math
posted by sarah on .
Which of the following four fundamental funtions have the same tangent line at x = 0 Ans:_________________ ?
y = x, y = x2, y = x3 and y = ex

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=e^{x} =>f'(x)=e^{x} => f'(0)=e^{0}=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 yintercepts 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