Let f(x) and g(x) be the differentiable functions graphed for here(The link). Point c is the point where the vertical distance between the curves is the greatest. Is there anything special about the tangents to the two curves at c? Give reasons for your answer.
圖在以下網址:
http://hk.geocities.com/abc2004y2k/1111.jpg
微積分1題(附圖)
2007-10-31 10:14 pm
回答 (3)
2007-10-31 10:32 pm
✔ 最佳答案
First of all, consider a function h(x) = f(x) - g(x) which represents the vertical distance between the curves f(x) and g(x) for a ≦ x ≦ b.Also, we are sure that h(x) ≧ 0 for a ≦ x ≦ b since the curve of y = f(x) is at a higher position than y = g(x) within this range of x.
Therefore, to maximize h(x), we have:
h'(x) = 0
h'(c) = 0
f'(c) - g'(c) = 0
f'(c) = g'(c)
So to speak, we can see that the slopes of tangents to the curves y = f(x) and y = g(x) are the same, implying that the tangents are parallel.
參考: My Maths knowledge
2007-11-04 12:05 am
First of all, consider a function h(x) = f(x) - g(x) which represents the vertical distance between the curves f(x) and g(x) for a ≦ x ≦ b.
Also, we are sure that h(x) ≧ 0 for a ≦ x ≦ b since the curve of y = f(x) is at a higher position than y = g(x) within this range of x.
Therefore, to maximize h(x), we have:
h'(x) = 0
h'(c) = 0
f'(c) - g'(c) = 0
f'(c) = g'(c)
So to speak, we can see that the slopes of tangents to the curves y = f(x) and y = g(x) are the same, implying that the tangents are parallel.
Also, we are sure that h(x) ≧ 0 for a ≦ x ≦ b since the curve of y = f(x) is at a higher position than y = g(x) within this range of x.
Therefore, to maximize h(x), we have:
h'(x) = 0
h'(c) = 0
f'(c) - g'(c) = 0
f'(c) = g'(c)
So to speak, we can see that the slopes of tangents to the curves y = f(x) and y = g(x) are the same, implying that the tangents are parallel.
參考: me
2007-11-01 3:37 am
First of all, consider a function h(x) = f(x) - g(x) which represents the vertical distance between the curves f(x) and g(x) for a ≦ x ≦ b.
Also, we are sure that h(x) ≧ 0 for a ≦ x ≦ b since the curve of y = f(x) is at a higher position than y = g(x) within this range of x.
Therefore, to maximize h(x), we have:
h'(x) = 0
h'(c) = 0
f'(c) - g'(c) = 0
f'(c) = g'(c)
So to speak, we can see that the slopes of tangents to the curves y = f(x) and y = g(x) are the same, implying that the tangents are parallel.
Also, we are sure that h(x) ≧ 0 for a ≦ x ≦ b since the curve of y = f(x) is at a higher position than y = g(x) within this range of x.
Therefore, to maximize h(x), we have:
h'(x) = 0
h'(c) = 0
f'(c) - g'(c) = 0
f'(c) = g'(c)
So to speak, we can see that the slopes of tangents to the curves y = f(x) and y = g(x) are the same, implying that the tangents are parallel.
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原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20071031000051KK01457