Aiemmin opiskeltua

sös

1.
a)

$2\cdot\left(-3\right)^2=2\cdot9=18$

$-2^4=-16$

$4\cdot2^{-3}=4\cdot\frac{1}{8}=\frac{1}{2}$

2.
a) $x^5+5x=x\left(x^4+5\right)$
b) $4x^3+2x^2=2x^2\left(2x+1\right)$
c) $2\left(x+1\right)+4x\left(x+1\right)=\left(x+1\right)\left(2+4x\right)$

3.
a) f(3)=2
b) f'(3)=1
c) f'(x)=0, kun x=1 tai x=4
d) f'(x)>0, kun $1<x<4$

4.
a)
$f\left(x\right)=x^3+x$

$f'\left(x\right)=3x^2+1$

$f\left(x\right)=\frac{1}{x^4}{,}\ x\ne0$
$f\left(x\right)=x^{-4}$
$f'\left(x\right)=-4x^{-5}=-\frac{4}{x^5}$

$f\left(x\right)=x^2\sin x$
$f'\left(x\right)=2x\sin x+x^2\cos x$

5.
a)
$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$
$f\left(x\right)=\left(-3x+2\right)^{10}$
$u\left(x\right)=x^{10}$
$s\left(x\right)=-3x+2$
$u'\left(x\right)=10x^9$
$s'\left(x\right)=-3$
$D\left(u\left(s\left(x\right)\right)\right)=-30\left(-3x+2\right)^9$

b)
$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$
$f\left(x\right)=\left(2x^2-3\right)^4$
$u\left(x\right)=x^4$
$s\left(x\right)=2x^2-3$
$u'\left(x\right)=4x^3$
$s'\left(x\right)=4x$
$D\left(u\left(s\left(x\right)\right)\right)=4\left(2x^2-3\right)^3\cdot4x$

6.
a)

$\sqrt{49}=\pm7$
b)

$\sqrt[3]{-8}=-2$

$\sqrt[4]{16}=\pm2$

$\sqrt{8}=8^{\frac{1}{2}}$

$\sqrt{a^2}=\left|a\right|$

7.
a)
2
b)
$\sqrt{0{,}1}$
c)
$\sqrt{x+1}$

8.
a)
$2^x=32$
$32=2\cdot2\cdot2\cdot2\cdot2=2^5$
$x=5$
b)
$8^x=4^6$
$4^6=4096$
$4096=8\cdot8\cdot8\cdot8=8^4$
$x=4$
c)
$2^x\cdot5^x=1000$
$2^x\cdot5^x=10^x$
$1000=10^x$
$1000=10\cdot10\cdot10=10^3$
$x=3$