GATE CS 2016 - Set 2 - 1 Mark Question
Duration: 3 min
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Inside: a video lesson and guided study material.
Module outline
- Engineering Mathematics: Linear Algebra, Calculus, Permutation and Combination, Probability, Statistics, Powers, Surds and Logarithms, Number System, Alphabet and Word Tests, Sequences and Series, Transformation of Shapes, Signal Space and Optimum Detection, Probability, Statistics, Series (Number and Letter Series) (Numerical Relations and Reasoning), Powers and Exponents (Surds and Indices)
AI summary & chapters
AI Summary
An AI-generated summary of this video lecture.
This educational video solves a calculus problem from the GATE CS 2016 exam regarding polynomial degrees and derivatives. The instructor demonstrates a practical method using a specific example function to determine the degree of a derived expression involving f(x) and g(x). The lesson covers properties of even and odd functions and how differentiation affects polynomial degrees.
Chapters
0:00 – 2:00 00:00-02:00
The video opens with a blackboard background displaying the word "CALCULUS" and various equations like cos x and sin x. The main problem is displayed in red text: "Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x) + f(-x)) is 10, then the degree of (g(x) - g(-x)) is _______." The text indicates this is from "(GATE CS 2016) (SET-2) (1 MARK)". The instructor, Yash Jain Sir, wearing a blue polo shirt with a "KG" logo, begins solving by assuming a specific polynomial to simplify the logic. He writes f(x) = x^10 on the digital whiteboard. He then calculates f(-x) = (-x)^10, simplifying it to x^10. He sums them to show f(x) + f(-x) = x^10 + x^10 = 2x^10, confirming the degree is 10. He then differentiates to find g(x), writing g(x) = 10x^9.
2:00 – 3:08 02:00-03:08
The instructor continues by evaluating g(-x). He writes g(-x) = 10(-x)^9, which simplifies to -10x^9. He then substitutes these into the expression g(x) - g(-x), resulting in 10x^9 - (-10x^9). This simplifies to 20x^9. He circles the exponent 9 and writes the final answer "9" in the blank space. The video concludes with a slide titled "Correct Answer: Option C". The solution text explains that f(x) must be an even function because f(x) + f(-x) has a degree. It derives that g(x) = -g(-x), leading to g(x) - g(-x) = 2f'(x). Since f(x) is degree 10, f'(x) is degree 9, making the final degree 9. The slide explicitly shows the step g(x) - g(-x) = f'(x) - (-f'(-x)).
The lesson progresses from a specific example to a general theoretical proof. By assuming f(x) = x^10, the instructor shows that g(x) becomes 10x^9. The final slide reinforces that if f(x) is even, its derivative g(x) is odd, meaning g(x) - g(-x) effectively doubles g(x), preserving the degree of 9.