5. For problems 1-4 in part 1, I had you consider a fixed V in order to comment on what happened to h.
   1. If we use V = 1 cup, use ideas from Section 3.1 to explain why we can make the approximation h ( V ) h. (Hint: We can approximate instantaneous rates of change using the average rate of change, and the average rate of change is the slope of a secant line -- what is the formula for slope?)
   2. Why is it easier to compare the effects of different bottle shapes if we consider a fixed V?

1. Consider plotting height of water in a bottle vs. the volume of the water in the bottle. That is, height is on the vertical axis (dependent variable) and volume is on the horizontal axis (independent variable). Steepness of the graph of the function h ( V ) is related to the cross-sectional area of the bottle: a steeper graph corresponds to a narrower bottle and a less steep graph corresponds to a wider bottle, as shown below Problem 1.
   1. Consider a fixed V, and the resulting hs in each of the two bottles. Why does the narrower bottle correspond to the steeper graph? Make sure to explain in terms of amounts of change in height (i.e., hs) and amounts of change in volume (i.e., V s).
   2. How does the relationship between h and V in the steeper graph compare to the relationship in the less steep graph? Explain using slope, clearly describing what the slope represents in this context.
   3. How does the relationship between h and V in the narrower bottle compare to the relationship in the wider bottle?

2. 1. Describe what bottle shapes could correspond to a straight line graph. Do not just give one example. What feature of their shape must all these bottles have in common?
   2. A linear graph represents a constant rate of change between the two quantities, height and volume. Explain why the bottles you described would have a straight line graph.
3. Consider a bottle that is wide at the bottom and narrow at the top (as indicated in the diagram below Problem 3).
   1. Draw a graph that could represent the function h ( V ) for this bottle. Be sure to label your axes.
   2. Explain, in terms of h and V , why the shape of your graph is correct. (Hint: Itll be easier if you consider different h's for the same, fixed V.)

4. Inflection points correspond to points where the bottle changes from getting narrower to getting wider (or vice-versa). This is because an inflection point on the graph occurs when the graph changes from getting steeper to becoming less steep (or vice-versa).
   1. For the bottle pictured below that is narrower in the middle: Explain what is happening at the inflection point both on the bottle and on the graph. Use language about amounts of change. (Hint: Consider the effect of a fixed V at different places in the bottle.)
   2. Do the same for the bottle that is wider in the middle.