2.11: Homework- Instantaneous Velocity

1. The position of a falling object follows the equation $f(t) = -16 t^2 + 64$ from $t = 0$ to $t = 2$.
   1. Verify that the points $(1, 48)$ and $(2, 0)$ are on the curve by computing $f(1)$ and $f(2)$ and verifying you get $48$ and $0$.
      This seems to work.

      ans
   2. Compute the slope of the line going through $(1, 48)$ and $(2, 0)$.
      -48

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   3. Verify that the points $(1, 48)$ and $(1 + h, -16 h^2 - 32 h + 48)$ lie on the curve. You’ve already done $(1, 48)$, so now just simplify $f(1 + h)$ and verify you get $-16 h^2 - 32 h + 48$.
      This seems to work.

      ans
   4. Compute the slope of the line through $(1, 48)$ and $(1 + h, -16 h^2 - 32 h + 48)$ (hint: you should get $-16h - 32$!)
2. The position of a falling object follows the equation $f(t) = -5 t^2 + 45$ from $t = 0$ to $t = 3$.
   1. Verify that the points $(2, 25)$ and $(3, 0)$ line on the curve, and compute the slope through these two points.
      The slope is $-25$

      ans
   2. Verify that the points $(2, 25)$ and $(2 + h, -5 h^2 - 20h + 25)$ lie on the curve, and compute the slope of the line through these two points.
   The slope is $-5h - 20$.

   ans
3. Let $g(t) = -10t^2 + 2000$ represent a population of goats, where $g(t)$ is measured in goats and $t$ is measured in years. Suppose $t$ only works on the range from $0$ to $10$. This population is stabilizing during this period.
   1. Sketch a graph of $g(t)$.
   2. Find the slope of the secant line hitting $g(t)$ at $t = 2$ and $t = 3$.
      $-50$ goats per year

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   3. The slope of the secant line hitting $g(t)$ at $t = 2$ and $t = 2+h$.
      $-40-10h$ goats per year

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   4. What is $\lim_{h \to 0}$ for your answer in part (c)?
      $-40$ goats per year.

      ans
   5. How quickly is the goat population growing at $t = 2$?
      $-40$ goats per year.

      ans