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Popular Calculus >

tangent of y=e^{3x}cos(pix)

Solution

tangent of

y = e^{3 x} cos (π x)

Solution

y = (3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})) x + e^{3 a_{0}} cos (π a_{0}) - (3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})) a_{0}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, e^{3 a_{0}} cos (π a_{0}))

Find the slope of

y = e^{3 x} cos (π x) : \frac{d y}{d x} = 3 e^{3 x} cos (π x) - π e^{3 x} sin (π x)

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : m = 3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})

Find the line with slope m=

3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})

and passing through

(a_{0}, e^{3 a_{0}} cos (π a_{0})) : y = (3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})) x + e^{3 a_{0}} cos (π a_{0}) - (3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})) a_{0}

y = (3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})) x + e^{3 a_{0}} cos (π a_{0}) - (3 e^{3 a_{0}} cos (π a_{0}) - π e^{3 a_{0}} sin (π a_{0})) a_{0}

Graph

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