Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$f(x)=3 e^{3 x / 2}-962$$

Step-by-Step Solution

Answer

The zero of the function \(f(x)=3 e^{3 x / 2}-962\) is approximately at \[x = ...\] (Enter the x-coordinate found in Step 3 to three decimal places.)

Step 1: Graph the Function

Start by plotting the given function \(f(x)=3 e^{3 x / 2}-962\), a transformed exponential function, on your graphing utility. Ensure your graph has a suitable range to capture the key details of the function.

Step 2: Locate the Graph's Intersection With the X-axis

Look for the point where the function crosses the x-axis. The exact location of this intersection represents the zero of the function.

Step 3: Approximate the Zero of the Function

Use your graphing utility to determine the x-coordinate of the intersection point you located in Step 2. Be sure to provide the x-coordinate accurate to three decimal places.

Key Concepts

Exponential Function Graphing

Graphing an exponential function is essential for understanding its characteristics, such as growth or decay, intercepts, and asymptotic behavior. The function in our exercise, f(x) = 3e^{3x/2} - 962, features an exponential part 3e^{3x/2} which determines the rapid increase of the function's value as x increases, modified by a vertical shift of -962 units.

When graphing an exponential function, it's important to select an appropriate range for the x-axis. This helps in capturing the key points where the function undergoes significant changes, such as the x-intercept, or zero, and where it starts to level off. Modern graphing utilities allow you to zoom in or out, which can be very helpful in finding these key features of the graph. Pay particular attention to the 'sharpness' of the curve, as exponential functions can rise or fall very rapidly. As we graph, locating key features becomes simpler and helps set the stage for further analysis, such as zero approximation.

Approximating Zeros

Approximating zeros, also known as finding the roots, is a crucial part of understanding where a function crosses the x-axis. Zeros are the points where the function has no value; in other words, f(x) = 0. This is especially significant for exponential functions, which often represent growth processes in fields such as biology and finance.

To approximate zeros accurately, use the 'zero' or 'root' feature on your graphing utility. The goal is to find the x-value where the function intersects the x-axis. After graphing the function, you may need to adjust the viewing window to pinpoint the zero more precisely. Remember that because the function rises and falls so sharply, the zero may only be visible within a certain range. Proof of a good approximation includes checking if the zero is consistent with the graph visually and if it satisfies the function numerically, by evaluating f(x) at the given point and confirming it's sufficiently close to zero.

Exponential Equations

Exponential equations involve an unknown variable in the exponent, which presents unique challenges and concepts. The equation from our exercise, f(x) = 3e^{3x/2} - 962, is one such example. Understanding how to manipulate and solve these equations is vital for mastering exponential growth and decay problems.

When dealing with an exponential equation, you often have to isolate the exponential expression to one side of the equation before you apply logarithms to solve for the variable. While graphing utilities offer a visual approach to finding zeros or solutions, algebraic techniques such as taking natural logarithms on both sides can provide exact analytic solutions. In the case of approximations, a graphing utility is uniquely efficient; it can provide a visual representation of the solution and an immediate numerical answer. For complex or real-world situations, graphing can become an invaluable method for estimation and immediate analysis.