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Slope of a Curve at a Point quiz #1

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  • Which of the black lines represents the tangent line that shows the slope of the curve at the point indicated by the dot?
    The black line that is tangent to the curve at the point indicated by the dot—touching the curve at only that point and not intersecting it elsewhere—shows the slope of the data at that point. The slope of this tangent line equals the slope of the curve at the dot.
  • What does the slope of a curve represent at a specific point?
    It represents how steep the curve is at that exact location, which can be found using the tangent line at that point.
  • Why does the slope of a curve change at different points?
    Because the curve is not a straight line, its steepness varies depending on where you measure it.
  • What is the first step in calculating the slope of a curve at a point using the point method?
    The first step is to draw a tangent line that touches the curve at only the point of interest.
  • How do you determine the slope of the tangent line once it is drawn?
    You identify two points on the tangent line, measure the vertical rise and horizontal run between them, and divide the rise by the run.
  • If the rise between two points on a tangent line is 2 and the run is also 2, what is the slope?
    The slope is 1, since 2 divided by 2 equals 1.
  • What does the tangent line indicate about the curve at the point where it touches?
    It shows the instantaneous rate of change or slope of the curve at that specific point.
  • Why is it important to use the tangent line when analyzing curved graphs?
    Because the tangent line allows us to measure the slope at a single point, which is crucial for understanding how the curve behaves locally.
  • What mathematical operation is used to calculate the slope from the rise and run?
    You divide the rise by the run to get the slope.
  • How does the point method help in understanding relationships in economics graphs?
    It helps by showing how the rate of change varies at different points, which is essential for analyzing non-linear relationships.