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Please answer this question to the best of your ability. You are welcome to re-watch parts of any of the videos to help you.

Problems 1 and 2 refer to the following context.

Li is out for a bike ride. He starts his ride from a bike shop. The function $y = v(t)$ expresses the relationship between Li’s velocity (in meters per minute) as he bikes and the number of minutes $t$ elapsed since he started biking.

What does the product $v(t) \times \Delta t$ approximate?

The average rate of change of Li’s velocity over a particular $\Delta t$ interval. The change in Li’s distance away from the bike shop over a particular $\Delta t$ interval. Li’s instantaneous velocity at a particular moment in time. Li’s acceleration over a particular $\Delta t$ interval. Li’s distance away from the bike shop after having run for $\Delta t$ minutes.

What does the sum $v(0) \times \dfrac {1}{2} + v\left (\dfrac {1}{2}\right ) \times \dfrac {1}{2} + v(1) \times \dfrac {1}{2} + v\left (\frac {3}{2}\right ) \times \dfrac {1}{2} + v(2) \times \dfrac {1}{2} + v\left (\frac {5}{2}\right ) \times \dfrac {1}{2}$ approximate?

Li’s distance away from the bike shop after having run for 2.5 minutes. The average rate of change of Li’s velocity over the interval of time from $t = 0$ to $t = \dfrac {5}{2}$ . The change in Li’s distance away from the bike shop over the interval of time from $t = 0$ to $t = 3$ . Li’s distance away from the bike shop after having run for 3 minutes. Li’s acceleration over the interval of time from $t = 0$ to $t = 3$ .

The function $y = r(t)$ expresses the relationship between the rate at which water drains from a tank (in gallons per minute) and the number of minutes $t$ elapsed since water started draining from the tank. What quantity does the sum of the areas of the blue rectangles on the graph of the function $y = r(t)$ below approximate?

The average rate at which water drained from the tank over the interval of time from 2 to 14 minutes after water started draining from the tank. The total amount of water drained from the tank over the interval of time from 2 to 14 minutes after water started draining from the tank. The constant rate at which water would have to drain for the tank to be empty 20 minutes after water started draining. The instantaneous rate at which water drains from the tank 14 minutes after water started draining. The initial amount of water in the tank.

The function $y = g(t)$ represents the relationship between the rate of change in the value of investment stocks (in dollars per month) and the number of months $t$ elapsed since the stocks were purchased. Which of the following sums approximates the change in the value of the stocks over the interval of time from 4 to 7 months after the stocks were purchased?

$\sum _{k=0}^{6} g(4 + 0.5 k) \times 0.5$ $\sum _{k=4}^7 g(k)$ $\sum _{k=0}^2 = g(4+k)$ $\sum _{k=4}^7 g(t) \Delta t$ $\sum _{k=0}^{14} g(4 + 0.2k) \times 0.2$

The table below shows the horizontal velocity $v(t)$ of a baseball (in feet per second) for various values of $t$ , which is the number of seconds elapsed since the baseball was thrown.


t	0	0.25	0.5	0.75	1	1.25	1.5	1.75	2

v(t)	132	130.03	128.08	126.16	124.27	122.41	120.58	118.78	117

Approximate the total distance the ball traveled during the first two seconds after it was thrown by computing a left-hand Riemann sum with four terms.

$\answer {252.465}$