explain why the instantaneous rate of change for g(x)=-x*2 cannot be the same for any 2 different values of x.
The function g(x)=-x^2 represents a downward parabolic curve. In other words, as x increases, g(x) decreases.
The instantaneous rate of change of a function at a specific point is represented by its derivative. The derivative of g(x)=-x^2 is g'(x)=-2x. This means that the slope of the tangent line at any point on the curve is given by -2x.
For any two different values of x, the slope of the tangent line at those points will vary. This is because the value of x changes, and the slope of the tangent line depends on the value of x.
In other words, the instantaneous rate of change for g(x)=-x^2 cannot be the same for any two different values of x because the slope of the tangent line at any given point varies as x changes.
Sketch a graph and you will see that the slope (change in y / change in x ) gets steeper and steeper as you get further from the origin.
You are correct. Apologies for the oversight in my previous explanation. Let's examine the graph of g(x)=-x^2 to further understand why the instantaneous rate of change cannot be the same for any two different values of x.
Here is a sketch of the graph of g(x)=-x^2:
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-3 -2 -1 0 1 2 3 x-axis
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As you can see, the graph is a downward opening parabola, with its vertex at (0, 0). At any point on the graph, the tangent line represents the instantaneous rate of change or the slope at that specific point.
Now, let's consider two different values of x, say x1 and x2, where x1 is smaller than x2. When you calculate the slope between these two points on the graph, you will notice that the line connecting the points becomes steeper as you move further from the origin.
For example, let's take x1 = -2 and x2 = 2. Using the formula for slope (change in y / change in x), we can calculate the slopes between these points:
- Slope between (-2, 4) and (2, -4):
(4 - (-4)) / (-2 - 2) = 8 / (-4) = -2
Here, the slope is constant, which gives us a consistent rate of change between x1 and x2.
However, if we consider another set of values, such as x1 = -1 and x2 = 1, and calculate the slopes:
- Slope between (-1, -1) and (1, -1):
(-1 - (-1)) / (-1 - 1) = 0 / (-2) = 0
- Slope between (-1, -1) and (1, -1):
(-1 - (-1)) / (-1 - 1) = 0 / (-2) = 0
Here, the slopes are both zero, indicating that the rate of change between x1 and x2 is zero, meaning the function remains constant.
As you move further from the origin on the graph, the change in y (vertical change) becomes larger, while the change in x (horizontal change) remains the same, resulting in steep slopes. This shows that the instantaneous rate of change for g(x)=-x^2 cannot be the same for any two different values of x.