Question

Find the x-intercept of the tangent line to the graph of f at the point P(0,f(0)) when f(x)= e^-x(5cosx+2sinx)

Answer 1

f ( 0 ) = e^0 * ( 5 cos 0 + 2 sin 0 = = 1 * 5 = 5
f ` ( x ) = - e^(-x) * ( 5 cos x + 2 sin x ) + e^(-x) * ( - 5 sin x + 2 cos x ) =
( using the chain rule )
= e^(-x) * ( - 5 cos x - 2 sin x - 5 sin x + 2 cos x ) =
= e^(-x) * ( - 3 cos x - 7 sin x )
f ` ( x ) = 1 * ( - 3 - 0 ) = - 3
The equation of the tangent line:
y - f ( 0 ) = f ` ( 0 ) * ( x - 0 )
y - 5 = - 3 x
y = - 3 x + 5
The x - intercept of the tangent line:
0 = - 3 x + 5
3 x = 5
x = 5 / 3
or ( 5/3, 0 )