Tangent Line Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

At what point(s) on

f(x) = x^3 - 3x

does the tangent line have slope

9

1
$f'(x) = 3x^2 - 3$ . Set equal to 9: $3x^2 - 3 = 9 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$ .
2
At $x = 2$ : $f(2) = 8 - 6 = 2$ , point $(2, 2)$ .
3
At $x = -2$ : $f(-2) = -8 + 6 = -2$ , point $(-2, -2)$ .

Answer

(2,\,2)

and

(-2,\,-2)

Set

f'(x)

equal to the desired slope and solve for

x

. Then evaluate

f(x)

at each solution to find the full point. There are two points because

f'(x) = 9

is a quadratic equation.

About Tangent Line

A line that touches a curve at exactly one point and has the same slope as the curve there.

Find the equation of the tangent line to [formula] at [formula].

Find the equation of the tangent line to [formula] at [formula], and use it to approximate [formula]

Find the equation of the tangent line to [formula] at [formula].