Actually the integral is pretty easy and would be calc 1. Most books don't get into sums until calc 2 though. e^ipi is just an identity you probably don't need until calc2/3 or Diff Eq, depending.
e^iPi isn't even a problem, its just something used to transform between various things. Its just -1, after all.
(that totally freaked me out when I first discovered that. Who would expect that taking e and raising it to i times Pi would be NEGATIVE ONE of all the damn things?!)
Overlay is probably overkill, but the answers are simple enough (I hope I got these right, it's been a while... -_-).
e^(i*Pi) = -1
62 + (2/3)
I forget the formal expression of this, but the limit approaches 2
4x^2 + 4x + 1
3 + (11/12)
2048
63
It might be overkill, but I did it anyway for the ones I understood in order to show readers how to get the correct answers. I was good at math when I was in school, but I didn't take university-level courses on these subjects and wasn't taught the more advanced concepts in high school, so I don't know the second and third ones (I found Euler's identity by accident while taking a Wiki Walk online some time ago).
Poor girl, being humiliated in front of her class then having it posted to the internet, whose first reaction is to start discussing her answers. Who would be tactless enough to do such a thing‽ Anyways, I added explanations to the second and third questions. Could someone smarter than me have a look at them?
GO TO grade schoolThe hell is this...Whaaaa...Even idiocy has limits.SERIOUSLY, DIE!This is way too stupidGarbageSo stupid it's amazingShe just added them lololol There's a pink arrow behind her back pointing to 9x7How do you even go shopping??The numerators and denominators of the fractions were incorrectly added together without first multiplying them to reach the least common multiple of the denominators. The correct answer is as follows:
2 (1 / 4) + 1 (2 / 3) = [Original equation]
2 (3 / 12) + 1 (8 / 12) = [Multiply 1 / 4 (both numbers, to keep its value the same) by 3 and multiply 2 / 3 (both numbers again) by 4 so that both denominators equal 12]
3 (11 / 12) [NOW you add the numerators of the fractions BUT NOT the denominators; if the numerator is greater than or equal to the denominator, subtract the denominator from the numerator and add 1 to the whole number as needed (not needed here)]9 * 7 = 63. 9 + 7 = 16.What is e?
The correct answer is -1. Euler's identity is the mathematical formula "e ^ (π * i) + 1 = 0" where e is Euler's number (the base for natural logarithms [ln]), π (pi) is a circle's diameter divided by its circumference (always almost equals 3.14), and i is the "imaginary unit" which provides "i ^ 2 = -1" with an answer.Division by the fraction was incorrectly applied to the base of the exponent instead of being properly converted to multiplication by an integer and applied to the exponent. The correct answer is as follows:
2 ^ 10 / 0.5 = [Original equation]
2 ^ 10 * 2 = [Division by a fraction equals multiplication by its reciprocal except for 0 (which has no reciprocal)]
2 ^ 11 = [Multiplying by the base of the exponent increases the exponent by 1; multiplying by a power of the exponent increases the exponent accordingly]
2048 [The correct answer]Assuming the little white tint at the tip of her hair isn't more writing on the chalkboard behind her, the exponent was incorrectly applied to 2 and x separately instead of treating 2x as a single unit. The correct answer is as follows:
(2x + 1) ^ 2 = [Original equation]
(2x + 1) * (2x + 1) = [Expanding the exponent into multiplication]
( (2x * 2x) + (1 * 2x) ) + ( (2x * 1) + (1 * 1) ) = [First multiply the first unit within the first parenthesis group by each unit within the second parenthesis group, then multiply the second unit within the first parenthesis group by each unit within the second parenthesis group), before finally adding the two groups together]
(4x^2 + 2x) + (2x + 1) = [Because 2 is an integer while x is a variable, 2 can be squared to 4 but x^2 must be presented in variable-and-exponent form]
4x^2 + 4x + 1 [The correct answer]This is a summation equation. The bottom states what i starts out equal to, and the top states when the summation stops (in this case, the summation never stops).
∑∞ᵢ ₌ ₀ ¹/₂
= (¹/₂)⁰ + (¹/₂)¹ + (¹/₂)² ⋯ + (¹/₂)^∞
= 1 + ¹/₂ + ¹/₄ ⋯
= 1 + ∑∞ᵢ ₌ 1 ¹/₂
Since ∑∞ᵢ ₌ ₁ ¹/₂ = 1,
= 1+1
= 2
Miss Ushiromiya has just substituted the i in (¹/₂)ⁱ with i=0. Her answer is also not simplified (it should be 1 as (¹/₂)⁰=1).This is a (definite) integration equation, part of calculus, which among other things can be used to find the area under a curve. When one integrates (definitely), we increase the exponent of 𝑥 by one then divide it by the new exponent (∫²₁ 𝑥d𝑥 = [¹/₂*𝑥²]²₁), then replace 𝑥 with the top number and bottom number and subtract the two ([¹/₂*𝑥²]²₁ = [¹/₂*2²]-[¹/₂*1²] = [2]-[¹/₂] = 1¹/₂).
∫³₋₁ (𝑥³+2𝑥²+6)d𝑥
By the addition rule of integration,
= ∫³₋₁ (𝑥³)d𝑥 + ∫³₋₁(2𝑥²)d𝑥 +∫³₋₁ (6𝑥⁰)d𝑥
By the constant multiple rule of integration,
= ∫³₋₁ (𝑥³)d𝑥 + 2∫³₋₁(𝑥²)d𝑥 + 6∫³₋₁ (𝑥⁰)d𝑥
= [¹/₄*𝑥⁴]³₋₁ + 2[¹/₃*2𝑥³]³₋₁ + 6[¹/₁*𝑥¹]³₋₁
By the constant multiple rule of integration,
= ¹/₄[𝑥⁴]³₋₁ + (2*¹/₃)[2𝑥³]³₋₁ + (6*1)[𝑥¹]³₋₁
= ¹/₄[3⁴-(-1)⁴] + ²/₃[3³-(-1)³] + 6[3¹-(-1)¹]
= ¹/₄[81-1] + ²/₃[27-(-1)] + 6[3-(-1)]
= ¹/₄[80] + ²/₃[28] + 6[4]
= 20 + ²⁸/₃ + 24
= ¹⁸⁸/₃
= 62²/₃
The addition rule of integration states that '∫ f(𝑥) + g(𝑥)d𝑥 = ∫ f(𝑥)d𝑥 + ∫ g(𝑥)d𝑥.' That is to say, an integral of a sum of functions is equal to the integrals of the individual bits of that sum.
The constant multiple rule of integration states that 'if k is constant, ∫ kf(𝑥)d𝑥 = k∫ f(𝑥)d𝑥.'
f(𝑥) represents a function of 𝑥. 𝑥, 7𝑥, (3𝑥+2)(𝑥-2) can all be represented by f(𝑥)
...I'm not entirely sure how Miss Ushiromiya here got 14.
