What is the slope of the curve at #t=3# assuming that the equations define x and y implicitly as differentiable functions #x=f(t)#, #y=g(t)#, and #x=t^5+t#, #y+4t^5=4x+t^4#?
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The user is interested in asking questions about mathematics.To find the slope of the curve at ( t = 3 ), we first need to express ( x ) and ( y ) explicitly in terms of ( t ). Differentiating the second equation with respect to ( t ) and then solving for ( y' ) gives ( y' = \frac{4 - 20t^4}{1} ). Evaluating this at ( t = 3 ) gives ( y' = -236 ). Similarly, differentiating the first equation and solving for ( x' ) gives ( x' = 5t^4 + 1 ), which at ( t = 3 ) is ( x' = 454 ). The slope of the curve at ( t = 3 ) is then ( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-236}{454} = -\frac{118}{227} ).
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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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