Posted by sarah on Wednesday, August 31, 2011 at 6:19pm.
Tangent line at x=0 means to evaluate the derivative at x=0, or f'(0).
For the tangent lines to be equal (coincident), f(0) must also be equal.
y=x => f'(x)=1 => f'(0)=1, f(0)=0
y=x²=> f'(x)=2x => f'(0)=0, f(0)=0
y=x³ => f'(x)=3x² => f'(0)=0, f(0)=0
y=ex =>f'(x)=ex => f'(0)=e0=1, f(0)=1
Therefore there are two sets of tangent lines with equal slopes at x=0, but only one set of them have equal y-intercepts and hence coincident lines.
For:
y=x
Technically there is no tangent line to a straight line because the definition of tangent means a line a touching at one point.
Answer is:
x^2 and x^3
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