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Timbo Slice

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  • Derivative d(a/b) w.r.t (a/b)?

    x = (a/b)^(2/3)

    What is dx / d(a/b) ?

    I don't understand the notation :O how can you take the derivative with respect to (a/b) ?

    1 AnswerMathematics12 months ago
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    Law of Large Numbers?

    Basically, the sample mean x̄ = (1/n)Σ(Xᵢ) tends to population mean µ, as the sample size n tends to infinity.

    I understand the concept in layman's terms. However I don't quite understand the math description:

    What in the world is this epsilon? I would think |x̄ - µ| = 0, not 1

    2 AnswersMathematics1 year ago

  • Find the Confidence Level of Interval?

    I'll omit the subscript so Z*α/2 becomes Z* for readability

    I have the following confidence interval from a difference of proportions

    [0.012, 0.148] = [0.08 - 0.0693Z*, 0.08 + 0.0693Z*]

    What are the confidence/significant levels?

    I believe you get Z* = 0.9809? Now I just need to find confidence (1 - α) and significance α using the right table. I'm just confused about which table, since we have the α/2 division of 2.

    1 AnswerMathematics1 year ago

  • Related Rate, Volume of cube?

    V = s^(3)

    s(t) = t

    Then s'(t) = 1 and V'(t) = 3t^(2)

    Now let V(t) = t

    Then V'(t) = 1

    And s'(t) = ?

    Why is s'(t) = (1 / (3t^(2/3))?

    Mathematics1 year ago
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    Why is E{[X-E(X)][(X-a)-E(X-a)]} = Var(X)?

    For the second last line, how does the numerator equal to Var(X)? 

    I know that Var(X) = E{[X-E(X)]²} 

    I can't quite figure out what's going on though. How does the constant a disappear?

    1 AnswerMathematics1 year ago
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    Proof of E(X + Y) explanation of step?

    1) In general, do we have 1 sigma for each random variable?2) In the fourth line, we can take the x (which is summed over i) out of the summation operator over j. Is this always the case, because there's no j attached to the x term?

    3) In the fifth/last line, why does the Pj disappear, and we're left with the Pi probability mass function part thing?

    1 AnswerMathematics2 years ago
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    B = (A∩B)⋃(A∩B̅)?

    Why is B = (A∩B)⋃(A∩B̅)?

    It looks to me like it equals A? This is tripping me up!A∩B is the intersectionA∩B̅ ? Its the intersection of A and everything outside of B, so shouldn't this be A minus its the intersection with B?Then you add them together, it's A? Where am I thinking wrong?

    1 AnswerMathematics2 years ago

  • Min and max slope? Max angle we can rotate graph?

    f(x) = e⁴ˢᶦⁿ⁽ˣ⁾ - 3x

    What is the maximum angles we can rotate this about the origin and still have a proper function?

    My work

    f'(x) = e⁴ˢᶦⁿ⁽ˣ⁾*4cos(x) - 3

    f''(x) = [ e⁴ˢᶦⁿ⁽ˣ⁾*4cos(x)*4cos(x) ] - [ 4e⁴ˢᶦⁿ⁽ˣ⁾*sin(x) ] = 0

    From symbolab online calculator, solving for x in the interval [-2pi, 2pi]

    x = 1.08175 + 2pi, and pi - 1.08175 + 2pi

    Plugging these into f'(x),

    The maximum slope = 61.19

    The minimum slope = -67.19

    My questions

    Q1) What is the maximum acute angle through which the graph can be rotated counter-clockwise? Why is it 90 - arctan (61.19) = 0.936?

    Q2) What is the maximum acute angel through which the graph can be rotated clockwise? Why is it 90 - arctan (-67.19579) = 179.147?TL;DR, Why do we calculate 90 - arctan for rotating the graph? I have no idea what rotating a graph even means...

    Mathematics2 years ago
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    Find f'(x) using the definition of the derivative?

    Given function f has the properties:

    f(x + y) = f(x) + f(y) + x²y + xy²

    lim{x-->0} = f(x)/x = 1

    Find f'(x)

    I typed out my questions and work in the picture.

    Also, why do we start out by testing x and y = 0? I really wouldn't know how to start without researching online.

    2 AnswersMathematics2 years ago

  • Can the limit be found without using L'hospital rule?

    lim x --> 1

    [ √(x) - x² ] ÷ [ 1 - √(x) ]

    Is it possible to solve this limit without using L'hospital rule, since it hasn't been taught in class yet?

    Thank you

    4 AnswersMathematics2 years ago

  • Probability of the first coin toss being a head?

    You have 3 coins tossed independently.

    If your first coin toss is a head, you get one dollar.

    Q) What is the probability you get one dollar, given that you get 3 heads?

    *The word given shouldn't be used, rather we're interested in the joint probability. Does the word given matter?

    The answer is 1/8

    I don't understand why, wouldn't it be 1, since if you have 3 coins and 3 heads, the first coin MUST be a head?

    1 AnswerMathematics2 years ago

  • Hyper-geometric distribution problem?

    From 20 components, 18 are good, 2 are defective.

    If I take 5, the probability all 5 are good is:

    r = 18, N-r = 2, n = 5

    P(Y = 5) = 0.553.

    Now, it takes 2 minutes to install these components if they are good, and 10 minutes to install if they are defective.

    What is the mean and variance of the time to install 5 components?

    E(X) = 4.5 good components

    4.5*2 + .5*10 = 9.5 minutes to install 5 components on average

    V(X) = ?

    The solution is V(T) =

    1 AnswerMathematics2 years ago

  • What is consumer surplus for the last buyer?

    1 AnswerEconomics3 years ago

  • DNA insertion mutation consequences? Protein and mRNA length?

    We have a 1000 base pair insertion in some gene.

    Consequences?

    The mRNA is longer by 1000 base pairs.

    1)The same amount of mRNA is transcribed. Why isn't it more?

    Protein may be longer or shorter depending on start/stop codons altered.

    Probably causes non-functional protein.

    2) The same amount of protein is translated. Why is it the same amount?

    1 AnswerBiology3 years ago

  • Gene linkage, find genotype/phenotype of parents given progeny?

    A plant has 3 genes? that control colour.

    P purple > p1 light purple > p white.

    The number of leaves is controlled by 2 genes?

    F five leafs > f four leafs

    I cross a purple leaf with five leafs to white flower with four leafs.

    F1 Progeny:

    35 purple and five leafs

    36 light purple and five leafs.

    1) What is the genotype of the parents crossed?

    I just had this on a test, hopefully I have all the information!

    I guessed the parents :(

    The dot means we don't know if genes are linked.

    My punnet square for the hypothesized parent genotype:

    P/p1 ⋅ F/F x p/p ⋅ f/f

    .........P ⋅ F...........P ⋅ F...........p1 ⋅ F............p1 ⋅ F

    p ⋅ f...P/p ⋅ F/f......P/p ⋅ F/f......p1/p ⋅ F/f......p1/p ⋅ F/f

    Hope the above is understandable!

    Progeny is 1/4 for all:

    P/p ⋅ F/f

    P/p ⋅ F/f

    p1/p ⋅ F/f

    p1/p ⋅ F/f

    So half are Purple with five leafs

    Other half are light purple with five leafs!

    Parents must be: P/p1 ⋅ F/F and p/p ⋅ f/f?

    1 AnswerBiology3 years ago
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    What is this question asking? Critical points of multi variable functions?

    I'm just struggling with understanding what the question is asking? O.o

    I know about critical points existing when the gradient (all derivatives are zero) is 0.

    I just don't understand what the question even wants lol.

    1 AnswerMathematics3 years ago
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    Polar coordinates, interval of integration?

    1) Why is the interval of integration for dθ from just 0 to π?

    It interval kinda looks like it's from -π/4 to π/4. So I can see the "duration" of the integral is π radians also. So...

    2) Why does it not matter where we start our integration? What am I missing?

    2 AnswersMathematics3 years ago
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    Explain multivariable differentiability definition?

    For z = f(x, y), the total differential of z is

    dz = fₓdx + fᵧdy, by definition.

    So I understand the above sort of. At least I can read it.

    I cannot understand the next definition for multivariable differentiability o.O I don't think any can! It looks like gibberish to me!

    1 AnswerMathematics3 years ago
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    Magnitude of magnetic field from current carrying wire?

    For point 1, the magnetic field due to the horizontal wire and diagonal wire add up. The answer I get is C.

    (μ₀×I) / (πd)

    This is correct.

    1) For point 2, I'm not sure how to find the magnitude of the magnetic field due to the diagonal wire? The answer is F apparently.

    2) Why did we not have to account for the angle for point 1, but now it matters for point 2? Because the horizontal wire is in between?

    1 AnswerPhysics3 years ago

  • What is ionizing radiation?

    Nuclear radiation energy is much larger than visible light energies.

    So instead of just being absorbed and increasing an objects temperature, nuclear radiation ionizes matter and breaks molecular bonds.

    Explanation was because:

    "These energies are much larger than ionization energies of atoms and molecules".

    "The ionization energies of an atom is about 10 eV, while alpha or beta particles are around 1 MeV".

    How does an atom's ionization energy and ionizing radiation fight? Atom ionization energy is energy needed to rid of an electron.

    Nuclear radiation just damages everything.

    I don't see how they connect?

    1 AnswerPhysics3 years ago