10.
Observe the curves numbered 1, 2, 3 and 4 in this plot. For each of these curves at several
evenly-spaced values of X, you measure the slope dY/dX. For which curve will you
observe the largest variance in measured values of dY/dX?
a. Curve 1
b. Curve 2
c. Curve 3
d. Curve 4
Curve 2 will show the largest variance in the measured values of the slope dY/dX, because its slope changes the most across X, while the other curves have either constant or nearly constant slopes.
Understanding the question
The plot shows four curves (1, 2, 3 and 4) of Y versus X.
At several evenly spaced X-values, the slope dY/dX (tangent slope) is measured on each curve, and the question asks which curve will give the largest variance in these slope values.
Variance is a statistical measure of how spread out a set of values is: if slopes change a lot from point to point, variance is large; if slopes are all similar, variance is small.
Why curve 2 is correct (option b)
Curve 2 is visibly curved: its slope starts positive and relatively steep, then decreases to nearly zero or slightly negative, and then becomes mildly positive again toward the right.
Because the slopes at different X positions on curve 2 range from relatively large positive to near zero and possibly negative, the collection of dY/dX values is highly spread out, giving the largest variance.
Mathematically, curve 2 has changing concavity and changing first derivative; this means dY/dX is not constant and varies substantially across X, which is exactly what produces a high variance in the measured slopes.
Why curve 1 is incorrect (option a)
Curve 1 is almost flat, a gently sloping straight line that is nearly horizontal across the entire X‑range.
Since its slope changes very little with X (dY/dX is almost the same everywhere), the set of measured slopes is tightly clustered around a single value, giving very low variance.
Even if there is a slight tilt, this produces only a small difference between slopes at different X points, so curve 1 cannot have the largest variance in dY/dX.
Why curve 3 is incorrect (option c)
Curve 3 is a straight line with a constant positive slope, drawn from the origin upward.
For a perfect straight line, dY/dX is exactly the same at every X; therefore the variance of the measured slopes is zero (all values identical).
Because the question compares variance across curves, any curve with constant slope immediately loses to any curve where the slope changes; hence curve 3 cannot give the largest variance in dY/dX.
Why curve 4 is incorrect (option d)
Curve 4 is also a straight line, but very shallow, nearly horizontal with a small constant slope.
Like curve 3, a straight line has the same slope everywhere, so repeated measurements of dY/dX at evenly spaced X values will all be equal (or extremely close), giving variance essentially zero.
Since curves 1, 3, and 4 have slopes that are constant or almost constant, only the strongly curved profile of curve 2 produces a wide range of slope values and thus the largest variance in dY/dX.
