"What is the equation of the line tangent..." was answered by our experts

Marshall Wolf

Answered question

2022-04-10

What is the equation of the line tangent to

f (x) = 3 x^{2} - 3 x + e^{1 - x^{2}}

at x=1?

Coupewopmergorlpt

Beginner2022-04-11Added 13 answers

Explanation:
Start by finding the corresponding y-coordinate of tangency.

f (1) = 3 {(1)}^{2} - 3 (1) + e^{1 - 1^{2}} = 3 - 3 = e^{0} = 1

Differentiate now, using

\frac{d}{d x} (x^{n}) = n x^{n - 1}

and

\frac{d}{d x} e^{f (x)} = f^{'} (x) e^{f (x)}

f^{'} (x) = 6 x - 3 - 2 x e^{1 - x^{2}}

The slope of the tangent is given by evaluating your point x=a into the derivative.

f^{'} (1) = 6 (1) - 3 - 2 (1) e^{1 - 1^{2}} = 6 - 3 - 2 = 1

Find the equation of the tangent.

y - y_{1} = m (x - x_{1})

y-1=1(x-1)
y-1=x-1
y=x

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