- #1
Sesse
- 5
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*opps fixing the latex code*
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. :)
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. :)
Last edited: