In Exercise, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example.
f(x) = x3 , x = −2
Example
Finding the Equation of the Tangent Line at a Given x Find the slope–point equation of the tangent line to the graph of f(x) = at x = 2.
SOLUTION In this problem, we are not given a point on the tangent line, but only its first coordinate x = 2. Since the point is on the graph of f(x) = we get the second coordinate by plugging the x-value into f(x):
Thus, is the point on the graph and the tangent line. Next, we find the slope of the tangent line. For this purpose, we compute f_(x) by using the power rule:
The slope of the tangent line when x = 2 is
In slope–point form, the equation of the tangent line is y
In general, to find the point–slope equation of the tangent line to the graph of y = f(x) at the point with first coordinate x = a, proceed as follows:
Step 1 Find the point of contact of the graph and the tangent line by evaluating f(x) at x = a. This yields the point (a, f(a)).
Step 2 Find the slope of the tangent line by evaluating the derivative f′(x) at x = a. This yields the slope m = f′(a).
Using the point (a, f(a)) and the slope m = f′(a), we obtain the equation of the tangent line:
You do not need to memorize formula (6), but you should be able to derive it in a given situation, as we did in Example.
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