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 Question from Ethan, a student: How many tangent lines to the curve y = x^3 pass through the point (2, 4)? For each such line, and the exact coordinates of the point of tangency on the curve.

Hi Ethan,

Suppose that $(a, b)$ is a point on the curve $y = x^3$ where the tangent to the curve at that point passes through $(2, 4).$ Since $(a, b)$ is on the curve you know that $b = a^3$.

Use the calculus you know to find the slope of the tangent to $y = x^3$ at the point $(a, b).$ Write the equation of the line with this slope that passes through $(2, 4).$ You know that the point $(a, b)$ lies on this line so substitute $x = a$ and $y = b = a^3$. Solve this equation for $a.$

If you get stuck write back and tell us what you did and where you are stuck,

Penny

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