Question

1. Use the graph of f(x) to answer the following questions, and note that f(x) means different things in different

parts:
0
y = f(x)
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10
11
12
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14
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(a) If f(x) describes the position of my dog Maya, where the position is given in feet from me at time x in
seconds, determine Maya's average velocity between 6 seconds and 14 seconds. Include units.
(b) If f(x) describes the same scenario as part (a), that is the position of my dog Maya, where the position
is given in feet from me at time x in seconds, estimate Maya's instantaneous velocity at 2 seconds, 5
seconds, and 10 seconds. Include units, and explain how you found your answers.
(c) Sketch an accurate graph of the derivative of f(x), including the correct sign, direction, relative heights,
and zeros.
(d) If f(x) instead was the velocity of my dog Maya given in feet per second at time x in seconds, (NOT THE
POSITION FOR THIS PART), what was Maya's total distance traveled in the first 6 seconds? Can you
find it exactly? Include units.
(e) If f(x) is still the velocity of my dog Maya given in feet per second at time x in seconds, (the same as in
part (d)), use a left Riemann sum with only 4 rectangles to approximate Maya's total change in distance
on the time interval 6 ≤1≤ 14. Include units. Note, this is a different time interval than in part (d).

Answer 1

Average velocity is found by dividing the change in position by the change in time. Instantaneous velocity is estimated using tangent slopes. The derivative graph indicates rates of change. Total distance is calculated using integrals. Riemann sums approximate changes in distance.

(a) To determine Maya's average velocity between 6 seconds and 14 seconds, calculate the change in position over the change in time. Average Velocity
`(\(v_{\text{avg}}\))` is given by the formula:

`\[v_{\text{avg}} = (\Delta f)/(\Delta x).\]`

(b) To estimate Maya's instantaneous velocity at 2 seconds, 5 seconds, and 10 seconds, use the slope of the tangent line at each point on the graph of (f(x)).

(c) Sketching the derivative involves indicating the sign, direction, relative heights, and zeros. The derivative represents the rate of change of position.

(d) If (f(x)) is the velocity, the total distance traveled is the area under the velocity-time graph. Calculate the definite integral of f(x) from 0 to 6 seconds to find the total distance traveled in the first 6 seconds.

(e) Use a left Riemann sum with 4 rectangles to approximate the total change in distance on the time interval 6 to 14 seconds. Multiply each rectangle's width by the corresponding velocity value and sum the results.