Gate CS 2015 - Set 2 - 1 Mark Question
Duration: 5 min
This video lesson is available to enrolled students.
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This educational video features a lecture solving a calculus problem from the GATE CS 2015 exam. The instructor analyzes the function f(x) = x^(-1/3) over the interval [-1, 1]. He evaluates three statements concerning the function's continuity, boundedness, and the finiteness of the area under the curve. Through board work and explanation, he demonstrates that while the function is discontinuous and unbounded at x=0, the improper integral converges to a finite value.
Chapters
0:00 – 2:00 00:00-02:00
The video opens with a title card displaying 'CALCULUS' amidst various mathematical formulas. The instructor introduces a problem from GATE CS 2015 involving the function f(x) = x^(-1/3). He writes the function on the whiteboard in two forms: f(x) = x^(-1/3) and f(x) = 1/x^(1/3). He sets up the integral for the area A bounded by the curve and the X-axis from x = -1 to 1, writing A = integral from -1 to 1 of (1/x^(1/3)) dx. He identifies the three statements to be evaluated: continuity, boundedness, and the nature of the area.
2:00 – 5:00 02:00-05:00
The instructor analyzes the first statement regarding continuity. He writes 'continuous in [-1, 1]' and marks it with an X, explaining that the function is undefined at x=0. He writes the limit expression lim(x->0+) 1/x^(1/3) = infinity to show the vertical asymptote. Next, he addresses the second statement, 'f is not bounded in [-1, 1]', marking it with a check. He explains that since the limit approaches infinity, the function is unbounded. He writes 'range f(x) = 1/x^(1/3)' to illustrate the unbounded range. Finally, he discusses the third statement about the area A, noting that despite the discontinuity, the integral converges.
5:00 – 5:28 05:00-05:28
The video concludes with a summary slide displayed on the screen. The text explicitly states that statement 1 is false because the function is not continuous. It confirms statement 2 is true as the function becomes infinite at x=0. It also confirms statement 3 is true, stating the area is bounded and calculable via integration. The instructor points to the options, indicating that C (2 and 3 only) is the correct answer. The final frame shows a 'THANKS FOR WATCHING' message.
The lecture effectively illustrates the properties of improper integrals for power functions. It clarifies that a function can be discontinuous and unbounded at a specific point within an interval yet still yield a finite area under the curve, provided the exponent of the singularity is greater than -1. This distinction is crucial for understanding convergence in calculus.