Area Function Of F(t) = 7t + 5 Calculation And Graphing

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Hey guys! Today, we're diving deep into the fascinating world of calculus, specifically focusing on area functions. We'll be exploring the function f(t) = 7t + 5 and the real number a = 0 to understand how area functions work and what they can tell us. Buckle up, because this is going to be an exciting journey through integrals, derivatives, and graphs!

Unveiling the Area Function: A(x) = ∫ₐˣ f(t) dt

Let's kick things off by understanding what an area function really is. In essence, the area function, denoted as A(x), calculates the area under the curve of a given function f(t) between a fixed point a and a variable point x. The integral symbol ∫ represents this calculation, and the limits of integration, a and x, define the interval over which we're finding the area. So, in simpler terms, A(x) gives us the cumulative area under the curve of f(t) as x changes, starting from the point a.

Now, let's apply this to our specific function, f(t) = 7t + 5, and the real number a = 0. This means we need to find the area function A(x) = ∫₀ˣ (7t + 5) dt. To do this, we'll use the fundamental theorem of calculus, which tells us that the definite integral of a function can be found by calculating the antiderivative of the function and evaluating it at the limits of integration.

The first step is to find the antiderivative of 7t + 5. Remember, the power rule for integration states that ∫tⁿ dt = (t^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule, we get:

∫(7t + 5) dt = 7∫t dt + 5∫dt = 7(t²/2) + 5t + C.

Now, we can evaluate the definite integral:

A(x) = ∫₀ˣ (7t + 5) dt = [7(x²)/2 + 5x] - [7(0²)/2 + 5(0)] = 7x²/2 + 5x.

So, the area function for f(t) = 7t + 5 with a = 0 is A(x) = 7x²/2 + 5x. This is a quadratic function, meaning its graph will be a parabola. Let's move on to graphing this function to visualize the area it represents.

Graphing the Area Function: Visualizing Cumulative Area

Now that we've found the area function, A(x) = 7x²/2 + 5x, it's time to visualize it. Graphing this function will give us a clear picture of how the area under the curve of f(t) = 7t + 5 changes as x varies.

Since A(x) is a quadratic function, its graph will be a parabola. To accurately graph it, we need to identify some key features, such as the vertex, intercepts, and general shape.

First, let's find the vertex of the parabola. The x-coordinate of the vertex of a parabola in the form ax² + bx + c is given by -b/(2a). In our case, a = 7/2 and b = 5, so the x-coordinate of the vertex is:

x_vertex = -5 / (2 * 7/2) = -5/7.

To find the y-coordinate of the vertex, we plug this value back into the area function:

A(-5/7) = 7(-5/7)²/2 + 5(-5/7) = 7(25/49)/2 - 25/7 = 25/14 - 25/7 = -25/14.

So, the vertex of the parabola is at (-5/7, -25/14). This point represents the minimum value of the area function since the parabola opens upwards (because the coefficient of x² is positive).

Next, let's find the x-intercepts of the graph. These are the points where A(x) = 0. So, we need to solve the equation 7x²/2 + 5x = 0. We can factor out an x to get:

x(7x/2 + 5) = 0.

This gives us two solutions: x = 0 and 7x/2 + 5 = 0, which means x = -10/7. So, the x-intercepts are at (0, 0) and (-10/7, 0).

Now, we have enough information to sketch the graph of A(x). It's a parabola that opens upwards, with a vertex at (-5/7, -25/14) and x-intercepts at (0, 0) and (-10/7, 0). As x increases from 0, the area function increases, representing the cumulative area under the curve of f(t) = 7t + 5 from t = 0 to t = x.

By graphing the area function, we've gained a visual understanding of how the area accumulates under the original function. Now, let's move on to the next part and explore the derivative of the area function.

Finding the Derivative: A'(x) and the Fundamental Theorem of Calculus

The final piece of our puzzle is finding the derivative of the area function, A'(x). This is where the fundamental theorem of calculus truly shines. The fundamental theorem of calculus has two parts, but the one we're interested in right now states that if A(x) = ∫ₐˣ f(t) dt, then A'(x) = f(x). In simpler terms, the derivative of the area function is just the original function, with the variable of integration t replaced by x.

This is a profound result because it connects the concepts of integration and differentiation. It tells us that differentiation and integration are, in a sense, inverse operations of each other. Finding the area function involves integration, and then differentiating that area function brings us back to the original function.

In our case, we found that A(x) = 7x²/2 + 5x. To find A'(x), we simply need to differentiate this function with respect to x. Using the power rule for differentiation, which states that d/dx(xⁿ) = nx^(n-1), we get:

A'(x) = d/dx (7x²/2 + 5x) = 7x + 5.

And there you have it! A'(x) = 7x + 5, which is exactly our original function, f(x). This beautifully illustrates the fundamental theorem of calculus in action. The rate of change of the area under the curve of f(t) at any point x is precisely the value of the function f(x) at that point.

By finding A'(x), we've completed our exploration of the area function for f(t) = 7t + 5. We've seen how to calculate the area function, graph it to visualize the accumulating area, and find its derivative to connect it back to the original function. This exercise has provided a solid understanding of the fundamental concepts of calculus and how they relate to each other.

Wrapping Up: Key Takeaways and Further Exploration

Alright guys, we've covered a lot of ground in this exploration of the area function! Let's recap the key takeaways from our journey:

  • The area function, A(x) = ∫ₐˣ f(t) dt, calculates the area under the curve of f(t) between a fixed point a and a variable point x.
  • For f(t) = 7t + 5 and a = 0, the area function is A(x) = 7x²/2 + 5x.
  • Graphing the area function provides a visual representation of how the area accumulates under the original function.
  • The fundamental theorem of calculus tells us that A'(x) = f(x), meaning the derivative of the area function is the original function.
  • Finding A'(x) for our example confirmed this theorem, giving us A'(x) = 7x + 5.

This exercise has provided a strong foundation for understanding area functions and the fundamental theorem of calculus. But the journey doesn't have to end here! You can further explore these concepts by:

  • Trying different functions for f(t), such as trigonometric functions, exponential functions, or more complex polynomials.
  • Changing the value of a and observing how it affects the area function and its graph.
  • Investigating the relationship between the area function and the definite integral in more depth.
  • Exploring applications of area functions in real-world scenarios, such as calculating displacement from velocity or accumulated revenue from a sales rate.

Calculus is a powerful tool for understanding change and accumulation, and area functions are a key concept in this field. By continuing to explore and practice, you can deepen your understanding and unlock even more of its potential.

So, keep exploring, keep questioning, and keep learning! The world of calculus is vast and full of exciting discoveries waiting to be made.