3.2 Numerinen derivointi

322
a)
h=-0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(1+\left(-0{,}5\right)\right)-f\left(1\right)}{-0{,}5}=\frac{65-55}{-0{,}5}=-20\ \frac{°C}{\min}=20\ \frac{°C}{\min}
h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(1+\left(0{,}5\right)\right)-f\left(1\right)}{0{,}5}=\frac{47-55}{0{,}5}=\frac{-8}{0{,}5}=-16\ \frac{°C}{\min}=16\ \frac{°C}{\min}
b)
h=-0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}5\right)\right)-f\left(2\right)}{-0{,}5}=\frac{47-40}{-0{,}5}=-14\ \frac{°C}{\min}=14\ \frac{°C}{\min}
h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(0{,}5\right)\right)-f\left(2\right)}{0{,}5}=\frac{36-40}{0{,}5}=\frac{-4}{0{,}5}=-8\ \frac{°C}{\min}=8\ \frac{°C}{\min}
c)
h=-0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(4+\left(-0{,}5\right)\right)-f\left(4\right)}{-0{,}5}=\frac{30-28}{-0{,}5}=-4\ \frac{°C}{\min}=4\ \frac{°C}{\min}
h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(4+\left(0{,}5\right)\right)-f\left(4\right)}{0{,}5}=\frac{27-28}{0{,}5}=\frac{-1}{0{,}5}=-2\ \frac{°C}{\min}=2\ \frac{°C}{\min}
 
324
h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx0.66816
h=0{,}01;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx0.67104
h=0{,}001;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx0.67107
 
325
h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}=\frac{3.5-2.4}{1}=1.1\ \frac{m}{s}
 
326
a)
h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+0{,}1\right)-f\left(2\right)}{0{,}1}=7.49638...\approx7.4964
h=0{,}01;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+0{,}01\right)-f\left(2\right)}{0{,}01}=6.840402...\approx6.8404
 h=0{,}001;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+0{,}001\right)-f\left(2\right)}{0{,}001}=6.77932...\approx6.7793
b)
h=-0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}1\right)\right)-f\left(2\right)}{-0{,}1}=6.14429...\approx6.1443
h=-0{,}01;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}01\right)\right)-f\left(2\right)}{-0{,}01}=6.70572...\approx6.7057 
h=-0{,}001;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}001\right)\right)-f\left(2\right)}{-0{,}001}=6.76585...\approx6.7659 
c)
h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx6.8203
h=0{,}01;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx6.7731
h=0{,}001;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx6.7726¨
d)
f'\left(2\right)\approx6.77259

329
h=0{,}1;\ \approx-0.3466
h=0{,}01;\ \approx-0.3466
h=0{,}001;\ \approx-0.3466
f'\left(x\right)=2^{-x}\cdot\left(-\ln2\right)
f'\left(1\right)=2^{-1}\cdot\left(-\ln2\right)=−0.34657...\approx-0{,}3466

331
Raja-arvo ei ole
 
332
f\left(x\right)=e^x
f'\left(x\right)=e^x
f'\left(2\right)=e^2

333
h=-0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(0+\left(-0{,}1\right)\right)-f\left(0\right)}{-0{,}1}\approx−1.0488 
h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(0+0{,}1\right)-f\left(0\right)}{0{,}1}\approx0.9487
h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx-0.0501 
Ei näytä oleva derivoituva

340
f'\left(0.5\right)\approx0.877586521890373
p=3;\ \approx0.877582415626633
p=4;\ \approx0.877582560427638
p=5;\ \approx0.87758256187287
p=6;\ \approx0.87758256189785
p=7;\ \approx0.877582561620295
p=8;\ \approx0.877582562175406
p=9;\ \approx0.877582562175405
p=10;\ \approx0.877582451153103
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