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Questions and Answers
What is the first step to apply Lagrange's Mean Value Theorem to the function $f(x) = x^1(1-2)(x-2)$ on the interval [0, 4]?
What is the first step to apply Lagrange's Mean Value Theorem to the function $f(x) = x^1(1-2)(x-2)$ on the interval [0, 4]?
After confirming the conditions of Lagrange's Mean Value Theorem, which equation would you set up to find the value of c?
After confirming the conditions of Lagrange's Mean Value Theorem, which equation would you set up to find the value of c?
What is the derivative $f'(x)$ of the function $f(x) = x^1(1-2)(x-2)$?
What is the derivative $f'(x)$ of the function $f(x) = x^1(1-2)(x-2)$?
If you evaluate $f(0)$ using the function $f(x) = x^1(1-2)(x-2)$, what value do you obtain?
If you evaluate $f(0)$ using the function $f(x) = x^1(1-2)(x-2)$, what value do you obtain?
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Once the average rate of change has been calculated for $f(x)$ over the interval [0, 4], how do you interpret the value of c derived from Lagrange's Mean Value Theorem?
Once the average rate of change has been calculated for $f(x)$ over the interval [0, 4], how do you interpret the value of c derived from Lagrange's Mean Value Theorem?
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Study Notes
Lagrange's Mean Value Theorem (LMVT) Application
- First step involves verifying the function's continuity and differentiability over the closed interval [0, 4].
- Function must be continuous on [0, 4] and differentiable on (0, 4) for LMVT to apply.
Finding the Value of c
- After confirming LMVT conditions, set up the equation:
( f'(c) = \frac{f(b) - f(a)}{b - a} )
Where ( a = 0 ) and ( b = 4 ).
Derivative of the Function
- The function given is ( f(x) = x(1-2)(x-2) ).
- Simplifying, ( f(x) = -x^2 + 2x ).
- The derivative is calculated as:
( f'(x) = -2x + 2 ).
Evaluating f(0)
- Calculate ( f(0) ) using the function:
( f(0) = 0(1-2)(0-2) ) results in ( f(0) = 0 ) because of the multiplication by zero.
Interpreting the Value of c
- After finding the average rate of change, ( f'(c) ) represents the slope of the tangent to the function at ( c ).
- The value of ( c ) indicates a point in the interval [0, 4] where the instantaneous rate of change (slope) equals the average rate of change over the interval.
Lagrange's Mean Value Theorem (LMVT) Application
- The first step in applying LMVT is verifying the function satisfies the theorem's conditions: continuity on a closed interval and differentiability on the open interval.
- For the function ( f(x) = x^1(1-2)(x-2) ) on the interval [0, 4], check if it is continuous and differentiable in that range.
Setting Up the Equation
- After confirming LMVT conditions, set up the equation:
[ f'(c) = \frac{f(b) - f(a)}{b - a} ]
where ( a = 0 ) and ( b = 4 ) for the interval [0, 4].
Derivative Calculation
- Compute the derivative ( f'(x) ) of the function:
[ f(x) = x^1(1-2)(x-2) = -x(x-2) = -x^2 + 2x ]
Thus,
[ f'(x) = -2x + 2 ]
Evaluating ( f(0) )
- Evaluate ( f(0) ): [ f(0) = -0^2 + 2(0) = 0 ] The value obtained is 0.
Interpretation of ( c )
- The value of ( c ) derived from LMVT represents a point in the interval [0, 4] where the instantaneous rate of change (derivative) equals the average rate of change over that interval.
- This means ( f'(c) ) will equal the value calculated from (\frac{f(4) - f(0)}{4 - 0}), providing valuable insight into the function's behavior at that specific point.
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Description
This quiz explores Lagrange's Mean Value Theorem with a specific function f(x) = x^1(1-2)(x-2) on the interval [0, 4]. You'll learn how to apply the theorem, calculate derivatives, evaluate function values, and interpret the obtained value of c. Test your understanding of this fundamental calculus concept with practical questions.
