# 2.11: Homework- Instantaneous Velocity

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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.
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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 $$-25$$
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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 $$-5h - 20$$.
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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.
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5. How quickly is the goat population growing at $$t = 2$$?
$$-40$$ goats per year.
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