Generalized Mean Value Theorem Suppose the functions f and g are continuous on ⌈ a, b ⌉ and differentiable on ( a, b ) , where g ( a ) ≠ g ( b ) . Then there is a point c in ( a , b ) at which f ( b ) − f ( a ) g ( b ) − g ( a ) = f ′ ( c ) g ′ ( c ) . This result is known as the Generalized (or Cauchy’s) Mean Value Theorem. a. If g ( x ) = x , then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem. b. Suppose f ( x ) = x 2 − l, g ( x ) = 4 x + 2, and [ a , b ] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.
Generalized Mean Value Theorem Suppose the functions f and g are continuous on ⌈ a, b ⌉ and differentiable on ( a, b ) , where g ( a ) ≠ g ( b ) . Then there is a point c in ( a , b ) at which f ( b ) − f ( a ) g ( b ) − g ( a ) = f ′ ( c ) g ′ ( c ) . This result is known as the Generalized (or Cauchy’s) Mean Value Theorem. a. If g ( x ) = x , then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem. b. Suppose f ( x ) = x 2 − l, g ( x ) = 4 x + 2, and [ a , b ] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.
Solution Summary: The author explains the Generalized Mean Value Theorem when reduce to the Mean Valuation Theory for the given function. The function is g(x)=x.
Generalized Mean Value Theorem Suppose the functions f and g are continuous on ⌈a, b⌉ and differentiable on (a, b), where g(a) ≠ g(b). Then there is a point c in (a, b) at which
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This result is known as the Generalized (or Cauchy’s) Mean Value Theorem.
a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem.
b. Suppose f(x) = x2 − l, g(x) = 4x + 2, and [a, b] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.
A 20 foot ladder rests on level ground; its head (top) is against a vertical wall. The bottom of the ladder begins by being 12 feet from the wall but begins moving away at the rate of 0.1 feet per second. At what rate is the top of the ladder slipping down the wall? You may use a calculator.
Explain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)
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