Problem 10 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(9-12,\) use a graph to find the zeros of the function. $$f(x)=e^{x}-4$$

Step-by-Step Solution

Answer

The zero for the function \(f(x) = e^{x}-4\) is \(x = ln(4)\).

Step 1: Understanding the Function

The given function is \(f(x) = e^{x}-4\). The base of the function, e, is the mathematical constant that is approximately equal to 2.71828. This is an exponential function which decreases in value as x becomes more negative and increases in value as x becomes more positive. It is shifted down by 4 units due to the -4.

Step 2: Sketch the Function

Draw the exponential function \(e^{x}\) and then shift it down by 4 units to get the function \(f(x) = e^{x}-4\). The graph will cross the x-axis where the function equals zero.

Step 3: Finding the Zeros

The zeros of the function are where the graph crosses the x-axis, or where \(f(x) = 0\). For the function \(f(x) = e^{x}-4\), this happens where the graph of \(e^{x}\) is equal to 4. This can be solved algebraically by setting \(e^{x} = 4\) and solving for x. Using a natural logarithm function, the solution is \(x = ln(4)\).

Key Concepts

Exponential Functions

Exponential functions play a significant role in various fields such as finance, computer science, and natural sciences due to their remarkable properties. The term 'exponential' refers to a function that increases or decreases at rates proportional to its current value. In mathematical terms, an exponential function can be expressed as f(x) = a^x, where a is a positive constant base and x is the exponent.

A unique and remarkable property of exponential functions is that they have a constant rate of growth or decay, which is represented by the base a. The function's growth or decay nature depends on whether the base is greater or lesser than 1. Applications of exponential functions include calculating interest, population growth models, radioactive decay, and even in algorithms of computer programs.

Basic features identified in the step-by-step solution of the function f(x) = e^{x} - 4 include the base e, which is an irrational mathematical constant known as Euler's number, and is approximately 2.71828. This base is special because its rate of change (the derivative) is equal to the value of the function itself, which makes it a natural choice for representing continuously growing processes.

Natural Logarithm

The natural logarithm is the inverse operation of taking an exponential function with base e. If you have a function y = e^{x}, then the natural logarithm of y, denoted as ln(y), will provide you with the exponent x. This operation is crucial for solving equations where the variable is an exponent in the context of an e-based exponential function, such as f(x) = e^{x} - 4.

The natural logarithm has several unique properties that simplify solving equations. For instance, ln(e) = 1 because e raised to the power of 1 is e. Moreover, the natural logarithm of a product is the sum of the logarithms, and the logarithm of a quotient is the difference of the logarithms. These properties often help reframe complex exponential relationships into simpler linear forms, thus making it easier to isolate the variable and solve for it.

In relation to our given problem, we used the natural logarithm to isolate x and find the zeros of the function by solving ln(e^{x}) = ln(4), leading to x = ln(4), an application showing how logarithms are vital tools in finding solutions to exponential equations.

Graphing Functions

Graphing is an invaluable tool in visualizing mathematical functions and their characteristics. By plotting a function on a coordinate plane, we can quickly identify features such as intercepts, asymptotes, intervals of increase or decrease, and local maxima or minima.

For exponential functions like f(x) = e^{x} - 4, graphing helps us find the zeros or roots, which are the points where the graph crosses the x-axis. The step-by-step solution provided begins with plotting the basic exponential curve of e^{x} and then shifting it vertically by 4 units downwards, which allows us to see that the zero of the function f(x) corresponds to the point where the shifted curve intersects the x-axis.

Graphing not only aids in understanding the solution but also offers a visual confirmation of the algebraic work involved in solving for zeros. Students often benefit from this dual approach as it solidifies their grasp of the relationship between algebraic expressions and their graphical representations.

Solving Exponential Equations

Solving exponential equations is about finding the value(s) of the variable for which the equation holds true. Exponential equations often require specific techniques involving logarithms to solve for the unknown exponent.

The process typically includes isolating the exponential part of the equation and then either applying logarithms to both sides or using properties of exponents to simplify. For example, as part of the step-by-step solution, when faced with f(x) = e^{x} - 4 = 0, we isolate e^{x} to get e^{x} = 4. We then apply the natural logarithm to both sides, effectively 'unpacking' the exponent and allowing us to solve for x. This gives us x = ln(4), which is our solution.

Understanding such techniques is vital not just for particular textbook exercises, but for any real-world problem where the rate of change is exponential in nature, hence, mastering solving exponential equations is foundational for advanced studies in mathematics, sciences, and engineering.