Explain why or why not Determine whether the following...

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The distance traveled by an object moving along a line is the same as the displacement of the object. b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal. c. Consider a tank that is filled and drained at a flow rate of $V^{\prime}(t)=1-t^{2} / 100(\operatorname{gal} / \min ),$ for $t \geq 0 .$ It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty. d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from $A$ units to $2 A$ units is greater than the cost of increasing production from $2 A$ units to $3 A$ units.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The distance traveled by an object moving along a line is the same as the displacement of the object.
b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.
c. Consider a tank that is filled and drained at a flow rate of $V^{\prime}(t)=1-t^{2} / 100(\operatorname{gal} / \min ),$ for $t \geq 0 .$ It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty.
d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from $A$ units to $2 A$ units is greater than the cost of increasing production from $2 A$ units to $3 A$ units.

Key Concepts

Displacement versus Distance Traveled

Displacement is the net change in position (a vector quantity) while distance traveled accounts for the total length of the path taken (a scalar quantity). In problems involving motion, the equality of displacement and distance traveled occurs when the movement is in a single direction, meaning the velocity does not change sign. Understanding this distinction is crucial when determining the true extent of an object’s movement versus its net position change.

Effect of Velocity Sign on Motion

The sign of the velocity function indicates direction. When velocity remains positive (or remains negative) over an interval, the object is moving in one direction only. In such cases, the total distance traveled and the displacement become identical because there is no reversal in direction. This concept is fundamental in analyzing motion where the integral of the velocity and the integral of its absolute value yield the same outcome.

Integration of Rate Functions and Accumulation

Many problems involving motion, flow rates, or marginal quantities use integration to accumulate a variable over time from its rate of change. Understanding that the integral of a rate function gives the net change in the accumulated quantity allows for assessment of whether a system is increasing or decreasing. This concept requires careful analysis of where the rate (derivative) is positive or negative and may involve determining intervals of increase or decrease.

Marginal Cost Functions in Economic Analysis

In economics, the marginal cost function represents the cost incurred by producing one additional unit of output. The relationship between changes in production and total cost is determined by integrating the marginal cost function over the desired production interval. Recognizing the properties of such functions—whether they are increasing, decreasing, or constant—is critical when comparing costs over different intervals of production.

Monotonicity and Function Behavior

Monotonicity refers to a function consistently increasing or decreasing over an interval. In contexts like velocity, flow rates, or marginal costs, understanding whether a function is monotonic helps predict the behavior of the accumulated quantity. For example, a consistently positive, monotonic function (always increasing or always decreasing) simplifies the comparison between different intervals of accumulation but does not directly imply that changes over equal or proportional intervals will exhibit similar differences.

Transcript

00:01 Okay, here we have a problem.

00:04 It's going to be a problem on, and it's going to be true or false problems.

00:10 So the first true or false problem asks us a specific detail about distance, the distance traveled by an object.

00:22 Is the distance traveled by an object the same as the displacement of that object? so we're going to look at that.

00:28 So we're asked, is the distance traveled, the same as the displacement? so on a line, displacement, so we have a line here.

00:45 So let's say this is our starting point.

00:47 So this is our zero.

00:49 Displacement is where it ends up.

00:51 So here, that's going to be 10.

00:54 So this is 10.

00:55 So our displacement in this case is 10.

00:57 So we're here.

00:58 Now distance traveled is not necessarily 10.

01:01 So let's say we got to 10.

01:03 But let's say we did it like this.

01:05 So we go over here.

01:06 We go six units to the right.

01:09 Then we go four units to the left and then back six units to the right.

01:21 Sorry, eight units to the right minus four plus six.

01:25 If we do that, so we did plus eight minus four plus six.

01:33 If we do that, our total distance travel, no, distance travel doesn't account for positive and negative.

01:39 So we don't cancel those.

01:41 If we're just looking at this and travel, we have eight plus four plus six.

01:44 That would be 18, which is not the same as the displacement, which is 10.

01:49 So 18 plus 10, or is not equal to 10.

01:52 Thus, the distance travel is not the same as the displacement.

01:55 So this would be a counter example.

01:57 But the second one, if the velocity is only positive, is the displacement and distance the same.

02:05 But in this case, so this time we have the velocity has to be greater than zero.

02:11 In this case, remember last time we had negative four in that case.

02:15 So we went backwards a bit, but we still have to add it because distance doesn't account for backwards and forwards.

02:22 So if the velocity is only greater than zero, if we have the same kind of problem, so we're moving 10 units once again, the total distance is 10.

02:33 If i start from here and i have a velocity that is only positive, i can only go to the right.

02:39 I can't turn around.

02:41 And thus, the distance that i have to take from here to here has to be equal to 10 because i can only go.

02:48 Go to the right.

02:48 I can only go in one direction...

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