Math Problem Statement
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ex = 8 − 7x, (0, 1)
Solution
To use the Intermediate Value Theorem (IVT) to show that there is a root of the equation in the interval , follow these steps:
Step 1: Rewrite the equation
We can rewrite the equation as a function where we move all terms to one side: This simplifies to: Our goal is to show that there is a root of in the interval .
Step 2: Check continuity of the function
The function is continuous because both and are continuous functions, and the sum of continuous functions is also continuous.
Step 3: Apply the Intermediate Value Theorem
The Intermediate Value Theorem states that if a continuous function takes values of opposite signs at two points and , then there exists at least one such that .
Now, evaluate the function at the endpoints of the interval :
-
Evaluate : So, .
-
Evaluate : So, .
Step 4: Conclusion using IVT
We have:
- (a negative value)
- (a positive value)
Since is continuous and changes from negative to positive between and , by the Intermediate Value Theorem, there must be some such that . Therefore, there is a root of the equation in the interval .
Would you like further details or have any questions about this solution?
Here are 5 additional questions you can explore related to the Intermediate Value Theorem:
- How would you find the exact value of the root using numerical methods like bisection or Newton's method?
- Can the Intermediate Value Theorem be applied if the function is not continuous?
- What are the key differences between the Intermediate Value Theorem and the Mean Value Theorem?
- How would you use the IVT to find roots for more complex functions?
- How does the behavior of influence the application of IVT to this particular problem?
Tip: The Intermediate Value Theorem only guarantees the existence of a root, not the exact value. To find the precise root, you’ll often need numerical methods like the bisection method.
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Math Problem Analysis
Mathematical Concepts
Calculus
Exponential Functions
Intermediate Value Theorem
Formulas
f(x) = e^x + 7x - 8
f(0) = e^0 + 7(0) - 8 = -7
f(1) = e^1 + 7(1) - 8 = e - 1 ≈ 1.718
Theorems
Intermediate Value Theorem
Suitable Grade Level
Grades 11-12
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