Question

(1 point) A person walking along a straight path has her velocity in miles per hour at time $t$ given by the function $v(t) = 0.25t^3 - 1.5t^2 + 3t + 0.25$, for times in the interval $0 le t le 2$. The graph of this function is also given in each of the three diagrams below. Note that in each diagram, we use four rectangles to estimate the area under $y = v(t)$ on the interval $[0, 2]$, but the method by which the four rectangles' respective heights are decided varies among the three individual graphs. Think about how the heights of the rectangles in the left-most diagram are being chosen. Determine the value of $S = A_1 + A_2 + A_3 + A_4$ by evaluating the function $y = v(t)$ at appropriately-chosen values and observing the width of each rectangle. Note, for example, that $A_3 = v(1) cdot frac{1}{2} = 2 cdot frac{1}{2} = 1$. $S = $ Use the rectangles in the middle diagram to find the value of $T = B_1 + B_2 + B_3 + B_4$. $T = $ Use the rectangles in the right-most diagram to find the value of $U = C_1 + C_2 + C_3 + C_4$. $U = $ Which estimate do you think is the best approximation of $D$, the total distance the person traveled on $[0, 2]$? Why?

          (1 point) A person walking along a straight path has her velocity in miles per hour at time $t$ given by the function $v(t) = 0.25t^3 - 1.5t^2 + 3t + 0.25$, for times in the interval $0 le t le 2$. The graph of this function is also given in each of the three diagrams below.

Note that in each diagram, we use four rectangles to estimate the area under $y = v(t)$ on the interval $[0, 2]$, but the method by which the four rectangles' respective heights are decided varies among the three individual graphs.

Think about how the heights of the rectangles in the left-most diagram are being chosen. Determine the value of $S = A_1 + A_2 + A_3 + A_4$ by evaluating the function $y = v(t)$ at appropriately-chosen values and observing the width of each rectangle. Note, for example, that $A_3 = v(1) cdot frac{1}{2} = 2 cdot frac{1}{2} = 1$.

$S = $

Use the rectangles in the middle diagram to find the value of $T = B_1 + B_2 + B_3 + B_4$.

$T = $

Use the rectangles in the right-most diagram to find the value of $U = C_1 + C_2 + C_3 + C_4$.

$U = $

Which estimate do you think is the best approximation of $D$, the total distance the person traveled on $[0, 2]$? Why?

$(1 point) A person walking along a straight path has her velocity in miles per hour at time t given by the function v(t) = 0.25t^3 - 1.5t^2 + 3t + 0.25, for times in the interval 0 le t le 2. The graph of this function is also given in each of the three diagrams below. Note that in each diagram, we use four rectangles to estimate the area under y = v(t) on the interval [0, 2], but the method by which the four rectangles' respective heights are decided varies among the three individual graphs. Think about how the heights of the rectangles in the left-most diagram are being chosen. Determine the value of S = A1 + A2 + A3 + A4 by evaluating the function y = v(t) at appropriately-chosen values and observing the width of each rectangle. Note, for example, that A3 = v(1) cdot frac12 = 2 cdot frac12 = 1. S = Use the rectangles in the middle diagram to find the value of T = B1 + B2 + B3 + B4. T = Use the rectangles in the right-most diagram to find the value of U = C1 + C2 + C3 + C4. U = Which estimate do you think is the best approximation of D, the total distance the person traveled on [0, 2]? Why?$

Added by Elizabeth T.

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