Which of the following signals is monotonic?
Answer & step-by-step solution
A monotonic signal consistently moves in only one direction (non-increasing or non-decreasing) over its domain. Simplify each candidate and/or inspect its derivative; the keyed choice is the expression that reduces to a monotonic function.
Concept & Reasoning
The safest method is to differentiate candidate continuous-time functions. A derivative that keeps one sign indicates monotonic behavior. Periodic sine/cosine functions are generally not monotonic over the entire real line, and log(cos t) is not even defined for all t. The trigonometric identity sin²(2t)+cos²(2t)=1 can reduce one candidate to a simple linear function, which is monotonic.
How to Solve It in the Exam
Simplify identities before differentiating.
Important Exam Point
Monotonic does not mean 'always positive'; it means consistently non-increasing or non-decreasing.
Quick Trick
Simplify identities before differentiating.
Common Mistake
Rejecting a decreasing straight line because it is not increasing.
निम्नलिखित में से कौन-सा संकेत एकदिष्ट (monotonic) है?
उत्तर और चरणबद्ध हल
C — sin²(2t) + cos²(2t) = 1, अतः x(t) = 1 − 2t, जिसका अवकलज −2 हमेशा ऋणात्मक है — यह एकदिष्ट ह्रासमान है। t³ − 2t, sin t और log(cos t) दिशा बदलते हैं।
विस्तृत हल (मूल भाषा में)
Concept & Reasoning
The safest method is to differentiate candidate continuous-time functions. A derivative that keeps one sign indicates monotonic behavior. Periodic sine/cosine functions are generally not monotonic over the entire real line, and log(cos t) is not even defined for all t. The trigonometric identity sin²(2t)+cos²(2t)=1 can reduce one candidate to a simple linear function, which is monotonic.
How to Solve It in the Exam
Simplify identities before differentiating.
Important Exam Point
Monotonic does not mean 'always positive'; it means consistently non-increasing or non-decreasing.
Quick Trick
Simplify identities before differentiating.
Common Mistake
Rejecting a decreasing straight line because it is not increasing.
નીચેનામાંથી કયો સંકેત એકદિશ (monotonic) છે?
જવાબ અને પગલાવાર ઉકેલ
C — sin²(2t) + cos²(2t) = 1, તેથી x(t) = 1 − 2t, જેનું વિકલજ −2 હંમેશા ઋણ છે — તે એકદિશ ઘટતો છે. t³ − 2t, sin t અને log(cos t) દિશા બદલે છે.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Concept & Reasoning
The safest method is to differentiate candidate continuous-time functions. A derivative that keeps one sign indicates monotonic behavior. Periodic sine/cosine functions are generally not monotonic over the entire real line, and log(cos t) is not even defined for all t. The trigonometric identity sin²(2t)+cos²(2t)=1 can reduce one candidate to a simple linear function, which is monotonic.
How to Solve It in the Exam
Simplify identities before differentiating.
Important Exam Point
Monotonic does not mean 'always positive'; it means consistently non-increasing or non-decreasing.
Quick Trick
Simplify identities before differentiating.
Common Mistake
Rejecting a decreasing straight line because it is not increasing.
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