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Parabolic Mean Value
Consider the parabola. Part 1: Determine the slope of the secant line between two points and. 
Arithmetic Triangle
The Arithmetic Triangle is a diagram that illustrates the binomial coefficients of the expansion of. Jia Xian (fl. 1050 AD) is known to have discussed the arithmetic triangle in his works. The combinatorial formula for 
Inscribed Circle of a Triangle
The following geometry problem is selected from Aida Yasuaki’s Sanpo tenseiho shinan 《算法天生法指南》 (1811): Suppose a circle is inscribed in a triangle with side lengths 13, 14 
Omar Khayyam's Cubic
Omar Khayyam (1048 – 1131 AD) solved cubic equations of the form by transforming the problem into finding the intersection of a circle and a parabola: where are positive numbers. 
Jiuzhang Suanshu (Fangcheng 13)
今有五家共井，甲二綆不足，如乙一綆；乙三綆不足，如丙一綆；丙四綆不足，如丁一綆；丁五 
Volume of Prismoids
The solids introduced to schoolchildren are called prismoids. The general formula for computing the volume of a prismoid is where is the height, i the area of the top, is the area of the midsection 
Archimedes' Triumph
In Volume I of On the Sphere and the Cylinder, Archimedes determined the volumetric ratio of a sphere to a circumscribed cylinder. The height and width of the cylinder is equal to the diameter of 
Skydiver
Consider a skydiver falling out of an airplane. He opens his parachute during the descent. Approximate the external force on the skydiver, given that air resistance is proportional to the velocity squared. Part 1: Use 
3D Quantum Particle in a Box
Imagine a box with zero potential enclosed in dimensions. Outside the box is the region where the particle’s wavefunction does not exist. Hence, the potential outside the box be infinite. Boundary Conditions 
Adiabatic Process
Physics concept: An adiabatic process is a type of thermodynamic process which occurs without transferring heat or mass between the system and its surroundings. Unlike an isothermal process, an adiabatic process transfers energy to the 
Wave Travelling Through Two Wires
A steel wire of length 33.0 m and a copper wire of length 24.0 m, both with 1.00mm diameters, are connected end to end and stretched to a tension of 170 
Product of Two Negative Numbers
The product of two negative real numbers is often counterintuitive when first introduced. Prove why the product of two negative real numbers is a positive real number. Let A and B be two positive 
Nitrous Acid Titration
A 100.0 mL sample of 0.150 M nitrous acid ( ) with is titrated with 0.300 M sodium hydroxide ( ). Part 1: Determine the pH of the acid solution when no sodium hydroxide is added. 
Molar Mass of Theobromine
Theobromine, also known as 3,7dimethylxanthine, is a compound found in chocolate. When caffeine is metabolized in the liver, around 12% becomes theobromine. The chemical formula of theobromine is. Determine the molar mass of 
One Sheet, Two Cylinders
Consider a sheet of paper where its length is longer than its width. There are two ways to roll up the sheet of paper into a cylinder: along its length, and along its width. Which 
Projectile Motion with Stokes' Flow
Assume the air resistance is proportional to the projectile's velocity. Determine the trajectory of a projectile fired at origin, and with initial velocity and initial position 
Simple Harmonic Oscillator
Consider the position function. Part 1: Determine the value of if this function solves the differential equation: 
Drag Force of a Falling Object
Consider a ball dropped form rest. The ball is subject to a drag force proportional to its velocity squared:. Part 1: Determine the terminal velocity. 
History of STEM
Biographies of Scientists Translations of Ancient Scientific Works 
2D Random Walks with Python
By: Tao Steven Zheng (郑涛) 1 import random 2 import matplotlib.pyplot as plt 3 import turtle 4 5 def randwalk(n): 6 x= 0 7 y= 0 8 step_x= [x] 9 step 
Dicey Primes
Problem created by Tao Steven Zheng (郑涛). A gambling game asks the player to select either a sixsided die (D6) or a twentysided die (D20). Is there a higher probability getting three 
Nine Islands
Problem created by Tao Steven Zheng (郑涛) This is a problem I created when studying polygon networks and combinatorial numbers. 
Volume of the Tricylinder
Consider the solid formed by intersecting three perpendicular cylinders: This solid is called the tricylinder. Determine the volume of the tricylinder. Express the volume in terms of the radius of the cylinder. Then express the 
Ratio Test
Part 1: Consider the following infinite series. Determine whether each infinite series converges or diverges. Part 2: Consider the following infinite series. Determine the radius of convergence and the interval of convergence. 
Capicitors in Series and Parallel
Consider two capacitors with different capacitance and. The formula for calculating the stored charge across a capacitor is. Part 1: Determine the equivalent capacitance when the two capacitors are connected in series.
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