Best solutions to - "How do you find the equation of the..."

auskunftsgkp

Answered question

2022-02-10

How do you find the equation of the tangent line to the curve

y = (x^{2}) e^{x + 2}

at x=2?

Jonathan Mckenzie

Beginner2022-02-11Added 14 answers

y = (x^{2}) e^{x + 2}

at x=2, we get

y = 4 e^{4}

, So our line goes through the point:

(2, 4 e^{4})

To find the slope, find y'. To find y' use the product rule and the derivative of exponentials (and the chain rule).

y^{'} = 2 c e^{x + 2} + x^{2} e^{x + 2} (1)

.... I included the (1) to remind us of the chain rule.
At x=2,

m = y^{'} ∣_{2} = 4 e^{4} + 4 e^{3} = 8 e^{4}

The line with

m = 8 e^{4}

through

(2, 4 e^{4})

has equation(s):

y - 4 e^{4} = 8 e^{4} (x - 2)

y - 4 e^{4} = 8 e^{4} x - 16 e^{4}

y = 8 e^{4} x - 12 e^{4}

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