From 4a82a1ed7ad4b9f2871b045e17ad3c62dbf8281a Mon Sep 17 00:00:00 2001 From: andyeisenberg Date: Fri, 9 Jan 2026 13:38:46 -0500 Subject: [PATCH] Better recitation formatting --- source/recitations/calculus-review.ptx | 175 +++++++++++++++---------- 1 file changed, 108 insertions(+), 67 deletions(-) diff --git a/source/recitations/calculus-review.ptx b/source/recitations/calculus-review.ptx index c63aa14..29eeaa8 100644 --- a/source/recitations/calculus-review.ptx +++ b/source/recitations/calculus-review.ptx @@ -11,75 +11,116 @@ All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.

- - - -

- The Power Rule for derivatives states: - - \frac{d}{dx}\left( x^n \right) = - -

- - - - -

- The Power Rule for antiderivatives states: - - \int\left( x^n \right)\ dx = - -

- - - - -

- Practice with the following derivatives: - - \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) \amp = - \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) \amp = - -

- - - - -

- Practice with the following antiderivatives: - - \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx \amp = - \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx \amp = - -

- - - - Fundamental Theorem of Calculus - - - + + + + +

- If F(x) is an antiderivative of f(x)i.e., if F'(x) = f(x)then: - - \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). - - If f(x) is a probability density function (don't worry about what that means for now), then integrals such as \int_a^b f(x)\ dx are used to calculate certain probabilities, and so are function values of the form F(x). + The Power Rule for derivatives states: + + \frac{d}{dx}\left( x^n \right) = +

- - - - - -

- Practice with the following integrals: - - \int_0^1 \frac{3}{2} x^{3/2} \ dx \amp = - \int_0^x 3e^{-3t} \ dt \amp = - -

- - + + + + +

+ The Power Rule for antiderivatives states: + + \int\left( x^n \right)\ dx = + +

+ + + + + +

+ Practice with the following derivatives: +

+ + + + +

+ \displaystyle \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) = +

+ + + + +

+ \displaystyle \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) = +

+ + + + + +

+ Practice with the following antiderivatives: +

+ + + + +

+ \displaystyle \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx = +

+ + + + +

+ \displaystyle \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx = +

+ + + + + + + + Fundamental Theorem of Calculus + + +

+ If F(x) is an antiderivative of f(x)i.e., if F'(x) = f(x)then: + + \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). + +

+ + + +

+ If f(x) is a probability density function (don't worry about what that means for now), then integrals such as \int_a^b f(x)\ dx are used to calculate certain probabilities, as are values of the form F(x). +

+ + + +

+ Practice with the following integrals: +

+ + + + +

+ \displaystyle \int_0^1 \frac{3}{2} x^{3/2} \ dx = +

+ + + + +

+ \displaystyle \int_0^x 3e^{-3t} \ dt = +

+ + + \ No newline at end of file