The height of bamboo is proportional to its age. Use the graph to determine how long it takes for bamboo to reach a height of 12
0 inches,
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The point (2, 5) is not on the curve; probably you meant to say (2, -5)?
Consider an arbitrary point Q on the curve to the right of P, , where
. The slope of the secant line through P and Q is given by the difference quotient,
where we are allowed to simplify because .
Then the equation of the secant line is
Taking the limit as , we have
so the slope of the line tangent to the curve at P as slope 2.
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We can verify this with differentiation. Taking the derivative, we get
and at , we get a slope of
, as expected.