GATE EC 2016 Set 1 — Question 15
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Engineering Mathematics → Calculus → Definite & Improper Integrals
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Question
Consider the plot of versus as shown below.
Suppose . Which one of the following is a graph of ?
Correct answer
(C) [figure]
Solution
The function is defined as the integral of from to : . By the Fundamental Theorem of Calculus, .
1.Initial Value: At , . All graphs except (D) satisfy this.
2.Interval : In this region, the graph of is below the x-axis (). Since , must be strictly decreasing from to .
3.Critical Point at : At , . Since changes from negative to positive at this point, reaches a local minimum at .
4.Interval : In this region, , so and must be strictly increasing.
5.Final Value: . The graph of is an odd function (), meaning the area from to is equal in magnitude but opposite in sign to the area from to . Thus, the total integral .
Comparing these characteristics with the given options, only graph (C) correctly represents as it starts at 0, decreases to a minimum at , and returns to 0 at .Continue learning with Success Tracker
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