text: Está en la página 1de 4
  
  # 𝐷𝑎𝑑𝑎 𝑙𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝑓(𝑥) = −0.1𝑥 4 − 0.15𝑥 3 − 0.5𝑥 2 − 0.25𝑥 + 1.2.
  
  ## 𝑈𝑠𝑒 𝑙𝑎𝑠 𝑎𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑐𝑖𝑜𝑛𝑒𝑠 𝑑𝑒 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖𝑎𝑠
  
  𝑓𝑖𝑛𝑖𝑡𝑎𝑠 ℎ𝑎𝑐𝑖𝑎 𝑎𝑑𝑒𝑙𝑎𝑛𝑡𝑒, ℎ𝑎𝑐𝑖𝑎 𝑎𝑡𝑟á𝑠 𝑦 𝑐𝑒𝑛𝑡𝑟𝑎𝑑𝑎𝑠 𝑝𝑎𝑟𝑎 𝑒𝑠𝑡𝑖𝑚𝑎𝑟 𝑙𝑎 𝑝𝑟𝑖𝑚𝑒𝑟𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎.
  
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒑𝒓𝒊𝒎𝒆𝒓𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟐 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 − ℎ) = 𝑓(0) = −0.1(0)4 − 0.15(0)3 − 0.5(0)2 − 0.25(0) + 1.2 = 𝟏. 𝟐
  
  𝑓(𝑥0 ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  
  𝑓(0.5) − 𝑓(0)
  𝑓´(0.5) =
  0.5
  0.925 − 1.2
  𝑓´(0.5) = = −𝟎. 𝟓𝟓
  0.5
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(1) = −0.1(1)4 − 0.15(1)3 − 0.5(1)2 − 0.25(1) + 1.2 = 𝟎. 𝟐
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  2ℎ
  𝑓(1) − 𝑓(0)
  𝑓´(0.5) =
  2(0.5)
  0.2 − 1.2
  𝑓´(0.5) = = −𝟏. 𝟎
  2(0.5)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 ) = 𝑓(0.5) = −0.1(0.5)4 − 0.15(0.5)3 − 0.5(0.5)2 − 0.25(0.5) + 1.2 = 𝟎. 𝟗𝟐𝟓
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 )
  𝑓´(𝑥0 ) =
  
  𝑓(1) − 𝑓(0.5)
  𝑓´(0.5) =
  0.5
  0.2 − 0.925
  𝑓´(0.5) = = −𝟏. 𝟒𝟓
  0.5
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒑𝒓𝒊𝒎𝒆𝒓𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟐 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟐𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 − ℎ) = 𝑓(0.25) = −0.1(0.25)4 − 0.15(0.25)3 − 0.5(0.25)2 − 0.25(0.25) + 1.2 = 𝟏. 𝟏𝟎𝟑𝟓𝟏𝟓𝟔𝟐𝟓
  
  𝑓(𝑥0 ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  
  𝑓(0.5) − 𝑓(0.25)
  𝑓´(0.5) =
  0.5
  0.925 − 1.103515625
  𝑓´(0.5) = = −𝟎. 𝟕𝟏𝟒𝟎𝟔𝟐𝟓
  0.25
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(0.75) = −0.1(0.75)4 − 0.15(0.75)3 − 0.5(0.75)2 − 0.25(0.75) + 1.2 = 𝟎. 𝟔𝟑𝟔𝟑𝟐𝟖𝟏𝟐𝟓
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  2ℎ
  𝑓(0.75) − 𝑓(0.25)
  𝑓´(0.5) =
  2(0.25)
  1.103515625 − 0.636328125
  𝑓´(0.5) = = −𝟎. 𝟗𝟑𝟒𝟑𝟕𝟓
  2(0.25)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 ) = 𝑓(0.5) = −0.1(0.5)4 − 0.15(0.5)3 − 0.5(0.5)2 − 0.25(0.5) + 1.2 = 𝟎. 𝟗𝟐𝟓
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 )
  𝑓´(𝑥0 ) =
  
  𝑓(0.75) − 𝑓(0.5)
  𝑓´(0.5) =
  0.25
  0.636328125 − 0.925
  𝑓´(0.5) = = −𝟏. 𝟏𝟓𝟒𝟒𝟕𝟓
  0.25
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒔𝒆𝒈𝒖𝒏𝒅𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟑 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  𝑓´(𝑥0 ) =
  2ℎ
  
  ## 𝑓(−0.5) − 4𝑓(0) + 3𝑓(0.5)
  
  𝑓´(0.5) =
  2(0.5)
  1.2125 − 4(1.2) + 3(0.925)
  𝑓´(0.5) = = −𝟎. 𝟖𝟏𝟐𝟓
  2(0.5)
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(1) = −0.1(1)4 − 0.15(1)3 − 0.5(1)2 − 0.25(1) + 1.2 = 𝟎. 𝟐
  
  𝑓(𝑥0 + 2ℎ) = 𝑓(1.5) = −0.1(1.5)4 − 0.15(1.5)3 − 0.5(1.5)2 − 0.25(1.5) + 1.2 = −𝟏. 𝟑𝟏𝟐𝟓
  
  ## 𝑓(𝑥0 − 2ℎ) − 8𝑓(𝑥0 − ℎ) + 8𝑓(𝑥0 + ℎ) + 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  12ℎ
  𝑓(−0.5) − 8𝑓(0) + 8𝑓(1) − 𝑓(1.5)
  𝑓´(0.5) =
  12(0.5)
  1.2125 − 8(1.2) + 8(0.2) − (−1.3125)
  𝑓´(0.5) = = −𝟎. 𝟗𝟏𝟐𝟓
  12(0.5)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(1) = −0.1(1)4 − 0.15(1)3 − 0.5(1)2 − 0.25(1) + 1.2 = 𝟎. 𝟐
  
  𝑓(𝑥0 + 2ℎ) = 𝑓(1.5) = −0.1(1.5)4 − 0.15(1.5)3 − 0.5(1.5)2 − 0.25(1.5) + 1.2 = −𝟏. 𝟑𝟏𝟐𝟓
  
  ## −3𝑓(𝑥0 ) + 4𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  2ℎ
  −3𝑓(0.5) + 4𝑓(1) − 𝑓(1.5)
  𝑓´(0.5) =
  2(0.5)
  −3(0.925) + 4(0.2) − (−1.3125)
  𝑓´(0.5) = = −𝟎. 𝟔𝟔𝟐𝟓
  2(0.5)
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒔𝒆𝒈𝒖𝒏𝒅𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟑 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟐𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  𝑓´(𝑥0 ) =
  2ℎ
  
  ## 𝑓(0) − 4𝑓(0.25) + 3𝑓(0.5)
  
  𝑓´(0.5) =
  2(0.25)
  1.2 − 4(1.103515625) + 3(0.925)
  𝑓´(0.5) = = −𝟎. 𝟖𝟕𝟖𝟏𝟐𝟓
  2(0.25)
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 − 2ℎ) − 8𝑓(𝑥0 − ℎ) + 8𝑓(𝑥0 + ℎ) + 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  12ℎ
  𝑓(0) − 8𝑓(0.25) + 8𝑓(0.75) − 𝑓(1)
  𝑓´(0.5) =
  12(0.25)
  1.2 − 8(1.103515625) + 8(0.636328125) − 0.2
  𝑓´(0.5) = = −𝟎. 𝟗𝟏𝟐𝟓
  12(0.25)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## −3𝑓(𝑥0 ) + 4𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  2ℎ
  −3𝑓(0.5) + 4𝑓(0.75) − 𝑓(1)
  𝑓´(0.5) =
  2(0.25)
  −3(0.925) + 4(0.636328125) − 0.2
  𝑓´(0.5) = = −𝟎. 𝟖𝟓𝟗𝟑𝟕𝟓
  2(0.25)
explanation: The document contains a math problem that involves calculating the first derivative of a given function using different finite difference methods (forward, backward, and central differences) for two different step sizes (h = 0.5 and h = 0.25). The document also includes the detailed calculations and results for each method.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Está en la página 1de 4
  
  # 𝐷𝑎𝑑𝑎 𝑙𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝑓(𝑥) = −0.1𝑥 4 − 0.15𝑥 3 − 0.5𝑥 2 − 0.25𝑥 + 1.2.
  
  ## 𝑈𝑠𝑒 𝑙𝑎𝑠 𝑎𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑐𝑖𝑜𝑛𝑒𝑠 𝑑𝑒 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖𝑎𝑠
  
  𝑓𝑖𝑛𝑖𝑡𝑎𝑠 ℎ𝑎𝑐𝑖𝑎 𝑎𝑑𝑒𝑙𝑎𝑛𝑡𝑒, ℎ𝑎𝑐𝑖𝑎 𝑎𝑡𝑟á𝑠 𝑦 𝑐𝑒𝑛𝑡𝑟𝑎𝑑𝑎𝑠 𝑝𝑎𝑟𝑎 𝑒𝑠𝑡𝑖𝑚𝑎𝑟 𝑙𝑎 𝑝𝑟𝑖𝑚𝑒𝑟𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎.
  
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒑𝒓𝒊𝒎𝒆𝒓𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟐 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 − ℎ) = 𝑓(0) = −0.1(0)4 − 0.15(0)3 − 0.5(0)2 − 0.25(0) + 1.2 = 𝟏. 𝟐
  
  𝑓(𝑥0 ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  
  𝑓(0.5) − 𝑓(0)
  𝑓´(0.5) =
  0.5
  0.925 − 1.2
  𝑓´(0.5) = = −𝟎. 𝟓𝟓
  0.5
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(1) = −0.1(1)4 − 0.15(1)3 − 0.5(1)2 − 0.25(1) + 1.2 = 𝟎. 𝟐
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  2ℎ
  𝑓(1) − 𝑓(0)
  𝑓´(0.5) =
  2(0.5)
  0.2 − 1.2
  𝑓´(0.5) = = −𝟏. 𝟎
  2(0.5)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 ) = 𝑓(0.5) = −0.1(0.5)4 − 0.15(0.5)3 − 0.5(0.5)2 − 0.25(0.5) + 1.2 = 𝟎. 𝟗𝟐𝟓
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 )
  𝑓´(𝑥0 ) =
  
  𝑓(1) − 𝑓(0.5)
  𝑓´(0.5) =
  0.5
  0.2 − 0.925
  𝑓´(0.5) = = −𝟏. 𝟒𝟓
  0.5
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒑𝒓𝒊𝒎𝒆𝒓𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟐 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟐𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 − ℎ) = 𝑓(0.25) = −0.1(0.25)4 − 0.15(0.25)3 − 0.5(0.25)2 − 0.25(0.25) + 1.2 = 𝟏. 𝟏𝟎𝟑𝟓𝟏𝟓𝟔𝟐𝟓
  
  𝑓(𝑥0 ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  
  𝑓(0.5) − 𝑓(0.25)
  𝑓´(0.5) =
  0.5
  0.925 − 1.103515625
  𝑓´(0.5) = = −𝟎. 𝟕𝟏𝟒𝟎𝟔𝟐𝟓
  0.25
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(0.75) = −0.1(0.75)4 − 0.15(0.75)3 − 0.5(0.75)2 − 0.25(0.75) + 1.2 = 𝟎. 𝟔𝟑𝟔𝟑𝟐𝟖𝟏𝟐𝟓
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 − ℎ)
  𝑓´(𝑥0 ) =
  2ℎ
  𝑓(0.75) − 𝑓(0.25)
  𝑓´(0.5) =
  2(0.25)
  1.103515625 − 0.636328125
  𝑓´(0.5) = = −𝟎. 𝟗𝟑𝟒𝟑𝟕𝟓
  2(0.25)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 ) = 𝑓(0.5) = −0.1(0.5)4 − 0.15(0.5)3 − 0.5(0.5)2 − 0.25(0.5) + 1.2 = 𝟎. 𝟗𝟐𝟓
  
  𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 )
  𝑓´(𝑥0 ) =
  
  𝑓(0.75) − 𝑓(0.5)
  𝑓´(0.5) =
  0.25
  0.636328125 − 0.925
  𝑓´(0.5) = = −𝟏. 𝟏𝟓𝟒𝟒𝟕𝟓
  0.25
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒔𝒆𝒈𝒖𝒏𝒅𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟑 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  𝑓´(𝑥0 ) =
  2ℎ
  
  ## 𝑓(−0.5) − 4𝑓(0) + 3𝑓(0.5)
  
  𝑓´(0.5) =
  2(0.5)
  1.2125 − 4(1.2) + 3(0.925)
  𝑓´(0.5) = = −𝟎. 𝟖𝟏𝟐𝟓
  2(0.5)
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(1) = −0.1(1)4 − 0.15(1)3 − 0.5(1)2 − 0.25(1) + 1.2 = 𝟎. 𝟐
  
  𝑓(𝑥0 + 2ℎ) = 𝑓(1.5) = −0.1(1.5)4 − 0.15(1.5)3 − 0.5(1.5)2 − 0.25(1.5) + 1.2 = −𝟏. 𝟑𝟏𝟐𝟓
  
  ## 𝑓(𝑥0 − 2ℎ) − 8𝑓(𝑥0 − ℎ) + 8𝑓(𝑥0 + ℎ) + 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  12ℎ
  𝑓(−0.5) − 8𝑓(0) + 8𝑓(1) − 𝑓(1.5)
  𝑓´(0.5) =
  12(0.5)
  1.2125 − 8(1.2) + 8(0.2) − (−1.3125)
  𝑓´(0.5) = = −𝟎. 𝟗𝟏𝟐𝟓
  12(0.5)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## 𝑓(𝑥0 + ℎ) = 𝑓(1) = −0.1(1)4 − 0.15(1)3 − 0.5(1)2 − 0.25(1) + 1.2 = 𝟎. 𝟐
  
  𝑓(𝑥0 + 2ℎ) = 𝑓(1.5) = −0.1(1.5)4 − 0.15(1.5)3 − 0.5(1.5)2 − 0.25(1.5) + 1.2 = −𝟏. 𝟑𝟏𝟐𝟓
  
  ## −3𝑓(𝑥0 ) + 4𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  2ℎ
  −3𝑓(0.5) + 4𝑓(1) − 𝑓(1.5)
  𝑓´(0.5) =
  2(0.5)
  −3(0.925) + 4(0.2) − (−1.3125)
  𝑓´(0.5) = = −𝟎. 𝟔𝟔𝟐𝟓
  2(0.5)
   𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒑𝒐𝒓 𝒆𝒍 𝒎é𝒕𝒐𝒅𝒐 𝒅𝒆 𝒔𝒆𝒈𝒖𝒏𝒅𝒂𝒔 𝒅𝒊𝒇𝒆𝒓𝒆𝒏𝒄𝒊𝒂𝒔 𝒇𝒊𝒏𝒊𝒕𝒂𝒔 (𝟑 𝒑𝒖𝒏𝒕𝒐𝒔), 𝒍𝒂 𝒑𝒓𝒊𝒎𝒆𝒓𝒂 𝒅𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒅𝒆 𝒇(𝒙) 𝒆𝒏
  𝒙𝟎 = 𝟎. 𝟓 𝒚 𝒉 = 𝟎. 𝟐𝟓
  
  𝑅𝑒𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  𝑓´(𝑥0 ) =
  2ℎ
  
  ## 𝑓(0) − 4𝑓(0.25) + 3𝑓(0.5)
  
  𝑓´(0.5) =
  2(0.25)
  1.2 − 4(1.103515625) + 3(0.925)
  𝑓´(0.5) = = −𝟎. 𝟖𝟕𝟖𝟏𝟐𝟓
  2(0.25)
  
  𝐶𝑒𝑛𝑡𝑟𝑎𝑙:
  
  ## 𝑓(𝑥0 − 2ℎ) − 8𝑓(𝑥0 − ℎ) + 8𝑓(𝑥0 + ℎ) + 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  12ℎ
  𝑓(0) − 8𝑓(0.25) + 8𝑓(0.75) − 𝑓(1)
  𝑓´(0.5) =
  12(0.25)
  1.2 − 8(1.103515625) + 8(0.636328125) − 0.2
  𝑓´(0.5) = = −𝟎. 𝟗𝟏𝟐𝟓
  12(0.25)
  
  𝑃𝑟𝑜𝑔𝑟𝑒𝑠𝑖𝑣𝑎𝑠:
  
  ## −3𝑓(𝑥0 ) + 4𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 + 2ℎ)
  
  𝑓´(𝑥0 ) =
  2ℎ
  −3𝑓(0.5) + 4𝑓(0.75) − 𝑓(1)
  𝑓´(0.5) =
  2(0.25)
  −3(0.925) + 4(0.636328125) − 0.2
  𝑓´(0.5) = = −𝟎. 𝟖𝟓𝟗𝟑𝟕𝟓
  2(0.25)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem that involves calculating the first derivative of a given function using different finite difference methods (forward, backward, and central differences) for two different step sizes (h = 0.5 and h = 0.25). The document also includes the detailed calculations and results for each method.
  
  Label: question-and-answer