Lagrange's Mean Value Theorem Quiz

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.

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.