Graph sketching (differentiation)

(1) Given the curve x=6t^2 , y=t^3 -2t where t is not = 0, show that dy/dx = (t^2 -1)/4t. Hence, find the point in the curve at which the tangents to the curve and parallel to the x-axis.

(2) y= (2x-5)/(x^2 -4) . Prove that y>=1 or y<= for all real values of x.

(3) Sketch the curve of y^2 = x^2 (1-x).

Please show me the steps about how can i solve these. Especially about the graph sketching part. Thanks.

Re: Graph sketching (differentiation)

There are many mistakes in your post

anyway to find the dy/dx differentiate separate first and find dy/dt then dx/dt and then find dy/dx=(dy/dt)/(dx/dt) . verify your post because you may have typed wrongly the parametric equations. anyway if you do corect the calculations you will find the dy/dx.

For the second question something is missing...the curve y=(2x-5)/(x^2-4) has as domain R-{-2,+2} and has 3 branches. The part of the curve which is defined in the domain (-2,+2) has range y biger or equal to 1.... with the lines x=-2 and x=+2 to be the vertical asymptotes .

I don't understand what you want...

For the last question try to find a graph calculator to see the shape of the graph and then try to graph it.

Re: Graph sketching (differentiation)

For (1), you need to use $\displaystyle \frac{dy}{dx}=\frac{dy/dt}{dx/dt}$. And of course, the curve is parallel to the x-axis when $\displaystyle \frac{dy}{dx}=0$.

For (2), you kind of need to sketch the graph to figure out what's going on. The numerator is positive for $\displaystyle x>\frac{5}{2}$ and negative for $\displaystyle x<\frac{5}{2}$. You can do the same thing for the denominator and then figure out what the function's doing.

For (3), you might start by describing what $\displaystyle x^2(1-x)$ does in the ranges $\displaystyle -\infty<x<0$, $\displaystyle 0<x<1$, and $\displaystyle 1<x<\infty$. If, for example, it is negative, then there will be no solutions with those values of x.

- Hollywood