1.次の平均変化率を求めなさい。
(1)\(y=-3x+1\)の\(x=0\)から\(x=3\)までの平均変化率
\(\displaystyle \frac{f(3)-f(0)}{3-0}\)
\(\displaystyle =\frac{-8-1}{3}\)
\(=-3\)
(2)\(y=2x\)の\(x=a\)から\(x=b\)までの平均変化率
\(\displaystyle \frac{f(b)-f(a)}{b-a}\)
\(\displaystyle =\frac{2(b-a)}{b-a}\)
\(=2\)
(3)\(y=-x^2\)の\(x=2\)から\(x=2+h\)までの平均変化率
\(\displaystyle \frac{f(2+h)-f(2)}{(2+h)-2}\)
\(\displaystyle =\frac{-h^2-4h}{h}\)
\(=-h-4\)
(4)\(y=x^2-4x\)の\(x=1\)から\(x=4\)までの平均変化率
\(\displaystyle \frac{f(4)-f(1)}{4-1}\)
\(\displaystyle =\frac{0-(-3)}{3}\)
\(=1\)
(5)\(y=x^2-2\)の\(x=-2\)から\(x=1\)までの平均変化率
\(\displaystyle \frac{f(1)-f(-2)}{1-(-2)}\)
\(\displaystyle =\frac{-1-2}{3}\)
\(=-1\)
2.次の極限値を求めなさい。
(1)\(\displaystyle \lim_{x \to 2}(2x-1)\)
\(=3\)
(2)\(\displaystyle \lim_{x \to -1}(3x^2+5x)\)
\(=-2\)
3.次の微分係数を求めなさい。
(1)\(f(x)=2x-3\ \ (x=0)\)
\(\displaystyle f'(0)=\lim_{h \to 0}\frac{f(0+h)-f(0)}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}\frac{2h-3-(-3)}{h}\)
\(\ \ \ \ \ \ \ \ =2\)
(2)\(f(x)=x^2\ \ (x=1)\)
\(\displaystyle f'(1)=\lim_{h \to 0}\frac{f(1+h)-f(1)}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}\frac{h^2+2h}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}(h+2)\)
\(\ \ \ \ \ \ \ \ =2\)
(3)\(f(x)=x^2-4x\ \ (x=3)\)
\(\displaystyle f'(3)=\lim_{h \to 0}\frac{f(3+h)-f(3)}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}\frac{h^2+2h}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}(h+2)\)
\(\ \ \ \ \ \ \ \ =2\)
(4)\(f(x)=x^2-2\ \ (x=-1)\)
\(\displaystyle f'(-1)=\lim_{h \to 0}\frac{f(-1+h)-f(-1)}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}\frac{h^2-2h}{h}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\lim_{h \to 0}(h-2)\)
\(\ \ \ \ \ \ \ \ =-2\)