# How do you find the tangent line approximation to #f(x)=e^x# near #x=0# ?

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To find the tangent line approximation to ( f(x) = e^x ) near ( x = 0 ), you use the formula for the equation of a tangent line: ( y = f'(a)(x - a) + f(a) ), where ( f'(a) ) is the derivative of ( f(x) ) evaluated at ( x = a ), and ( f(a) ) is the value of ( f(x) ) at ( x = a ).

First, find the derivative of ( f(x) = e^x ), which is ( f'(x) = e^x ).

Evaluate ( f'(x) ) at ( x = 0 ) to find the slope of the tangent line: ( f'(0) = e^0 = 1 ).

Next, find ( f(0) ) to get the y-coordinate of the point of tangency: ( f(0) = e^0 = 1 ).

Now, plug these values into the formula for the equation of the tangent line: ( y = 1 \cdot (x - 0) + 1 ), which simplifies to ( y = x + 1 ).

Thus, the tangent line approximation to ( f(x) = e^x ) near ( x = 0 ) is ( y = x + 1 ).

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