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Question
If `f(x) = 1 - x + x^2 - x^3 + ... -x^99 + x^100`, then f'(1) is equal to ______.
Options
150
– 50
– 150
50
Solution
If `f(x) = 1 - x + x^2 - x^3 + ... -x^99 + x^100`, then f'(1) is equal to 50.
Explanation:
Given that `f(x) = 1 - x + x^2 - x^3 + ... - x^99 + x^100`
f'(x) = `-1 + 2x - 3x^2 + .... - 99x^98 + 100x^99`
∴ f'(1) = `-1 + 2 + 3 + ... - 99 + 100`
= `(-1 - 3 - 5 ... - 99) + (2 + 4 + 6 + ... + 100)`
= `50/2 [2 xx - 1 + (50 - 1)(-2)] + 50/2 [2 xx 2 - (50 - 1)2]`
= `25[-2 - 98] + 25[4 + 98]`
= `25 xx -100 + 25 xx 102`
= `25[-100 + 102]`
= `25 xx 2`
= 50
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