Showing posts with label Sweet Sixteen. Show all posts
Showing posts with label Sweet Sixteen. Show all posts
Thursday, March 31, 2022

March Calculus Madness Sweet Sixteen - Day 16: A Parametric Integral Example

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 ------------ Welcome to March Calculus Madness! ------------ Let: x(t) = a * cos t y(t) = b * sin t Then: ∫ y(x) dx = ∫ y(t) * x'(t) dt...
Wednesday, March 30, 2022

March Calculus Madness Sweet Sixteen - Day 15: r = α * (1 - cos Θ)

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 ------------ Welcome to March Calculus Madness! ------------ r = α * (1 - cos Θ) r = α * (1 - cos Θ) r^2 = α^2 * (1 - 2 * cos Θ + cos^2 Θ) ...
Tuesday, March 29, 2022

March Calculus Madness Sweet Sixteen - Day 14: The Arc Length of a Spiral

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 ------------ Welcome to March Calculus Madness! ------------ The Length of a Spiral from 0 ≤ Θ ≤ m The equation of a spiral:  r = α * Θ The...
Monday, March 28, 2022

March Calculus Madness Sweet Sixteen - Day 13: Some Double Integrals

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 ------------ Welcome to March Calculus Madness! ------------ Double Integration Time (I)  ∫ ∫ sin(a*x + b*y) dx dy = ∫ -1/a * cos(a*x + b*y...
Sunday, March 27, 2022

March Calculus Madness Sweet Sixteen - Day 12: Arc Length of Sine and Cosine

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 ------------ Welcome to March Calculus Madness! ------------ What is the arc length of y = sin x and y = cos x from x = 0 to x = 2 * π The ...
Saturday, March 26, 2022

March Calculus Madness Sweet Sixteen - Day 11: ∫ int(x) dx

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 ------------ Welcome to March Calculus Madness! ------------ f(x) = IP(x), integer function, HP Prime int(x):  integer part function Domain...
Friday, March 25, 2022

March Calculus Madness Sweet Sixteen - Day 10: ∫ frac(x) dx

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 ------------ Welcome to March Calculus Madness! ------------ frac(x) - HP Prime frac(x):  fractional part function Domain:  0 ≤ frac(x) ...
Thursday, March 24, 2022

March Calculus Madness Sweet Sixteen - Day 9: ∫ e^x/(e^x + 1) dx and ∫ e^x/(e^x - 1) dx

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 ------------ Welcome to March Calculus Madness! ------------ ∫ e^x/(e^x + 1) dx Let z = e^x Then:  ln z = x 1/z dz = dx ∫ e^x/(e^x + 1) dx ...
Wednesday, March 23, 2022

March Calculus Madness Sweet Sixteen - Day 8: ∫ ln^2 x dx and ∫ ln^3 x dx

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 ------------ Welcome to March Calculus Madness! ------------ Integration by parts to the rescue! ∫ ln^2 x dx u = ln^2 x,   dv = dx du = 2 *...
Tuesday, March 22, 2022

March Calculus Madness Sweet Sixteen - Day 7: ∫ 1/(1 + cos x) dx and ∫ 1/(1 + sin x) dx

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 ------------ Welcome to March Calculus Madness! ------------ ∫ 1/(1 + cos x) dx and ∫ 1/(1 + sin x) dx To tackle these integrals, we will m...
Sunday, March 20, 2022

March Calculus Madness Sweet Sixteen - Day 5: x^n ∙ √(1 + x)

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 ------------ Welcome to March Calculus Madness! ------------ d/dx x^n ∙ √(1 + x) Here we can make use the multiplication rule: d/dx f(x) ∙ ...
Saturday, March 19, 2022

March Calculus Madness Sweet Sixteen - Day 4: ∫x^2 / √(1 - x^2) dx

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 ------------ Welcome to March Calculus Madness! ------------ ∫x^2 / √(1 - x^2)  dx Substitute:  z = arcsin x sin z = x sin^2 z = x^2 and dz...
Friday, March 18, 2022

March Calculus Madness Sweet Sixteen - Day 3: Derivative and Integral of x^n∙(1+x)^2

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 ------------ Welcome to March Calculus Madness! ------------ x^n ∙ (1 + x)^2 = x^n ∙ (1 + 2 ∙ x + x^2) = x^n + 2 ∙x^(n+1) + x^(n+2) d/dx x^...
Thursday, March 17, 2022

March Calculus Madness Sweet Sixteen - Day 2: Derivative and Integral of the Absolute Value Function

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 ------------ Welcome to March Calculus Madness! ------------ What is the derivative and the indefinite integral of the absolute value? By d...
Wednesday, March 16, 2022

March Calculus Madness Sweet Sixteen - Day 1: Double Integration

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------------ Welcome to March Calculus Madness! ------------ For the two-variable function f(x,y), can we assume that ∫ ∫ f(x,y) dx dy = ∫∫ ...