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Where the tangent crosses

CalculusEasy

Let $g (x) = x^{3} - 3 x + 5$ . The tangent line to the graph of $g$ at $x = 2$ crosses the $y$ -axis at the point $(0, b)$ . Find $b$ . Give the answer as an integer.

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Hint

Find the point of tangency and the slope there, then follow the line back to $x = 0$ .

Solution

The point of tangency has $g (2) = 8 - 6 + 5 = 7$ .
The derivative is $g^{'} (x) = 3 x^{2} - 3$ , so the slope is $g^{'} (2) = 12 - 3 = 9$ .
The tangent is $y = 7 + 9 (x - 2)$ ; at $x = 0$ it gives $y = 7 - 18$ .
Equivalently, $b = g (2) - 2 g^{'} (2) = 7 - 18 = - 11$ , so $b = - 11$ .

Answer: $- 11$

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