Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?
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This question asks for the derivative of a function involving the greatest integer function at a specific point within a given domain. Let's break down the problem step by step.
Understanding the Greatest Integer Function
The greatest integer function, denoted by [x], gives the largest integer less than or equal to x. For example, [3.7] = 3, [-2.1] = -3, and [5] = 5.
An important property of the greatest integer function is that it is constant between consecutive integers. Its graph consists of horizontal line segments with jumps at integer values.
Analyzing y = [x + 1] in the Interval -4 < x < -3
The given function is y = [x + 1] and the domain is restricted to -4 < x < -3.
Let's consider the range of values for x + 1 within this domain. If -4 < x < -3, then by adding 1 to all parts of the inequality, we get:
-4 + 1 < x + 1 < -3 + 1
-3 < x + 1 < -2
So, for any value of x in the interval (-4, -3), the corresponding value of x + 1 is in the interval (-3, -2).
Now, let's find the greatest integer for any number in the interval (-3, -2). For any number z such that -3 < z < -2, the greatest integer less than or equal to z is -3. For instance, [-2.5] = -3, [-2.1] = -3, [-2.99] = -3.
Therefore, for all x in the interval -4 < x < -3, the value of [x + 1] is always -3.
So, in the domain -4 < x < -3, the function y is a constant function: y = -3.
Derivative of y at x = -3.5
We need to find the derivative of y with respect to x at x = -3.5. The point x = -3.5 lies within the given domain -4 < x < -3.
As we established, for all x in the interval (-4, -3), the function is y = -3.
The derivative of a constant function is always zero. That is, if y = C where C is a constant, then \frac{dy}{dx} = \frac{d}{dx}(C) = 0.
Since y = -3 for all x in the interval containing x = -3.5, the function is a constant in the neighborhood of x = -3.5. Therefore, the derivative of y with respect to x at x = -3.5 is 0.
\left.\frac{dy}{dx}\right|_{x=-3.5} = \frac{d}{dx}(-3) = 0
In summary, within the specified interval, the function y = [x + 1] simplifies to a constant value, and the derivative of any constant function is zero.
| Range of x | Range of x + 1 | Value of [x + 1] | Function y |
|---|---|---|---|
| -4 < x < -3 | -3 < x + 1 < -2 | -3 | y = -3 |
Conclusion on Derivative
At x = -3.5, which is in the interval (-4, -3), the function y = [x + 1] is equal to -3. The derivative of this constant function at any point in this interval is 0.
Revision Table: Greatest Integer Function Derivative
| Concept | Explanation | Relevance to Question |
|---|---|---|
| Greatest Integer Function [x] | Largest integer less than or equal to x. | The base function involves this concept. |
| Function y = [x + 1] | Defined by adding 1 to the input before taking the greatest integer. | The specific function given in the problem. |
| Domain -4 < x < -3 | The range of x values for which the function is considered. | Crucial for determining the value of [x + 1]. |
| Value of y in the domain | For -4 < x < -3, y = -3. | Shows that the function is constant in this interval. |
| Derivative of a Constant | The derivative of any constant is 0. | Applied directly to find the derivative of y = -3. |
| Derivative at a Point | The slope of the tangent line at that point. For a constant function, it's always 0. | The required calculation at x = -3.5. |
Additional Information: Differentiability of Step Functions
Functions like the greatest integer function, which have 'jumps' or discontinuities, are generally not differentiable at the points of discontinuity (the integer values). However, in this question, we are asked for the derivative at x = -3.5, which is *between* the jump points (-4 and -3 for x in the domain, which correspond to jumps in x+1 at -3 and -2 respectively).
In the open interval -4 < x < -3, the function y = [x + 1] is equivalent to the constant function y = -3. A constant function is continuous and differentiable everywhere.
The derivative exists at x = -3.5 because the function is locally constant around this point. The derivative of a constant function is always 0.
If the question had asked for the derivative at x = -4 or x = -3 (if included in the domain), or at any point where x + 1 is an integer (like x = -3, x = -2, etc.), the derivative would not exist because the greatest integer function has discontinuities at these points.
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