Find the point P on the curve r(t) that lies closest to Po and state the distance between Po and P. r(t) = i+ 2tj + 2tk; Po(4,14,13) (? -4)2 + (2t – 14)2 + (2t – 13)2 Distance = If the distance is minimized, then the square of the distance is also minimized. To find the minimum distance, differentiate the square of the distance with respect to t and find the value of t for which the result is zero. The square of the distance is (P - 4)2 + (2t – 14)2 + (2t – 13)2. Differentiate this expression with respect to t. +(2t - 14)2 + (2t – 13)2] = 413 – 108 Now set the derivative equal to zero to find the value of t that minimizes the distance. 4t3 - 108 413 = 108 t = 3

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: