Is #f(x)=4x-e^(3x-2) # increasing or decreasing at #x=-2 #?
Differentiating the function (this is where the chain rule comes in):
graph{[-13.19, 12.12, -6.53, 6.13]}-e^(3x-2)
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To determine whether ( f(x) = 4x - e^{3x - 2} ) is increasing or decreasing at ( x = -2 ), you need to evaluate the sign of the derivative of ( f(x) ) at that point. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.
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Find the derivative of ( f(x) ) with respect to ( x ): [ f'(x) = 4 - 3e^{3x - 2} ]
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Evaluate the derivative at ( x = -2 ): [ f'(-2) = 4 - 3e^{3(-2) - 2} = 4 - 3e^{-8} ]
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Determine the sign of ( f'(-2) ): Since ( e^{-8} ) is positive (since it's the reciprocal of a large positive number), ( f'(-2) ) will be positive.
Conclusion: At ( x = -2 ), ( f(x) ) is increasing.
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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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- Is #f(x)=cos^2x+sin2x# increasing or decreasing at #x=pi/6#?
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