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Ch 15

Ch 15. 偏微分 Partial Derivatives. 學習內容. 多變數函數. 15.1 Functions of Several Variables 15.3 Partial Derivatives 15.4 Tangent Planes and Linear Approx. 15.5 Chain Rule 15.6 Directional Derivatives and Gradient 15.7 Maximum and Minimum Values. 偏微分. 切平面與線性估計. 鎖鏈法則. 方向導函數與梯度向量. 極大值與極小值.

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Ch 15

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  1. Ch 15 偏微分 Partial Derivatives

  2. 學習內容 多變數函數 • 15.1 Functions of Several Variables • 15.3 Partial Derivatives • 15.4 Tangent Planes and Linear Approx. • 15.5 Chain Rule • 15.6 Directional Derivatives and Gradient • 15.7 Maximum and Minimum Values 偏微分 切平面與線性估計 鎖鏈法則 方向導函數與梯度向量 極大值與極小值

  3. 15.3 Partial Derivatives 偏微分

  4. 學習內容 • 知道多變數函數偏微分的定義 • 能以幾何的觀點解釋偏微分 • 能求高階的偏微分

  5. 偏微分的幾何觀點 • 對x的偏微分 • 延著x軸的切線斜率 • 對y的偏微分 • 延著y軸的切線斜率 11.3

  6. 延著x軸的切線斜率 曲線 切線

  7. 對x的偏微分 當y固定在y0時，延著x軸的切線斜率。

  8. 延著y軸的切線斜率 切線 曲線

  9. 對y的偏微分 當x固定在x0時，延著y軸的切線斜率。

  10. 延y軸的切線斜率 延x軸的切線斜率 延x軸的切線 延y軸的切線

  11. Example 2 1 1 1

  12. Q2 (a) (b) (c) (d)

  13. Example 2 1 1 1

  14. Example 3

  15. Q3 (a) (b) (c) (d)

  16. Example 3

  17. Example 4 隱函數

  18. Q4 (a) (c) (b) (d)

  19. Example 5 More Variables

  20. Example 5 More Variables

  21. Q5 (a) (b) (c) (d)

  22. Example 5 More Variables

  23. Example 6 Higher Derivatives

  24. Example 7 Calculate fxxyz if f(x, y, z) = sin(3x + yz) fx = 3 cos(3x + yz) fxx = –9 sin(3x + yz) fxxy = –9z cos(3x + yz)

  25. Example 7 • fxxy = –9z cos(3x + yz) • fxxyz = –9 cos(3x + yz) + (-9z) y (-sin(3x + yz))

  26. Q6 fxxy = –9z cos(3x + yz), Calculate fxxyz (a) fxxyz = –9 cos(3x + yz) + 9yz sin(3x + yz) (b) fxxyz = 9 cos(3x + yz) + 9yz sin(3x + yz) (c) fxxyz = –9 cos(yz) + 9yz sin(yz) (d) fxxyz = 9yz sin(3x + yz)

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