Honors Calculus Using Rates to get a Change in Amount page 6-1 solutions

You could argue that there are really only two important ideas in this entire course. One of the ideas is that we can measure the “slope” of a curve at a point and often think of this slope as an instantaneous rate (ft/sec for example). The other idea involves the graph of a “rate” function and the area between the curve and the x-axis.

1. In a moment of carelessness a motorist has accidentally set the cruise control to 75 mph. The motorist

continues for 1.5 hours; parks the car by the side of the road for 30 minutes for unknown reasons;

and then continues their journey for another hour with the cruise control now set at 65 mph. Sketch a graph

of speed vs time below. What is the area between the graph & the x-axis? What does this area represent

(think about the units)?

2. The rate of snowfall as a function of time is shown in the graph below. Would it make more sense to name the graphed function S(t) or S’(t)?

a) Is the snow falling more or less heavily as time passes? Explain.

b) Estimate the area between the curve and the x-axis between times 3 & 4.

Why does it make sense that this area corresponds to the amount of snow fall between time 3 & 4?

c) About how much snow fell during the middle 4 hours of the storm? Explain.

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You could argue that there are really only two important ideas in this entire course. One of the ideas is that we can measure the “slope” of a curve at a point and often think of this slope as an instantaneous rate (ft/sec for example). The other idea involves the graph of a “rate” function and the area between the curve and the x-axis.

1. In a moment of carelessness a motorist has accidentally set the cruise control to 75 mph. The motorist

continues for 1.5 hours; parks the car by the side of the road for 30 minutes for unknown reasons;

and then continues their journey for another hour with the cruise control now set at 65 mph. Sketch a graph

of speed vs time below. What is the area between the graph & the x-axis? What does this area represent

(think about the units)?

The area is 177.5 & represents the distance traveled (look at the unit): 75 mile/hr ∙ 1.5 hr = 112.5 miles

65 mile/hr ∙ 1.0 hr = 65.0 miles

177.5 miles

2. The rate of snowfall as a function of time is shown in the graph below. Would it make more sense to name the graphed function S(t) or S’(t)?

a) Is the snow falling more or less heavily as time passes? Explain.

More Heavily – rate increases from .5 inch/hr to 3 inch/hr.

b) Estimate the area between the curve and the x-axis between times 3 & 4.

Why does it make sense that this area corresponds to the amount of snow fall between time 3 & 4?

≈ 2 inches (look at the units used to get area: inch/hr ∙hrs inches)

c) About how much snow fell during the middle 4 hours of the storm? Explain.

≈ 9 inches (the area under this part of the curve is roughly 9 units)