Question

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

Answer 1

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)=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.

Answer 2

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