Problem

If $$F(x)=f(g(x)), where f(-2)=8, f^{\prime}(-2)=…

Problem 52 Hard Difficulty

Find an equation of the tangent line to the curve
$y=\sqrt{3+x^{2}}$ that is parallel to the line $x-2 y=1$

Answer

$y=\frac{1}{2} x+\frac{3}{2}$

Related Courses

Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 2

DERIVATIVES

Section 5

The Chain Rule

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Video Transcript

Okay, This question asks us to find a tangent line to this curve that's parallel to this other line. Nick of us and parallel lines are solely determined by their slow. So let's find the slope from our original line. So X minus two y equals one. So that means negative two y is equal to one minus X, So why is equal to negative 1/2 plus 1/2 X? So now we see the slope of this line is what half. So we want to find where the slope of our curve is equal to 1/2. So to do that, let's just take a derivative of our function. Said it equal to 1/2. So why prime for a square root is just one over two times the route times the derivative of the inside. So why prime is equal to X over the square root of three plus ax squared, and and this problem we're looking for where the slope is equal to 1/2 so 1/2 is equal to X over the square root of three plus ax squared, so multiplying we get 1/2 times the square root of three plus X squared is equal to X or square root three plus X squared is equal to two X and then squaring both sides. We get three plus X squared is equal to four X squared or three X squared, minus three is equal to zero. So three times X squared minus one is it with zero so X is equal to plus or minus one. So let's check to see which one of these points will work. Just to make sure we get a line that's parallel. So trying X equals one. So why is equal to the square root of three plus X squared? Then why have one would just be, too. So now, using point slope form, we get why minus two is equal to 1/2 times X minus one. So we'd get why is equal to 1/2 X unless three halves, because we get tu minus 1/2 and that is indeed parallel toe original line. So we're done

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