Final answer:
To find the smallest slope on a curve, you calculate the first derivative of the curve's equation, find points where this derivative is zero or changes sign, then compare the slopes to identify the smallest one.
Step-by-step explanation:
To find the smallest slope on a curve, you need to analyze the curve's behavior by examining its derivative. The slope of the curve at any point is equal to the value of the derivative at that point. This can be a point where the derivative equals zero, which typically indicates a local maximum or minimum on a smooth, continuous curve, or a point where the first derivative changes sign, which can indicate a point of inflection.
The smallest slope is therefore not necessarily at the point with the lowest y-coordinate, the steepest ascent, nor the highest x-coordinate. Instead, to accurately find the smallest slope:
- Calculate the first derivative of the equation of the curve.
- Analyze the first derivative to find points where the derivative is zero or where the derivative changes sign. These points may represent local minima, maxima, or points of inflection.
- Compare the slope values at these points to identify the smallest slope, which might be zero or the smallest positive or negative value depending on the context.