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Tangent Line

Q:  Find the derivative of function, then find the equation of the line that is tangent to its graph for the specified value x=c

f(x) = x²; c=1

A:  First, we take the derivative of f(x) using the power rule:

f(x) = x²

f ‘(x) = 2x

So, part (1) is done.  The derivative is 2x.

Now, to find the equation of the tangent line when x = c and c = 1… Well, this means x = 1.

Let’s find the slope (m)… The derivative tells us our slope, so plug in x = 1:

f ‘(x) = 2x

f ‘(1) = 2(1)

f ‘(1) = 2

So, the equation of our tangent line looks like:

y = mx + b

y = 2x + b

Now, we need an (x, y) point… We know x = 1, so we need to solve for y.  We do this by using the original equation:

f(x) = x²

Plug in x = 1:

f(1) = 1²

f(1) = 1

So, y = 1.  Our point is (1, 1).

Back on over to the equation of our tangent line:

y = 2x + b

Plug in (1, 1)

1 = 2(1) + b

1 = 2 + b

-1 = b

So, the equation of our tangent line is:

y = 2x – 1

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