative of the function at x = a. This connection to derivatives is why understanding the difference quotient is so crucial.

Why is the Difference Quotient Important?

The difference quotient serves as the foundation for understanding derivatives, a cornerstone of calculus. Derivatives, in turn, are used to find instantaneous rates of change, which have wide-ranging applications across various fields. Think about finding the velocity of an object at a specific moment, optimizing business processes, or even modeling population growth – derivatives are at the heart of these calculations.

By mastering the difference quotient, you're not just learning a formula; you're gaining a stepping stone to more advanced calculus concepts and their real-world applications. It allows you to analyze how functions change and opens doors to solving complex problems in science, engineering, economics, and more. So, spending the time to understand it thoroughly is an investment in your mathematical and analytical skills.

Step-by-Step: Finding f(a), f(a+h), and the Difference Quotient

Okay, let's get practical. We'll walk through the steps to find f(a), f(a + h), and the difference quotient for our function, f(x) = 7x² + 3. We'll take it one piece at a time, so it's super clear.

Step 1: Find f(a)

Finding f(a) is the first easy step. All we need to do is substitute x with a in our function:

f(x) = 7x² + 3

So, replacing x with a, we get:

f(a) = 7a² + 3

That's it! We've found f(a). This is the value of the function at the point x = a. It's like finding the y-coordinate on the graph of the function when the x-coordinate is a. Keep this expression handy, as we'll need it for the next steps.

Step 2: Find f(a + h)

Next, we need to find f(a + h). This means we'll substitute x with (a + h) in our original function:

f(x) = 7x² + 3

Replacing x with (a + h), we get:

f(a + h) = 7(a + h)² + 3

Now, we need to expand and simplify this expression. Remember to use the correct order of operations (PEMDAS/BODMAS). First, we'll expand the square:

(a + h)² = (a + h)(a + h) = a² + 2ah + h²

Now, substitute this back into our expression:

f(a + h) = 7(a² + 2ah + h²) + 3

Distribute the 7:

f(a + h) = 7a² + 14ah + 7h² + 3

Great! We've found f(a + h). This represents the value of the function at the point x = a + h. This expression might look a bit long, but it's a crucial part of calculating the difference quotient.

Step 3: Calculate the Difference Quotient (f(a + h) - f(a)) / h

Now for the main event: calculating the difference quotient. Remember the formula:

(f(a + h) - f(a)) / h

We've already found f(a + h) and f(a), so let's plug those in:

[(7a² + 14ah + 7h² + 3) - (7a² + 3)] / h

Notice the parentheses are important here! They ensure we subtract the entire expression for f(a).

Now, let's simplify. Distribute the negative sign in the numerator:

(7a² + 14ah + 7h² + 3 - 7a² - 3) / h

Now, we can cancel out some terms. Notice that the 7a² and -7a² cancel out, and the +3 and -3 cancel out as well:

(14ah + 7h²) / h

We're almost there! Now, we can factor out an h from the numerator:

h(14a + 7h) / h

Finally, we can cancel out the h in the numerator and denominator (remember, h ≠ 0):

14a + 7h

Boom! We've found the difference quotient for f(x) = 7x² + 3. The difference quotient is 14a + 7h. This simplified expression is super important because it allows us to easily see how the average rate of change of the function changes as h varies.

Putting it All Together: An Example

Let's solidify our understanding with an example. Suppose we want to find the difference quotient at a = 2. We'll use the result we just derived:

Difference quotient = 14a + 7h

Plug in a = 2:

Difference quotient = 14(2) + 7h = 28 + 7h

So, the difference quotient at a = 2 is 28 + 7h. This means that the average rate of change of the function f(x) = 7x² + 3 near the point x = 2 is approximately 28 + 7h. The smaller h is (i.e., the closer we get to the point x = 2), the closer this value gets to the instantaneous rate of change (the derivative) at that point.

Common Mistakes to Avoid

When calculating the difference quotient, there are a few common pitfalls that students often stumble into. Let's highlight these so you can avoid them!

  • Incorrectly Expanding (a + h)²: This is a classic mistake. Remember that (a + h)² = (a + h)(a + h) = a² + 2ah + h², not a² + h². Don't forget the middle term, 2ah! Failing to expand this correctly will throw off your entire calculation.
  • Forgetting Parentheses: When substituting f(a) into the difference quotient formula, remember to put it in parentheses: [(f(a + h) - (f(a))] / h. This ensures you distribute the negative sign correctly. Forgetting the parentheses can lead to incorrect signs and a wrong final answer.
  • Not Simplifying Completely: The goal is to simplify the difference quotient as much as possible. This usually involves canceling out terms and factoring out an h from the numerator. Make sure you've done all the simplification steps to get the most concise answer.
  • Dividing by Zero: Remember that h ≠ 0 in the difference quotient. If you end up with a denominator of zero, you've likely made a mistake somewhere in your calculations. Go back and carefully check each step.

By being aware of these common errors, you can increase your accuracy and confidence when working with the difference quotient.

The Difference Quotient and the Derivative

Here's where things get really cool. The difference quotient is intimately connected to the derivative of a function. In fact, the derivative is defined as the limit of the difference quotient as h approaches zero:

f'(x) = lim (h->0) [f(x + h) - f(x)] / h

Where f'(x) represents the derivative of f(x).

What this means is that the derivative gives us the instantaneous rate of change of the function at a specific point, while the difference quotient gives us the average rate of change over a small interval. As we make the interval smaller and smaller (i.e., as h approaches zero), the average rate of change gets closer and closer to the instantaneous rate of change.

In our example, we found the difference quotient for f(x) = 7x² + 3 to be 14a + 7h. To find the derivative at x = a, we would take the limit of this expression as h approaches zero:

lim (h->0) (14a + 7h) = 14a

So, the derivative of f(x) = 7x² + 3 is f'(x) = 14x. This connection between the difference quotient and the derivative is a fundamental concept in calculus and is essential for understanding how functions change.

Conclusion

Alright, guys! We've covered a lot in this article. We've explored the difference quotient, how to calculate it for the function f(x) = 7x² + 3, and why it's so important in calculus. We broke down the steps to find f(a), f(a + h), and the difference quotient (f(a + h) - f(a)) / h. We also looked at common mistakes to avoid and how the difference quotient relates to the derivative. I hope you found this breakdown helpful and that you're feeling more confident in tackling these types of problems.

Remember, the difference quotient is a building block for understanding more advanced calculus concepts, so mastering it is a great investment in your mathematical journey. Keep practicing, and you'll be a pro in no time! Now, go forth and conquer those calculus problems!