Find 4 consecutive even integers whose sum is 244

1 answer:

5 0

X, x+2, x+4, x+6 - four consecutive even integers

$x+x+2+x+4+x+6=244 \\
4x+12=244 \\
4x=244-12 \\
4x=232 \\
x=\frac{232}{4} \\
x=58 \Leftarrow \hbox{the first integer} \\ \\
\hbox{the second integer: } x+2=58+2=60 \\ \\
\hbox{the third integer: } x+4=58+4=62 \\ \\
\hbox{the fourth integer: } x+6=58+6=64$

The integers are 58, 60, 62, 64.

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Limit question: lim x-->pi ((e^sinx)-1)/(x-pi)

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$\displaystyle\lim_{x\to\pi}\dfrac{e^{\sin x}-1}{x-\pi}$

Notice that if $f(x)=e^{\sin x}$ , then $f(\pi)=e^{\sin\pi}=e^0=1$ . Recall the definition of the derivative of a function $f(x)$ at a point $x=c$ :

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So the value of this limit is exactly the value of the derivative of $f(x)=e^{\sin x}$ at $x=\pi$ .

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$f'(x)=\cos x\,e^{\sin x}\implies f'(\pi)=\cos\pi\,e^{\sin\pi}=-1$