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What are the exact coordinates of the point of tangency?
The line y=mx is tangent to the curve y=2^x. What are the exact coordinates of the point of tangency?
2 Answers
- atsuoLv 66 years agoFavorite Answer
Assume that the xy-coordinates of the point is (p,q) . Of course , q=2^p .
(d/dx)2^x = (ln2)2^x , and you said the tangent is the line y =mx , so
q = mp
2^p = ((ln2)2^p)p
1 = (ln2)p
p = 1 / ln2
q = 2^p = 2^(1 / ln2)
ln(q) = ln(2^(1 / ln2))
....... = (1 / ln2)ln2
....... = 1
So q = e ---> the xy-coordinates are (1 / ln2 , e)
- 6 years ago
y = mx has a slope of m
y = 2^x
y' = (2^x) * ln(2)
m = ln(2) * 2^(x)
m/ln(2) = 2^x
ln(m / ln(2)) = x * ln(2)
x = (1/ln(2)) * ln(m / ln(2))
y = mx
y = (m/ln(2)) * ln(m / ln(2))
((1/ln(2)) * ln(m / ln(2)) , (m / ln(2)) * ln(m / ln(2)))
There's your point.
