GATE CS 2016 - Set 2 - 1 Mark Question
Duration: 3 min
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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.