Matematik opg 204-208 210 218-219 223

Indholdsfortegnelse

1 Opg 204
2 Opg 205
3 Opg 206
4 Opg 207
5 Opg 208
6 Opg 210
7 Opg 218
8 Opg 219
9 Opg 223

Opg 204

$LaTex: s(t)=3t^2, t\geq 0$

Fart i [3;4]: $LaTex: \Delta y = s(4)-s(3) = 3\cdot 4^2-3\cdot 3^2 = 21\\ h = 4-3 = 1\\ a_{sekant} = \frac{\Delta y}{h} = 21\\$

Fart i [3;3,5]: $LaTex: \Delta y = s(3,5)-s(3) = 3\cdot 3,5^2-3\cdot 3^2 = 9,75\\ h = 3,5-3 = 0,5\\ a_{sekant} = \frac{\Delta y}{h} = 19,5\\$

Fart i [2,5;3]: $LaTex: \Delta y = s(3)-s(2,5) = 3\cdot 3^2-3\cdot 2,5^2 = 8,25\\ h = 3-2,5 = 0,5\\ a_{sekant} = \frac{\Delta y}{h} = 16,5\\$

Opg 205

$LaTex: O(x)=-x^2 + 600\cdot x +8000, 0 \leq x \leq 150\\ O(50) = 35500\\ \frac{O(x_0+h)-O(x_0)}{h}$

Gennemsnitlig prisstigning for 50-55: $LaTex: \lim_{h\to 5}\frac{O(50+5)-O(50)}{5} = 495\\$

Grænseværdier for omkostninger ved 50 og 55: $LaTex: \lim_{h\to 0}\frac{O(50+h)-O(50)}{h} = 500\\ \lim_{h\to 0}\frac{O(55+h)-O(55)}{h} = 490\\$

Opg 206

$LaTex: f(x)=2x-1$

Funktions tilvækst: $LaTex: x_0 = 1\\ \Delta y = 2\cdot (h+1) -1 - (2\cdot 1-1) = 2h\\$

Differentialkvotienten: $LaTex: \frac{2h}{h} = 2\\ \lim_{h\to 0}\frac{\Delta y}{h} = 2\\ f'(1)=2\\$

Differentialkvotienten (xo): $LaTex: \Delta y = 2\cdot (x_0+h)-1 -(2\cdot x_0 -1) = 2h\\ \frac{2h}{h} = 2\\ \lim_{h\to 0}\frac{\Delta y}{h} = 2\\ f'(x_0) = 2\\$

Opg 207

$LaTex: f(x) = 2x^2-3x+4$

$LaTex: \Delta y = f(h+2)-f(2) = 2(h+2)^2-3(h+2)+4-(2\cdot 2^2-3\cdot 2+4) = 2h^2+5h\\ \frac{\Delta y}{h} = 2h+5\\ f'(2)=\lim_{h\to 0}\frac{\Delta y}{h} = 5\\$

Opg 208

delopgave 1: $LaTex: f(x)=3x-1\\ x_0=7\\ \frac{f(x_0+h)-f(x_0)}{h} = \frac{3\cdot 7+3h-1-20}{h}=3\\ \lim_{h\to 0}\frac{3h}{h} = 3\\$

delopgave 2: $LaTex: f(x)=x^{-1}\\ x_0 = 2\\ \frac{f(x_0+h)-f(x_o)}{h}=\frac{\frac{1}{2+h}-\frac{1}{2}}{h} = \frac{\left(-\frac{h}{2\cdot h+4}\right)}{h}\\ \lim_{h\to 0}\frac{f(2+h)-f(2)}{h} = -\frac{1}{4}\\$