Question
Download Solution PDFLet 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?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Greatest Integer Function: (Floor function)
The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.
The domain of [x] is R and the range is I.
Note:
Any function is differentiable only if it is continuous.
The floor function f(x) = ⌊x⌋ is differentiable in every open interval between integers, (n, n + 1) for any integer n.
Calculation:
Given that,
y = [x + 1], -4 < x < -3
We have to determine the derivative at y = [x + 1] at x = -3.5
We know that the floor function is differentiable at all points except integer points.
Hence, y = [x + 1] is differentiable at x = -3.5
⇒ y = [-3.5 + 1] = [-2.5] = -3
⇒ dy/dx = 0
∴ The derivative of y with respect to x at x = -3.5 is 0.
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