"울프람알파의 활용"의 두 판 사이의 차이
(피타고라스님이 이 페이지의 이름을 울프람알파의 활용로 바꾸었습니다.) |
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| 5번째 줄: | 5번째 줄: | ||
| − | + | <h5>문제풀이와 울프람알파</h5> | |
| + | |||
| + | * [http://blog.wolframalpha.com/2009/12/01/step-by-step-math/ Step-by-Step Math] | ||
| 11번째 줄: | 13번째 줄: | ||
| − | <h5 | + | <h5>그래프 그리기</h5> |
'''10.3. 56''' | '''10.3. 56''' | ||
| 23번째 줄: | 25번째 줄: | ||
[/pages/4176465/attachments/2110527 MSP74719784071iad69b2900005cid5ibigi61dg94.gif] | [/pages/4176465/attachments/2110527 MSP74719784071iad69b2900005cid5ibigi61dg94.gif] | ||
| − | + | [http://www.wolframalpha.com/input/?i=r%3Dcos+%28theta/3%29 http://www.wolframalpha.com/input/?i=r%3Dcos+(theta/3)] | |
| 33번째 줄: | 35번째 줄: | ||
[/pages/4176465/attachments/2110537 MSP11271978443hf7aa7h2f00002bd5bh2db919dc1f.gif] | [/pages/4176465/attachments/2110537 MSP11271978443hf7aa7h2f00002bd5bh2db919dc1f.gif] | ||
| − | + | [http://www.wolframalpha.com/input/?i=r%3D1%2B2+cos+%28theta%29 http://www.wolframalpha.com/input/?i=r%3D1%2B2+cos+(theta)] | |
(e) | (e) | ||
| 39번째 줄: | 41번째 줄: | ||
<math>r=2+\sin 3\theta</math> | <math>r=2+\sin 3\theta</math> | ||
| − | http://www.wolframalpha.com/input/?i=r%3D2%2Bsin(3thetA) | + | [http://www.wolframalpha.com/input/?i=r%3D2%2Bsin%283thetA%29 http://www.wolframalpha.com/input/?i=r%3D2%2Bsin(3thetA)] |
[/pages/4176465/attachments/2110539 MSP47019784919haafi5hc00002h164hgd33f4i46i.gif] | [/pages/4176465/attachments/2110539 MSP47019784919haafi5hc00002h164hgd33f4i46i.gif] | ||
| 47번째 줄: | 49번째 줄: | ||
<math>r=1+2\sin 3\theta</math> | <math>r=1+2\sin 3\theta</math> | ||
| − | + | [http://www.wolframalpha.com/input/?i=r%3D1%2B2sin%283theta%29 http://www.wolframalpha.com/input/?i=r%3D1%2B2sin(3theta)] | |
[/pages/4176465/attachments/2110535 MSP6981978475e4574h83h00003a7ga0a9g5d06c13.gif] | [/pages/4176465/attachments/2110535 MSP6981978475e4574h83h00003a7ga0a9g5d06c13.gif] | ||
| 59번째 줄: | 61번째 줄: | ||
'''10.4.44.''' | '''10.4.44.''' | ||
| − | http://www.wolframalpha.com/input/?i=(r-(8%2B8sin+theta))(r+sin+theta+-4)%3D0 | + | [http://www.wolframalpha.com/input/?i=%28r-%288%2B8sin+theta%29%29%28r+sin+theta+-4%29%3D0 http://www.wolframalpha.com/input/?i=(r-(8%2B8sin+theta))(r+sin+theta+-4)%3D0] |
[/pages/4176465/attachments/2110589 1MSP761197847h1hg25e7he00004g3ehh9ag590ea81.gif] | [/pages/4176465/attachments/2110589 1MSP761197847h1hg25e7he00004g3ehh9ag590ea81.gif] | ||
| 85번째 줄: | 87번째 줄: | ||
<math>48\pi-96\alpha+128\cos \alpha+16\sin 2\alpha-(32\cos \alpha+16 \sin 2\alpha)=48\pi-96\alpha+96\cos \alpha\approx 204.16</math> | <math>48\pi-96\alpha+128\cos \alpha+16\sin 2\alpha-(32\cos \alpha+16 \sin 2\alpha)=48\pi-96\alpha+96\cos \alpha\approx 204.16</math> | ||
| − | http://www.wolframalpha.com/input/?i=N[48pi+-96(arcsin{(sqrt+3+-+1)/2})%2B96(sqrt((sqrt+3)/2)),10] | + | [http://www.wolframalpha.com/input/?i=N%5B48pi+-96%28arcsin%7B%28sqrt+3+-+1%29/2%7D%29%2B96%28sqrt%28%28sqrt+3%29/2%29%29,10%5D http://www.wolframalpha.com/input/?i=N[48pi+-96(arcsin{(sqrt+3+-+1)/2})%2B96(sqrt((sqrt+3)/2)),10]] |
| 99번째 줄: | 101번째 줄: | ||
[/pages/4176465/attachments/2110541 MSP71919784049c09bb98700001095e8fadhf5gc4d.gif] | [/pages/4176465/attachments/2110541 MSP71919784049c09bb98700001095e8fadhf5gc4d.gif] | ||
| − | http://www.wolframalpha.com/input/?i=r^2%3Dcos+(2theta) | + | [http://www.wolframalpha.com/input/?i=r%5E2%3Dcos+%282theta%29 http://www.wolframalpha.com/input/?i=r^2%3Dcos+(2theta)] |
| 105번째 줄: | 107번째 줄: | ||
| − | <h5 style="line-height: 3.428em | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">사용예</h5> |
| − | * http://www.wolframalpha.com/input/?i=r^2<br> | + | * [http://www.wolframalpha.com/input/?i=r%5E2 http://www.wolframalpha.com/input/?i=r^2]<br> |
* http://www.wolframalpha.com/input/?i=<br> | * http://www.wolframalpha.com/input/?i=<br> | ||
* http://www.wolframalpha.com/input/?i=<br> | * http://www.wolframalpha.com/input/?i=<br> | ||
* http://www.wolframalpha.com/input/?i=<br> | * http://www.wolframalpha.com/input/?i=<br> | ||
| − | < | + | |
| + | |||
| + | <h5>관련된 항목들</h5> | ||
| + | |||
| + | * [[#]] | ||
2009년 12월 30일 (수) 04:09 판
- 울프람 알파
- 연산능력을 갖춘 지식엔진
- http://www.wolframalpha.com
문제풀이와 울프람알파
그래프 그리기
10.3. 56
(c)
\(r=\cos \frac{\theta}{3}\)
[/pages/4176465/attachments/2110527 MSP74719784071iad69b2900005cid5ibigi61dg94.gif]
http://www.wolframalpha.com/input/?i=r%3Dcos+(theta/3)
(d)
\(r=1+2\cos\theta\)
[/pages/4176465/attachments/2110537 MSP11271978443hf7aa7h2f00002bd5bh2db919dc1f.gif]
http://www.wolframalpha.com/input/?i=r%3D1%2B2+cos+(theta)
(e)
\(r=2+\sin 3\theta\)
http://www.wolframalpha.com/input/?i=r%3D2%2Bsin(3thetA)
[/pages/4176465/attachments/2110539 MSP47019784919haafi5hc00002h164hgd33f4i46i.gif]
(f)
\(r=1+2\sin 3\theta\)
http://www.wolframalpha.com/input/?i=r%3D1%2B2sin(3theta)
[/pages/4176465/attachments/2110535 MSP6981978475e4574h83h00003a7ga0a9g5d06c13.gif]
10.4.44.
http://www.wolframalpha.com/input/?i=(r-(8%2B8sin+theta))(r+sin+theta+-4)%3D0
[/pages/4176465/attachments/2110589 1MSP761197847h1hg25e7he00004g3ehh9ag590ea81.gif]
Note that \(2\sin^2 \alpha +2\sin \alpha-1=0\).
Since \(0< \alpha <\frac{\pi}{2}\), \(\sin \alpha =\frac{\sqrt 3 -1}{2}\) and \(\cos \alpha =\sqrt{\frac{\sqrt 3}{2}}\)
\(\frac{1}{2}\int_{\alpha}^{\pi-\alpha}r^2 d\theta=\int_{\alpha}^{\pi/2}r^2 d\theta=\int_{\alpha}^{\pi/2}64(1+\sin\theta)^2d\theta\)
\(=64\int_{\alpha}^{\pi/2}(1+\sin\theta)^2d\theta=64\int_{\alpha}^{\pi/2}1+2\sin\theta+\sin^2\theta d\theta\)
\(=64[\frac{3}{2}\theta - 2 \cos \theta- \frac{1}{4}\sin 2\theta]_{\alpha}^{\pi/2}=[96\theta - 128 \cos \theta- 16\sin 2\theta]_{\alpha}^{\pi/2}\)
\(=[96\theta - 128 \cos \theta- 16\sin 2\theta]_{\alpha}^{\pi/2}=48\pi-96\alpha+128\cos \alpha+16\sin 2\alpha\)
We need to subtract from this the area of the triangle which is given by \[2\times \frac{1}{2}\cdot 4 (8+8\sin\alpha)\sin (\pi/2-\alpha)=32(1+\sin\alpha)\cos \alpha=32\cos \alpha+16 \sin 2\alpha\]
So the area will be given by \[48\pi-96\alpha+128\cos \alpha+16\sin 2\alpha-(32\cos \alpha+16 \sin 2\alpha)=48\pi-96\alpha+96\cos \alpha\approx 204.16\]
http://www.wolframalpha.com/input/?i=N[48pi+-96(arcsin{(sqrt+3+-+1)/2})%2B96(sqrt((sqrt+3)/2)),10]
10.4.55.
(b)
\(r^2=\cos 2\theta\)
[/pages/4176465/attachments/2110541 MSP71919784049c09bb98700001095e8fadhf5gc4d.gif]
http://www.wolframalpha.com/input/?i=r^2%3Dcos+(2theta)
사용예
- http://www.wolframalpha.com/input/?i=r^2
- http://www.wolframalpha.com/input/?i=
- http://www.wolframalpha.com/input/?i=
- http://www.wolframalpha.com/input/?i=