*opps fixing the latex code*(adsbygoogle = window.adsbygoogle || []).push({});

I have a test coming up so I tried solving some review questions but couldn't solve all of them. I will also post the ones which I am not sure whether they are correct or wrong so please spare some time to check them too.

1.

[tex] F(x)=

\left{

\begin{array}{cc}

-x, & \mbox{ if } x<-1\\

x^2, & \mbox{ if } -1 \leq x<1\\

\frac{1}{x}, & \mbox{ if } x\geq 1

\end{array}

\right

[/tex]

Use limit definition of the derivative to see if the fuction is differentiable at [tex]x=-1[/tex] and [tex]x=1[/tex]

At [tex]x=-1[/tex]

[tex]

\lim_{x\rightarrow -1^-} \frac{f(x)-f(-1)}{x-(-1)}=-1[/tex]

[tex]\lim_{x\rightarrow -1^+} \frac{f(x)+1}{x+1} = \frac{(x+1)(x-1)}{x+1}=x-1=-2[/tex]

If this solution for [tex]x=-1[/tex] is correct then it is also correct for [tex]x=1[/tex] so I won't show my solution for that.

So the fuction is not differentiable at both points?

2. Probably correct.

3. [tex]y=(t^{2/3}).e^{sin(t)}+t[/tex]

Find the slope at (0,0)

[tex]\frac{dy}{dx}=\left(\frac{2}{3}t^{-1/3}e^{sin(t)}+t^{2/3}e^{sin(t)}cos(t)\right)+1[/tex]

[tex]\frac{dy}{dx}=1?[/tex]

More will come. :)

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# Limits and Imp/Explicit Differentiation

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