Mathematics

Mathematics

Problem

Consider the tangent to the curve y=x3+4x2+x−6y = x^3 + 4x^2 + x - 6. a. Find the equation of this tangent at the point where x=−1x = -1. b. Find the coordinates of the point where this tangent meets the curve again.
Intuitive explanation

The key is recognizing that a tangent line's slope at any point comes from the derivative. Once you find dydx\frac{dy}{dx}, you can evaluate it at a specific xx-value to get the slope, then use point-slope form to write the full equation. The challenge is often identifying which point you're tangent to—watch for clues like "at x=1x = 1" or "with slope mm."

Intuitive explanation

The key is recognizing that a tangent line's slope at any point comes from the derivative. Once you find dydx\frac{dy}{dx}, you can evaluate it at a specific xx-value to get the slope, then use point-slope form to write the full equation. The challenge is often identifying which point you're tangent to—watch for clues like "at x=1x = 1" or "with slope mm."

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