Solve the systems equations for x and y and check:<br> y=2x+10<br> x=3y

$\bf \textit{circumference of a semi-circle}\\\\
C=\pi r\qquad 
\begin{cases}
r=radius\\
-----\\
C=5.14
\end{cases}\implies 5.14=\pi r\implies \boxed{\cfrac{5.14}{\pi }=r}\\\\
-------------------------------\\\\
\textit{area of a semi-circle}\\\\
A=\cfrac{\pi r^2}{2} \implies A=\cfrac{\pi }{2}r^2\implies A=\cfrac{\pi }{2}\left( \boxed{\cfrac{5.14}{\pi }} \right)^2
\\\\\\
A=\cfrac{\pi }{2}\implies \cfrac{5.14^2}{\pi ^2}\implies A=\cfrac{26.4196}{2\pi }\implies A=\cfrac{13.2098}{\pi }$

5 0

3 years ago

Derivative, by first principle<br><img src="https://tex.z-dn.net/?f=%20%5Ctan%28%20%5Csqrt%7Bx%20%7D%20%29%20" id="TexFormula1"

vampirchik [111]

$\displaystyle\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan x}h$

Employ a standard trick used in proving the chain rule:

$\dfrac{\tan\sqrt{x+h}-\tan x}{\sqrt{x+h}-\sqrt x}\cdot\dfrac{\sqrt{x+h}-\sqrt x}h$

The limit of a product is the product of limits, i.e. we can write

$\displaystyle\left(\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan x}{\sqrt{x+h}-\sqrt x}\right)\cdot\left(\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt x}h\right)$

The rightmost limit is an exercise in differentiating $\sqrt x$ using the definition, which you probably already know is $\dfrac1{2\sqrt x}$ .

For the leftmost limit, we make a substitution $y=\sqrt x$ . Now, if we make a slight change to $x$ by adding a small number $h$ , this propagates a similar small change in $y$ that we'll call $h'$ , so that we can set $y+h'=\sqrt{x+h}$ . Then as $h\to0$ , we see that it's also the case that $h'\to0$ (since we fix $y=\sqrt x$ ). So we can write the remaining limit as

$\displaystyle\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan\sqrt x}{\sqrt{x+h}-\sqrt x}=\lim_{h'\to0}\frac{\tan(y+h')-\tan y}{y+h'-y}=\lim_{h'\to0}\frac{\tan(y+h')-\tan y}{h'}$

which in turn is the derivative of $\tan y$ , another limit you probably already know how to compute. We'd end up with $\sec^2y$ , or $\sec^2\sqrt x$ .

So we find that

$\dfrac{\mathrm d\tan\sqrt x}{\mathrm dx}=\dfrac{\sec^2\sqrt x}{2\sqrt x}$

7 0

3 years ago

Solve the systems equations for x and y and check: y=2x+10 x=3y