How do you find the value of c that satisfy the equation #(f(b)-f(a))/(b-a)=f'(c)# in the conclusion of the mean value theorem for the function #f(x)=4x^2+4x-3# on the interval [-1,0]#?
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For the problem under consideration, we have
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To find the value of ( c ) that satisfies the equation ((f(b)-f(a))/(b-a)=f'(c)) in the conclusion of the Mean Value Theorem for the function ( f(x)=4x^2+4x-3 ) on the interval ([-1,0]), follow these steps:
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Determine the values of ( f(a) ) and ( f(b) ) for the given interval ([-1,0]). Plug in ( a = -1 ) and ( b = 0 ) into the function ( f(x) = 4x^2 + 4x - 3 ) to get: ( f(-1) = 4(-1)^2 + 4(-1) - 3 = 4 - 4 - 3 = -3 ) ( f(0) = 4(0)^2 + 4(0) - 3 = 0 - 0 - 3 = -3 )
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Calculate the derivative of ( f(x) ) to find ( f'(x) ). The derivative of ( f(x) = 4x^2 + 4x - 3 ) is: ( f'(x) = 8x + 4 )
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Use the Mean Value Theorem formula: (\frac{f(b)-f(a)}{b-a} = f'(c)) Plug in the values of ( f(a) ), ( f(b) ), and ( f'(x) ) into the equation: (\frac{f(0)-f(-1)}{0-(-1)} = f'(c)) (\frac{-3 - (-3)}{0 - (-1)} = f'(c)) (\frac{0}{1} = f'(c)) (0 = f'(c))
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Now, solve for ( c ) by setting ( f'(c) ) equal to 0: ( f'(c) = 0 ) ( 8c + 4 = 0 ) ( 8c = -4 ) ( c = -\frac{1}{2} )
Therefore, the value of ( c ) that satisfies the equation in the conclusion of the Mean Value Theorem is ( c = -\frac{1}{2} ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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