Suppose a curve is plotted on a graph, and a dot marks a specific point on the curve. Four black lines are drawn: one is tangent to the curve at the dot, and the others are secant lines passing through the dot and other points on the curve. Which black line represents the slope of the curve at the point indicated by the dot?

• A. The horizontal line passing through the dot
• B. The tangent line to the curve at the dot
• C. The vertical line passing through the dot
• D. The secant line connecting the dot to another point on the curve

Answer: (B)

Verified step by step guidance

1. Understand that the slope of a curve at a specific point is defined as the slope of the tangent line to the curve at that point.
2. Recall that a tangent line touches the curve at exactly one point and represents the instantaneous rate of change of the curve at that point.
3. Recognize that secant lines pass through two points on the curve and represent the average rate of change between those points, not the instantaneous slope.
4. Note that a horizontal line has a slope of zero and a vertical line has an undefined slope, so neither generally represents the slope of the curve at the point unless the curve itself is flat or vertical there.
5. Conclude that the black line representing the slope of the curve at the dot is the tangent line to the curve at that point.