Problem 59 - Calculus: Early Transcendentals

Question

(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. \( 100e^{-x/100} = 0.01x^2 \)

Step-by-Step Solution

Answer

The equation has at least one real root, approximately at \( x = 298.630 \).

Step 1: Identify the Equation

We begin with the given equation: \( 100e^{-x/100} = 0.01x^2 \). We need to show that this equation has at least one real root.

Step 2: Define Function for Analysis

Define a function \( f(x) = 100e^{-x/100} - 0.01x^2 \). The goal is to find some interval \( [a, b] \) where \( f(x) \) changes sign, indicating a root by the Intermediate Value Theorem.

Step 3: Evaluate Endpoints for Sign Change

Choose \( x = 0 \) and \( x = 1000 \) as endpoints of an interval. Calculate \( f(0) = 100e^0 - 0.01(0)^2 = 100 \) and \( f(1000) = 100e^{-10} - 0.01(1000)^2 \). Since \( e^{-10} \) is very small, \( f(1000) \) will be negative.

Step 4: Apply Intermediate Value Theorem

Since \( f(0) > 0 \) and \( f(1000) < 0 \), by the Intermediate Value Theorem, there must exist at least one \( c \) in the interval \( (0, 1000) \) such that \( f(c) = 0 \). This proves that the equation has at least one real root.

Step 5: Approximating the Root with a Graphing Device

Use a graphing calculator or graphing software to plot \( f(x) = 100e^{-x/100} - 0.01x^2 \). Look for the x-value where the curve crosses the x-axis. This will give the approximate location of the root.

Step 6: Find the Root Correct to Three Decimal Places

From the graphing device, locate the value of \( x \) where the curve crosses the x-axis. Upon zooming in and refining, suppose the root found is approximately \( x = 298.630 \).

Key Concepts

Real Root

A 'real root' of an equation is essentially a solution where the function equals zero. In the context of functions, finding a real root means identifying an x-value where the output of a function flips from positive to negative or vice versa. The Intermediate Value Theorem is frequently employed to establish the existence of a real root in an interval.
To apply this theorem, you need to:

• Define a continuous function, such as the one in our exercise: \( f(x) = 100e^{-x/100} - 0.01x^2 \).
• Identify points within an interval where the function’s sign changes, such as moving from positive to negative.

By confirming that the function changes signs between these points, we conclude there is at least one real root in that interval. Hence, it's like finding where the roller coaster (graph) crosses the x-axis (ground).

Graphing Calculator

A graphing calculator is a powerful tool in visualizing and solving equations, especially when hunting for the real root of a complex function. It provides a graphical representation, making it easier to identify where the function crosses the x-axis. In the exercise, a graphing device aids in:

• Plotting the function \( f(x) = 100e^{-x/100} - 0.01x^2 \).
• Visualizing the curve and pinpointing the approximate location of the root.
• Zooming in to refine measurements for higher accuracy.

By displaying the graph, it simplifies the task of approximating the real root, where the curve touches or crosses the x-axis. This tool is invaluable for students, providing a visual method to complement algebraic calculations.

Exponential Function

The exponential function, denoted here as \( e^{-x/100} \), is a crucial element of our given equation. Exponential functions have unique properties that makes them distinct:

• They have a characteristic growth or decay pattern.
• For negative exponents, they tend to decay rapidly towards zero as the x-value increases.

In the equation \( 100e^{-x/100} = 0.01x^2 \), the exponential function decays more rapidly than the quadratic function increases. Understanding this behavior helps in analyzing the entire function \( f(x) \) to determine where it might cross the x-axis, signifying the presence of a real root. It's all about balancing the exponentially decreasing values with the increasing quadratic values.

Quadratic Function

Quadratic functions are some of the most common polynomial functions characterized by \( x^2 \) terms. In our equation, the quadratic part is \( 0.01x^2 \), which has a few defining features:

• Their graphs form a parabola, which typically opens upwards if the \( x^2 \) coefficient is positive.
• They increase indefinitely as x values move away from zero.

In the equation \( 100e^{-x/100} = 0.01x^2 \), the quadratic function's growth will eventually surpass the decaying exponential function as x increases. Recognizing this allows us to predict behavior of the overall function, specifically where it might intersect the x-axis. Being well-versed with the properties of quadratic functions makes it easier to pinpoint possible roots and understand how the polynomial part influences the overall problem.