Question

45 - 46. analyzing slopes use the points a, b, c, d, and e in the following graphs to answer these questions.
a. at which points is the slope of the curve negative?
b. at which points is the slope of the curve positive?
c. using a - e, list the slopes in decreasing order.
45.

Explanation:

Step1: Recall slope - sign rules

The slope of a curve at a point is the slope of the tangent line at that point. If the curve is increasing at a point, the slope of the tangent is positive. If the curve is decreasing at a point, the slope of the tangent is negative.

Step2: Find points with negative slope

The curve is decreasing at points $C$ and $E$. So the points with negative slope are $C$ and $E$.

Step3: Find points with positive slope

The curve is increasing at points $A$ and $D$. So the points with positive slope are $A$ and $D$.

Step4: Analyze steepness for slope order

The steepness of the curve at a point is related to the magnitude of the slope. The steeper the curve, the larger the magnitude of the slope. At point $A$, the curve is quite steep and increasing. At point $D$, the curve is less steep and increasing. At point $C$, the curve is less steep and decreasing. At point $E$, the curve is steeper and decreasing. The slope at $B$ is 0 (since it is a local maximum). The slopes in decreasing order are $A$, $D$, $B$, $C$, $E$.

Answer:

a. $C$, $E$
b. $A$, $D$
c. $A$, $D$, $B$, $C$, $E$