Question

(a) Graph some representative integral curves of the function \(\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}} / 2\) (b) Find an equation for the integral curve that passes through the point \((0,1)\).

Short answer

Answer: The equation of the integral curve is \(g(x) = \frac{e^x}{2} + \frac{1}{2}\).

Step-by-step solution

Graph the function

Plot \(f(x) = \frac{e^x}{2}\). You can use a graphing software, calculator, or online graphing tool to sketch the function. Notice that the function is always positive and increasing, since \(e^x\) is always positive for any x.

Sketch some integral curves

Next, sketch some integral curves of the function. Remember that these curves represent the antiderivative of the function with different constant values. You can use the same graphing tool you used in the previous step and add some curves with different constant values.

Find the antiderivative of the function

Now, find the antiderivative of the function \(f(x) = \frac{e^x}{2}\). To do this, integrate the function with respect to x: \(\int \frac{e^x}{2} dx = \frac{e^x}{2} + C\), where C is the integration constant.

Find the constant corresponding to the given point

Now we need to find the constant C that corresponds to the point \((0,1)\). Plug the point into the antiderivative we found in step 3: \(1 = \frac{e^0}{2} + C\). Since \(e^0 = 1\), we have \(1 = \frac{1}{2} + C\). Solve for C: \(C = \frac{1}{2}\).

Write the equation for the integral curve

Finally, substitute the value of C we found in step 4 into the antiderivative equation: \(g(x) = \frac{e^x}{2} + \frac{1}{2}\). This is the equation of the integral curve passing through the point \((0,1)\).

Key concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is the reverse process of differentiation. In calculus, finding an antiderivative means looking for a function whose derivative gives the original function. For example, if we have a function represented by \( f(x) \), an antiderivative would be a function \( F(x) \) such that \( F'(x) = f(x) \).

Let's take a practical look at how to find an antiderivative using the function \( f(x) = \frac{e^x}{2} \). First, you should recall that the derivative of the exponential function \( e^x \) is itself, making it straightforward to determine the antiderivative. So, the antiderivative of \( f(x) \) simply involves integrating the function with respect to \( x \), resulting in \( F(x) = \frac{e^x}{2} + C \), where \( C \) is the integration constant representing an infinite number of possible antiderivatives for \( f(x) \).

To apply this concept to a specific problem, it is necessary to have additional information, such as a point through which the curve passes, to find the value of the integration constant and thus determine the precise antiderivative.

Exponential Functions

Exponential functions are among the most influential and widely used mathematical concepts, particularly in calculus. These functions have the form \( f(x) = a^x \) where \( a \) is a constant base and \( x \) is the exponent. One of the most important characteristics of exponential functions is that their rate of growth (or decay if \( 0 < a < 1 \)) is proportional to their current value.

The function \( f(x) = \frac{e^x}{2} \) is a classic example of an exponential function where the base is the natural exponential \( e \), an irrational constant approximately equal to 2.71828. This function continuously increases as \( x \) increases and never touches the x-axis, meaning it never becomes zero or negative. Understanding the behavior of exponential functions is crucial in the study of calculus and can also be of paramount importance in fields such as physics, biology, and finance, where these functions model growth and decay processes.

Definite Integration

Definite integration is the process of calculating the area under the curve of a function between two points, referred to as limits of integration. In contrast to indefinite integration, which provides a general antiderivative, definite integration computes a specific numerical value representing the area.

In the context of definite integration, we would be looking at integrating the function \( f(x) = \frac{e^x}{2} \) from one point to another, say \( a \) to \( b \), which gives us the net area under the curve between these two points. The result is obtained using the fundamental theorem of calculus, which states that if \( F \) is an antiderivative of \( f(x) \), then the definite integral of \( f(x) \) from \( a \) to \( b \) is \( F(b) - F(a) \).

Our focus, however, is on indefinite integration and finding the antiderivative, which does not provide a net area but rather the family of functions that can trace the curve of \( f(x) \) when plotted on a graph.

Integration Constant

The integration constant, represented by \( C \), is a critical part of the antiderivative in calculus. Whenever you integrate a function, you add a constant \( C \) because integration is the inverse of differentiation and differentiating a constant yields zero. Therefore, there is no way to identify unique values for \( C \) from the integral alone.

However, in the application of antiderivatives to real problems, we often have additional information that allows us to solve for \( C \). This is evident in the exercise where we find the equation for the integral curve that passes through the point \((0,1)\). Substituting these values into the general antiderivative formula \( g(x) = \frac{e^x}{2} + C \), we find that \( C = \frac{1}{2} \), thus specifying the unique antiderivative that fits the given condition.

The introduction of an integration constant is a fundamental concept in integral calculus, as it encapsulates the notion that there are infinitely many antiderivatives for any given function—the set of all these solutions differing only by a constant.