Q. 2 - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) The graph of a function whose associated slope function $f^{\text{'}}$ is positive on (−∞, 2) and negative on (2,∞).

(b) The graph of a function with the following three properties: The average rate of change of f on [0, 2] is 3, the average rate of change of f on [0, 1] is −1, and the average rate of change of f on [−2, 2] is 0. 

(c) The graph of a function f with the following three properties: The instantaneous rate of change of f at x = 2 is zero, the average rate of change of f on [1, 2] is 2, and the average rate of change of f on [2, 4] is 1. 

Step-by-Step Solution

Answer

Part (a)

Part (b)

Part (c)

Part (a) Step 1. Given information.

We have to construct the following :

The graph of a function whose associated slope function $f^{\text{'}}$ is positive on (−∞, 2) and negative on (2,∞). (b) The graph of a function with the following three properties: The average rate of change of f on [0, 2] is 3, the average rate of change of f on [0, 1] is −1, and the average rate of change of f on [−2, 2] is 0. (c) The graph of a function f with the following three properties: The instantaneous rate of change of f at x = 2 is zero, the average rate of change of f on [1, 2] is 2, and the average rate of change of f on [2, 4] is 1. 

Part (a) Step 2. Graph a function whose associated slope function f' is positive on (−∞, 2) and negative on (2,∞)

The corresponding graph can be drawn as:

Part (b) Step 1. Graph the function with the given three properties.

The average rate of change of f on [0, 2] is 3, 

the average rate of change of f on [0, 1] is −1, 

and the average rate of change of f on [−2, 2] is 0.

The corresponding graph can be drawn as:

Part (c) Step 1. Graph the function with the given three properties.

The instantaneous rate of change of f at x = 2 is zero, 

the average rate of change of f on [1, 2] is 2, 

and the average rate of change of f on [2, 4] is 1.