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Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a
(a) constant function
(b)continuous function
(c) continuous function but not differentiable (
d) neither continuous nor differentiable?
(a) constant function
(b)continuous function
(c) continuous function but not differentiable (
d) neither continuous nor differentiable?
Most Upvoted Answer
Let f / R -> R be a continuous function with period p > 0 Then g(x) = ...
Understanding the Function g(x)
To analyze the function g(x) defined as:
g(x) = ∫[x to x + p] f(t) dt
where f is a continuous function with period p, we need to explore its properties.
1. Periodicity of f
- Since f is periodic with period p, we have:
- f(t + p) = f(t) for all t in R.
2. Evaluating g(x + p)
- We can compute g(x + p):
- g(x + p) = ∫[x + p to x + 2p] f(t) dt.
- By changing the variable of integration (let u = t - p):
- g(x + p) = ∫[x to x + p] f(u + p) du = ∫[x to x + p] f(u) du = g(x).
3. Conclusion on g(x)
- Since g(x + p) = g(x), g is periodic with the same period p.
4. Continuity of g(x)
- g(x) is defined as an integral of a continuous function f(t) over a closed interval.
- By the properties of definite integrals, g(x) is continuous because the integral of a continuous function over a variable interval remains continuous.
5. Differentiability of g(x)
- To determine differentiability, we can apply the Fundamental Theorem of Calculus:
- g'(x) = f(x + p) - f(x) (using Leibniz's rule).
- Since f is periodic, this doesn't guarantee that g is differentiable everywhere, as f could have points of non-differentiability.
Final Assessment
- g(x) is a continuous function due to the continuity of f and the properties of integrals.
- However, g(x) is not guaranteed to be differentiable everywhere.
Correct Answer
- g(x) is a continuous function but not necessarily differentiable (option c).
To analyze the function g(x) defined as:
g(x) = ∫[x to x + p] f(t) dt
where f is a continuous function with period p, we need to explore its properties.
1. Periodicity of f
- Since f is periodic with period p, we have:
- f(t + p) = f(t) for all t in R.
2. Evaluating g(x + p)
- We can compute g(x + p):
- g(x + p) = ∫[x + p to x + 2p] f(t) dt.
- By changing the variable of integration (let u = t - p):
- g(x + p) = ∫[x to x + p] f(u + p) du = ∫[x to x + p] f(u) du = g(x).
3. Conclusion on g(x)
- Since g(x + p) = g(x), g is periodic with the same period p.
4. Continuity of g(x)
- g(x) is defined as an integral of a continuous function f(t) over a closed interval.
- By the properties of definite integrals, g(x) is continuous because the integral of a continuous function over a variable interval remains continuous.
5. Differentiability of g(x)
- To determine differentiability, we can apply the Fundamental Theorem of Calculus:
- g'(x) = f(x + p) - f(x) (using Leibniz's rule).
- Since f is periodic, this doesn't guarantee that g is differentiable everywhere, as f could have points of non-differentiability.
Final Assessment
- g(x) is a continuous function due to the continuity of f and the properties of integrals.
- However, g(x) is not guaranteed to be differentiable everywhere.
Correct Answer
- g(x) is a continuous function but not necessarily differentiable (option c).
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Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable?
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Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable?.
Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / R -> R be a continuous function with period p > 0 Then g(x) = integrate f(t) dt from x to x + p is a(a) constant function(b)continuous function(c) continuous function but not differentiable (d) neither continuous nor differentiable?.
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