1. A rectangle is inscribed under the parabola \( y = 12 - x^2 \), with its base on the \( x \)-axis. Find the dimensions of the rectangle that maximize its area.
2. Find the point on the line \( y = 2x + 3 \) that is closest to the point \( P(3, 4) \).
3. A box is to be made from a square piece of cardboard of side length 12 cm by cutting equal squares from each corner and folding up the sides. Find the size of the squares that minimizes the surface area of the box.
4. The population of a town is modeled by the function \( P(t) = 500e^{0.03t} \), where \( t \) is the time in years. Find the rate of change of the population after 10 years.
5. Find the equation of the tangent and normal to the curve \( y = x^3 - 4x^2 + 6x - 2 \) at the point \( (2, 2) \).
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