Question

Working with area functions Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c) .\) Interpret the results using the graphs of part \((b)\) $$f(x)=e^{x} ; a=0, b=\ln 2, c=\ln 4$$

Short answer

Answer: A(b) = 1 and A(c) = 3

Step-by-step solution

Find the area function A(x)

To find the area function \(A(x) = \int_a^x f(t) dt\), we will apply the Fundamental Theorem of Calculus to the given function \(f(x) = e^x\). \(a=0\) is given, so we have: $$ A(x) = \int_0^x e^t dt $$ Now, we integrate: $$ A(x) = e^t \Big|_0^x = e^x - e^0 = e^x - 1 $$ So, the area function is \(A(x) = e^x - 1\).

Graph f and A

We know that the domain of both \(f(x)\) and \(A(x)\) is \((-\infty, +\infty)\). Using graphing software, we can plot the graphs of \(f(x) = e^x\) and \(A(x) = e^x - 1\). Observe the the function \(A(x)\) lies one unit below \(f(x)\).

Evaluate A(b) and A(c) and interpret the results

We have \(b = \ln 2\) and \(c = \ln 4\). We need to calculate \(A(b)\) and \(A(c)\). Using the area function \(A(x) = e^x - 1\), we have: $$ A(b) = A(\ln 2) = e^{\ln 2} - 1 = 2 - 1 = 1 \\ A(c) = A(\ln 4) = e^{\ln 4} - 1 = 4 - 1 = 3 $$ So, \(A(b) = 1\) and \(A(c) = 3\). Now, let's interpret the results using the graphs of \(f\) and \(A\). The graph of the area function \(A(x)\) gives the net value of the area bounded by the graph of \(f(x)\) and the x-axis, from \(a=0\) to \(x = b\) or \(x = c\). Thus, \(A(b)\) tells us that the net area between the graph of \(f(x) = e^x\), the \(x\)-axis, and the vertical lines \(x = 0\) and \(x = \ln 2\) is equal to 1. Similarly, \(A(c)\) tells us that the net area between the graph of \(f(x)\), the \(x\)-axis, and the vertical lines \(x = 0\) and \(x = \ln 4\) is equal to 3.

Key concepts

Area Function

Understanding an area function is crucial when studying calculus, particularly when dealing with the application of integrals. It represents the accumulated area under the curve of a given function from a certain starting point, usually denoted as a, up to a variable endpoint x. In simpler terms, if you imagine the area under a curve from a fixed point onwards, the area function gives you the value of this area as the endpoint moves.

In the exercise provided, the area function A(x) calculated from a to x for the function f(t) = e^t took the form A(x) = e^x - 1. This is arrived at through the process of integration, and it represents the net area bounded by the graph of f(t), the x-axis, from a = 0 to the variable x. It's worth noting that when evaluating at specific points like b or c, the area function gives the actual area accumulated up to those points, offering a geometrical representation of the integral.

Integration

Integration is a fundamental concept in calculus that can be thought of as the reverse operation of differentiation or the process of finding the total accumulation of a quantity. When you integrate a function, you're essentially finding the area under its curve within a specific interval. It's a powerful tool that allows mathematicians and scientists to determine not just areas, but also other concepts such as displacement or the total growth over time.

The Fundamental Theorem of Calculus, which plays a pivotal role in integration, links the concept of differentiation with that of integration. It states that if a function is continuous over an interval and has an antiderivative within that interval, then the definite integral of the function can be obtained by evaluating its antiderivative at the endpoints of the interval. The solution to the given exercise applied this theorem to find the area function A(x), demonstrating a practical application of integration in finding areas.

Exponential Functions

In the realm of mathematics, exponential functions are unique in their behavior and have significant applications across various disciplines. An exponential function is of the form f(x) = a^x, where a is a constant called the base and x is the exponent. These functions grow at a rate proportional to their value, which makes them particularly important when modeling growth or decay processes, such as population growth, radioactive decay, or compound interest.

The function f(x) = e^x used in the exercise is a classic example of an exponential function, where the base is the mathematical constant e, approximately equal to 2.71828. This function increases rapidly as x increases, which is reflected in the integration results: the area under the curve of an exponential function is also an exponential function. The exercise provided a clear instance of how exponential functions can be integrated, and the calculations for A(b) and A(c) showed how the integral of an exponential function can be interpreted and understood in a geometrical sense as an area under the curve.