11.find The Values Of C That Satisty The Mean Value Theorem. Show Steps
Understanding the Mean Value Theorem (MVT) is fundamental in calculus, especially when analyzing the behavior of functions. The process of finding the specific values of \( c \) that satisfy the theorem helps in comprehending how functions behave between two points. In this article, we will explore the steps to find these values of \( c \) that satisfy the Mean Value Theorem, complete with detailed explanations and examples to solidify your understanding.
What Is the Mean Value Theorem?
The Mean Value Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that:\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
This theorem essentially guarantees that at some point \( c \), the instantaneous rate of change (the derivative) equals the average rate of change over \([a, b]\).
Steps to Find Values of \( c \) That Satisfy the MVT
Finding the specific \( c \) values involves a systematic process. Let’s break down the steps:Step 1: Verify the Conditions of the MVT
Before proceeding, ensure that:- The function \( f \) is continuous on the closed interval \([a, b]\).
- The function \( f \) is differentiable on the open interval \((a, b)\).
If these conditions are not satisfied, the MVT does not apply, and you cannot find such \( c \).
Step 2: Calculate the Average Rate of Change
Compute the difference quotient:\[ \frac{f(b) - f(a)}{b - a} \]
This value represents the average rate of change of \( f \) from \( a \) to \( b \).
Step 3: Find the Derivative \( f'(x) \)
Differentiate the function \( f(x) \) to obtain \( f'(x) \). This derivative represents the instantaneous rate of change at any point \( x \).Step 4: Set \( f'(c) \) Equal to the Average Rate of Change
Solve the equation:\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
for \( c \). This involves substituting \( c \) into the derivative \( f'(x) \).
Step 5: Solve for \( c \) in the Interval \((a, b)\)
Find all solutions \( c \) that satisfy the equation within the interval \((a, b)\). These are the points where the tangent line is parallel to the secant line connecting \( (a, f(a)) \) and \( (b, f(b)) \).Example: Find \( c \) for a Specific Function
Let’s apply these steps to a concrete example for clarity.Suppose \( f(x) = x^2 + 2x \), with \( a = 1 \) and \( b = 3 \).
Step 1: Verify Conditions
- \( f(x) = x^2 + 2x \) is a polynomial, which is continuous and differentiable everywhere.
- Therefore, MVT applies on \([1, 3]\).
Step 2: Calculate the Average Rate of Change
\[ f(3) = 3^2 + 2 \times 3 = 9 + 6 = 15 \\ f(1) = 1^2 + 2 \times 1 = 1 + 2 = 3 \] \[ \frac{f(3) - f(1)}{3 - 1} = \frac{15 - 3}{2} = \frac{12}{2} = 6 \]Step 3: Find \( f'(x) \)
\[ f'(x) = 2x + 2 \]Step 4: Set \( f'(c) = 6 \) and solve for \( c \)
\[ 2c + 2 = 6 \\ 2c = 4 \\ c = 2 \]Step 5: Verify \( c \) is in \((a, b)\)
\[ a = 1, \quad b = 3, \quad c = 2 \] Since \( 1 < 2 < 3 \), the value \( c = 2 \) satisfies the Mean Value Theorem.Additional Tips for Finding \( c \) Values
- Always double-check the conditions of the theorem before proceeding.
- When solving \( f'(c) = \) average rate of change, carefully manipulate the algebraic expressions.
- Remember that the solution for \( c \) must lie strictly within \((a, b)\), so discard any solutions outside this interval.
- For more complex functions, consider using numerical methods or graphing tools to estimate solutions.
Common Mistakes to Avoid
- Applying MVT to functions that are not continuous or differentiable on the interval.
- Forgetting to verify the interval for the solution \( c \).
- Mistakenly solving for \( c \) outside the open interval \((a, b)\).
- Miscalculating derivatives or the average rate of change.
Conclusion
Finding the values of \( c \) that satisfy the Mean Value Theorem involves a clear, step-by-step process: verify the theorem's conditions, compute the average rate of change, differentiate the function, set the derivative equal to this value, and solve within the interval. Mastering this process enables you to analyze functions' behavior between two points confidently and is essential for higher-level calculus topics.Whether working with simple polynomial functions or more complex ones, these steps serve as a reliable guide to identify the points where the function's instantaneous rate matches its average rate over an interval. Practice with various functions to become proficient in applying the Mean Value Theorem effectively.
