Approximate the interval(s) over which the acceleration is increasing. b. Approximate the interval(s) over which the acceleration is decreasing. c. How many turning points does the graph show? d. Based on the number of turning points, what is the minimum degree of a polynomial function that could be used to model acceleration versus time? Would the leading coefficient be positive or negative? e. Approximate the time when the acceleration was the greatest. f. Approximate the value of the maximum acceleration.
The Correct Answer and Explanation is:
To answer these questions, we need to analyze the graph of the acceleration versus time. Without the actual graph, I can provide general explanations and strategies for how to approach these types of problems. Let’s break it down:
a. Approximate the interval(s) over which the acceleration is increasing:
The acceleration is increasing when the slope of the acceleration-time graph is positive. In other words, we look for sections of the graph where the tangent lines (slopes) are slanting upward. If the graph is concave up (curving upwards), acceleration is increasing. To find the interval(s), you would need to identify the points where the graph starts to slope upwards and stops sloping upwards.
b. Approximate the interval(s) over which the acceleration is decreasing:
Acceleration is decreasing when the slope of the acceleration-time graph is negative. This corresponds to areas where the graph is sloping downward (concave down). So, you would need to locate the portions of the graph where the slope is negative, and the acceleration is decreasing.
c. How many turning points does the graph show?
A turning point is where the graph changes direction, i.e., where the graph switches from increasing to decreasing or vice versa. Count the number of times the graph changes from an increasing slope to a decreasing slope or from a decreasing slope to an increasing slope. Each of these points represents a turning point.
d. Based on the number of turning points, what is the minimum degree of a polynomial function that could be used to model acceleration versus time? Would the leading coefficient be positive or negative?
The number of turning points gives us a clue about the degree of the polynomial. If a polynomial has nnn turning points, the degree of the polynomial is at least n+1n + 1n+1. This is because a polynomial of degree nnn can have at most n−1n – 1n−1 turning points. As for the leading coefficient, if the acceleration is increasing to a maximum value and then decreasing, the graph likely represents a cubic or higher degree polynomial with a positive leading coefficient (which would produce an “M”-shaped or “W”-shaped curve).
e. Approximate the time when the acceleration was the greatest:
The time when the acceleration was greatest corresponds to the point on the graph where the acceleration reaches its maximum value. Look for the highest point (local maximum) on the acceleration-time graph and approximate the time at which it occurs.
f. Approximate the value of the maximum acceleration:
The value of the maximum acceleration is the highest point on the acceleration-time graph. This is the y-coordinate (acceleration) at the time when the graph reaches its peak.
In summary, to solve these types of problems, you would look closely at the graph, identify increasing and decreasing intervals, count the turning points, and analyze the behavior of the polynomial to determine its degree and leading coefficient.
