Below is a velocity (meters second) versus time (seconds) graph for two bikers. Compare the velocity of Bike 1 (solid line) at t = 3 seconds and t = 8 seconds. At which moment is Bike 1 moving more slowly? When is Bike 1 moving the slowest? How can you tell? Compute the average rate of change for the velocity of Bike 1 from t = 3 to t = 8. Include units. Show how you get your answer. Estimate the instantaneous rate of change for the velocity for Bike 1 at t = 8. Explain.
![Definite integrals are useful to science because we can often describe natural phenomena using definite integrals. However, integrating functions arising in real-life science can be rather challenging. In this exercise, we see how having only the graph of some functions is enough to answer some questions in science. The physical interpretation of an integral is a prerequisite knowledge you learned in MATH 1013. For a refresher, see the Net Change Theorem in Section 5.4 of the textbook. A group of researchers studies the population of arctic foxes in Northern Canada. Initially, the researchers identified a population of 1,000 arctic foxes. Due to various factors, the population of arctic foxes changes over time. The graph of the birthrate B(t) and the death rate D(t), for the population, are shown below. Time t is measured in years. (1) In a sentence or two, describe the meaning of the function f(t) = t 0 B(x) - D(x)dx in the context of this problem. (2) According the graph, is there a time t between 0 and 20 at which the population is equal to the initial population? If so, provide a rough estimate of this time, and write one or two sentences explaining how you found it. Your estimate should be a single number, not an interval. (3) Recall from Quiz 1 the definition of the average value of a function f(x). The average value of a function f(x) over an interval [a, b] is denoted by favg and is defined to be favg = 1 b - a b a f(x)dx. Suppose you know that the average birth rate over the first 20 years is 90 arctic foxes per year and the average death rate over the first 20 years is 100 arctic foxes per year. What is the population of the arctic foxes at t = 20 ? Show your work.](https://www.doubtrix.com/uploads/editor/1288831183fhExDQGbmw.png)
![The velocity-time graph of a car moving along a straight road is shown in Figure 3 below. [Assumption: The car is travelling in a straight line. ] Figure 3 (Not drawn to scale) a) Determine the velocity of the car at time t = 1 s. b) Determine the acceleration of the car at time t = 5 s. c) Determine the rate of change of velocity of the car at time t = 7 s. d) Determine the displacement of the car from time t = 4 s to t = 6 s. e) Determine the average velocity of the car from time t = 0 s to t = hs, leave your answer in terms of h. f) When the car starts moving from rest at time t = 0 s, a lorry travelling with a uniform constant velocity of vm/s passes by the car. At t = 4 s, the car and lorry have the same displacement. Determine the value of v. g) When the car starts moving from rest at time t = 0 s, a motorcycle travelling with a uniform constant velocity of 16 m/s passes by the car. The motorcycle maintains this uniform velocity for 2 s before decelerating uniformly to rest at t = 10 s. The car overtakes the motorcycle at time t = ws, where 4 < w < 6. Determine the value of w.](https://www.doubtrix.com/uploads/editor/7775664077JfqMMcGvFj.png)
