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Problem 67 Hard Difficulty

Let $ \displaystyle F(x) = \int^x_2 e^{t^2} \, dt $. Find an equation of the tangent line to the curve $ y = F(x) $ at the point with $ x $-coordinate 2.

Answer

$e^{4} x-2 e^{4}$

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Video Transcript

Okay. The first thing we can do is we can differentiate using the fundamental form of calculus. We get heated, the X squared. Now plug in half. Promise to We have each of the two squared, which is either the fourth. So, you know, the slope of the tangent is each of the fourth. Now, half of two is the integral from 2 to 2. E t. Square, D t. You know, this is your right. The tangent process for two comma zero. Therefore, the equation is going to be Why might a syrup is you? The fourth, The slope Times X minus two. This is the point slope form equation that you probably did years ago in algebra. So this is our final solution.