Question

1. a rock is stuck in the tire of a bicycle wheel that is traveling at a constant rate. as the tire moves forward the distance the rock is from the ground can be modeled by a graph where the y - axis represents the distance the rock is from the ground and the x - axis represents the time since wheel started moving. which of the following graphs models the situation where rock has gone around the tire 4 times? (no calculation)

(a)
(b)
(c)
the graph of the function f is given for - 2.2≤x≤3.2. on which of the following intervals is the graph of f concave up? (no calculation)
(a) -2, -1
(b) 2, 3
(c) -6, 2
(d) 0, 1

Explanation:

First problem - Analyze the motion of the rock in the tire:

The motion of a rock stuck in a bicycle - tire as the tire moves forward follows a cycloid pattern. A cycloid is a smooth, curved - shape. Option (a) has a series of sharp - peaked shapes which is not a cycloid. Option (b) has angular, non - smooth shapes which is also not a cycloid. Option (c) has a smooth, curved pattern which represents the cycloid motion of the rock in the tire as the tire rotates and moves forward.

Second problem - Determine the concave - up interval of the function:

The graph of a function \(y = f(x)\) is concave up when the second - derivative \(f''(x)>0\), or visually, when the graph "holds water". Looking at the given graph of the function \(f\):

• In the interval \([-2,-1]\), the graph is concave down.
• In the interval \([2,3]\), the graph is concave down.
• In the interval \([-6,2]\), the graph has both concave - up and concave - down parts.
• In the interval \([0,1]\), the graph is concave up.

Answer:

1. c
2. d