선형근사와 미분소

Use $y = x^2$; $dy = 2x\cdot dx$ with $x=2$ and $dx = 0.05$. Thus $dy = .2$; knowing $2^2=4$, we have $2.05^2 \approx 4.2$.

Use $y = x^2$; $dy = 2x\cdot dx$ with $x=6$ and $dx = -0.07$. Thus $dy = -0.84$; knowing $6^2=36$, we have $5.93^2 \approx 35.16$.

Use $y = x^3$; $dy = 3x^2\cdot dx$ with $x=5$ and $dx = 0.1$. Thus $dy = 7.5$; knowing $5^3=125$, we have $5.1^3 \approx 132.5$.

Use $y = x^3$; $dy = 3x^2\cdot dx$ with $x=7$ and $dx = -0.2$. Thus $dy = -29.4$; knowing $7^3=343$, we have $6.8^3 \approx 313.6$.

Use $y = \sqrt{x}$; $dy = 1/(2\sqrt{x})\cdot dx$ with $x=16$ and $dx = 0.5$. Thus $dy = .0625$; knowing $\sqrt{16}=4$, we have $\sqrt{16.5} \approx 4.0625$.

Use $y = \sqrt{x}$; $dy = 1/(2\sqrt{x})\cdot dx$ with $x=25$ and $dx = -1$. Thus $dy = -0.1$; knowing $\sqrt{25}=5$, we have $\sqrt{24} \approx 4.9$.

Use $y = \sqrt[3]{x}$; $dy = 1/(3\sqrt[3]{x^2})\cdot dx$ with $x=64$ and $dx = -1$. Thus $dy = -1/48\approx 0.0208$; we could use $-1/48\approx -1/50=-0.02$; knowing $\sqrt[3]{64}=4$, we have $\sqrt[3]{63} \approx 3.98$.

Use $y = \sqrt[3]{x}$; $dy = 1/(3\sqrt[3]{x^2})\cdot dx$ with $x=8$ and $dx = 0.5$. Thus $dy = 1/24\approx 1/25=0.04$; knowing $\sqrt[3]{8}=2$, we have $\sqrt[3]{8.5} \approx 2.04$.

Use $y = \sin x$; $dy = \cos x\cdot dx$ with $x=\pi$ and $dx \approx -0.14$. Thus $dy = 0.14$; knowing $\sin \pi = 0$, we have $\sin 3 \approx 0.14$.

Use $y = e^x$; $dy = e^x\cdot dx$ with $x=0$ and $dx =0.1$. Thus $dy = 0.1$; knowing $e^0=1$, we have $e^{0.1} \approx 1.1$.

$dy = (2x+3)dx$

$dy = (7x^6-5x^4)dx$

$dy = \frac{-2}{4x^3}dx$

$dy = 2(2x+\sin x)(2+\cos x)dx$

$dy = \big(2xe^{3x}+3x^2e^{3x}\big)dx$

$dy = \frac{-16}{x^5}dx$

$dy = \frac{2(\tan x+ 1)-2x\sec^2x}{(\tan x+1)^2}dx$

$dy = \frac{1}{x}dx$

$dy = (e^x\sin x+ e^x\cos x)dx$