Mechanical Engineering Exam > Mechanical Engineering Questions > The function f(x) = |x+1| on the interval [-2...
Start Learning for Free
The function f(x) = |x+1| on the interval [-2, 0]
- a)Continuous and differentiable
- b)Continuous on the integral but not differentiable at all points
- c)Neither continuous nor differentiable
- d)Differentiable but not continuous
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and diff...
f(x ) = x+ 1
f is continuous in [−2, 0]
but not differentiable at
x =−1 because we can draw
infinite number of tangents at x = −1
Most Upvoted Answer
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and diff...
Explanation:
The function f(x) = |x 1| on the interval [-2, 0] is a piecewise function, which means it has a different formula for different parts of the interval. In this case, the function is defined as:
f(x) = -x-1 for x < />
f(x) = x+1 for x >= -1
To determine if the function is continuous and differentiable on the interval [-2, 0], we need to check the properties of continuity and differentiability at every point in the interval.
Continuity:
A function is continuous at a point x if the limit of the function at x exists and is equal to the value of the function at x. A function is continuous on an interval if it is continuous at every point in the interval.
In this case, the function is continuous on the interval [-2, 0] because the limit of the function exists and is equal to the value of the function at every point in the interval. Specifically:
lim x->-1- f(x) = lim x->-1- (-x-1) = 0
lim x->-1+ f(x) = lim x->-1+ (x+1) = 0
f(-1) = 0
Therefore, f(x) is continuous at x = -1.
At x = -2 and x = 0, the function is also continuous because it is constant (f(-2) = -1 and f(0) = 1).
Differentiability:
A function is differentiable at a point x if the limit of the difference quotient exists as x approaches the point. A function is differentiable on an interval if it is differentiable at every point in the interval.
In this case, the function is not differentiable at x = -1 because the left and right limits of the difference quotient are not equal. Specifically:
f'(x) = -1 for x < />
f'(x) = 1 for x > -1
Therefore, f(x) is not differentiable at x = -1.
At x = -2 and x = 0, the function is differentiable because it is constant (f'(-2) = 0 and f'(0) = 0).
Conclusion:
Based on the above analysis, we can conclude that the function f(x) = |x 1| on the interval [-2, 0] is continuous on the interval but not differentiable at x = -1. Therefore, the correct answer is option 'B'.
The function f(x) = |x 1| on the interval [-2, 0] is a piecewise function, which means it has a different formula for different parts of the interval. In this case, the function is defined as:
f(x) = -x-1 for x < />
f(x) = x+1 for x >= -1
To determine if the function is continuous and differentiable on the interval [-2, 0], we need to check the properties of continuity and differentiability at every point in the interval.
Continuity:
A function is continuous at a point x if the limit of the function at x exists and is equal to the value of the function at x. A function is continuous on an interval if it is continuous at every point in the interval.
In this case, the function is continuous on the interval [-2, 0] because the limit of the function exists and is equal to the value of the function at every point in the interval. Specifically:
lim x->-1- f(x) = lim x->-1- (-x-1) = 0
lim x->-1+ f(x) = lim x->-1+ (x+1) = 0
f(-1) = 0
Therefore, f(x) is continuous at x = -1.
At x = -2 and x = 0, the function is also continuous because it is constant (f(-2) = -1 and f(0) = 1).
Differentiability:
A function is differentiable at a point x if the limit of the difference quotient exists as x approaches the point. A function is differentiable on an interval if it is differentiable at every point in the interval.
In this case, the function is not differentiable at x = -1 because the left and right limits of the difference quotient are not equal. Specifically:
f'(x) = -1 for x < />
f'(x) = 1 for x > -1
Therefore, f(x) is not differentiable at x = -1.
At x = -2 and x = 0, the function is differentiable because it is constant (f'(-2) = 0 and f'(0) = 0).
Conclusion:
Based on the above analysis, we can conclude that the function f(x) = |x 1| on the interval [-2, 0] is continuous on the interval but not differentiable at x = -1. Therefore, the correct answer is option 'B'.
Attention Mechanical Engineering Students!
To make sure you are not studying endlessly, EduRev has designed Mechanical Engineering study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Mechanical Engineering.
|
Explore Courses for Mechanical Engineering exam
|
|
Similar Mechanical Engineering Doubts
Top Courses for Mechanical EngineeringView all
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer?
Question Description
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer?.
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer?.
Solutions for The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering.
Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
|
|
Explore Courses for Mechanical Engineering exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup