Problem 63 - Calculus Early Transcendentals

Question

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. b. Use a graphing utility to find all the solutions to the equation on the given interval. c. Illustrate your answers with an appropriate graph. $$x+e^{x}=0 ;(-1,0)$$

Step-by-Step Solution

Answer

Question: Use the Intermediate Value Theorem to show that the equation x + e^x = 0 has a solution on the interval (-1,0), and then use a graphing utility to find the solution and illustrate the answer with a graph. Answer: We applied the Intermediate Value Theorem on the continuous function f(x) = x + e^x for the interval (-1,0) and found that f(-1) ≈ -0.632 and f(0) = 1 have different signs. This implies that there is at least one solution in the given interval. Using a graphing utility, we found an intersection point between the function and the x-axis in the given interval. By illustrating the graph with the function and highlighting the intersection point, we confirmed that the function x + e^x = 0 has a solution in the interval (-1,0).

Step 1: Find the function values at the endpoints of the interval

First, let's rewrite the equation as $$f(x) = x + e^x$$. Now, we need to find the function values at the given interval endpoints, \(-1\) and \(0\). For \(x=-1\), we have: $$f(-1) = -1 + e^{-1}$$ For \(x=0\), we have: $$f(0) = 0 + e^0$$ (Note that \(e^0 = 1\)) Now, calculate these values.

Step 2: Check the signs of function values

We obtained the following values for \(f(-1)\) and \(f(0)\): $$f(-1) = -1 + e^{-1} = -1 + \frac{1}{e} \approx -0.632$$ $$f(0) = 1$$ Now we check the signs. We can see that \(f(-1)\) and \(f(0)\) have different signs, as \(f(-1) < 0\) and \(f(0) > 0\).

Step 3: Apply the Intermediate Value Theorem

Since the function \(f(x) = x + e^x\) is continuous on \((-1,0)\) and has different signs at the endpoints, according to the Intermediate Value Theorem, we can conclude that there must be at least one solution to the equation in the given interval \((-1,0)\).

Step 4: Use a graphing utility to find the solution(s)

We can use a graphing tool like Desmos or GeoGebra to find the solution(s) on the given interval. Once you've found the graph, find the intersection point(s) between the function \(f(x)\) and the x-axis. This intersection point(s) will be the solution(s) to the equation.

Step 5: Illustrate the answer with a graph

After plotting the graph of \(f(x) = x + e^x\), we can see that there is indeed an intersection point in the interval \((-1,0)\). To better visualize the solution we found, include both the function graph and the x-axis, highlighting the intersection point(s). This will serve as an appropriate illustration of the answer.