How do you differentiate the following parametric equation: # x(t)=1/t, y(t)=lnt #?
and if you like
we can check this by de-parameterising the equation
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To differentiate the given parametric equations ( x(t) = \frac{1}{t} ) and ( y(t) = \ln(t) ) with respect to ( t ), you would differentiate each equation separately using the chain rule and the rules of differentiation for natural logarithm.
( \frac{dx}{dt} = \frac{d}{dt}\left(\frac{1}{t}\right) = -\frac{1}{t^2} )
( \frac{dy}{dt} = \frac{d}{dt}\left(\ln(t)\right) = \frac{1}{t} )
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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.
- For #f(t)= (sin^2t,cos^2t)# what is the distance between #f(pi/4)# and #f(pi)#?
- What is the arclength of #f(t) = (cos2t-sin2t,tan^2t)# on #t in [pi/12,(5pi)/12]#?
- What is the arclength of #f(t) = (sin^2t-cos2t,t/pi)# on #t in [-pi/4,pi/4]#?
- What is the slope of #f(t) = (t^2+2t,2t-3)# at #t =-1#?
- What is the derivative of #f(t) = (e^(t^2-1)+3t, -t^3+t ) #?

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