How do you differentiate the following parametric equation: # x(t)=-te^t-2t, y(t)= 3t^3-4t #?
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To differentiate the given parametric equations (x(t) = -te^t - 2t) and (y(t) = 3t^3 - 4t), you differentiate each equation with respect to (t) separately. Here are the steps:
- Differentiate (x(t)) with respect to (t) using the product rule and the chain rule:
[ \frac{dx}{dt} = \frac{d}{dt}(-te^t - 2t) = -e^t - te^t - 2 ]
- Differentiate (y(t)) with respect to (t):
[ \frac{dy}{dt} = \frac{d}{dt}(3t^3 - 4t) = 9t^2 - 4 ]
These are the derivatives of the parametric equations (x(t)) and (y(t)) with respect to (t).
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To differentiate the given parametric equations (x(t) = -te^t - 2t) and (y(t) = 3t^3 - 4t), we will differentiate each equation with respect to (t) separately.
The derivative of (x(t)) with respect to (t) is found using the product rule and the chain rule:
[ \frac{dx}{dt} = \frac{d}{dt}(-te^t) - \frac{d}{dt}(2t) ] [ = (-t\frac{d}{dt}e^t - e^t) - 2 ] [ = (-t(e^t) - e^t) - 2 ] [ = -te^t - e^t - 2 ]
Similarly, the derivative of (y(t)) with respect to (t) is calculated:
[ \frac{dy}{dt} = \frac{d}{dt}(3t^3) - \frac{d}{dt}(4t) ] [ = 9t^2 - 4 ]
Therefore, the derivatives of the parametric equations are: [ \frac{dx}{dt} = -te^t - e^t - 2 ] [ \frac{dy}{dt} = 9t^2 - 4 ]
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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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