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... Lesson 1: Functions and plots. - Science and math experience The inside story of what functions really are. Getting control of the functions of math and science by interacting with plot commands. Analyzing the US population data. Growth and percentage growth. Lesson 2: Make your own functions to get the job done - Science and math experience Calculating what you have to pay: College tuition, credit card payoff and car loans. Managing a catfish pond. Predator-prey interaction. Avoiding a DUI. Inflation.Discharge of a battery. War games including a simulation of the battle of Iwo Jima in World War II. Chaos - Just as in Jurassic Park. Lesson 3: Functions folks use for calculation in science and enginering: Linear and exponential functions - Science and math experience Linear functions are those that grow by a fixed amount everytime x goes up by h,: Exponential functions are those that grow by a fixed percentaget everytime x goes up by h, The base e. Consumer math: Exponential functions, bank accounts and compound interest. Use linear functions for centimeter-inch conversions, estimating the height of a tree, Chevy versus Honda and Airplane mileage. Use exponential functions for battery discharging, carbon dating, rock dating, underwater illumination, motor vehicle registrations and inflation awareness. Experiments with the natural logarithm. Linear and exponential data analysis. Lesson 4: Functions folks use for calculation in science and engineering: The oscillating functions Sin[x] and Cos[x] - Science and math experience Sin[x] and Cos[x] both oscillate between -1 and 1 and repeat themselves everytime x goes up by 2π. Cos[x] is a shifted version of Sin[x]. Even and odd functions. Frequency and period. The engineering oscillators: Harmonic oscillators, damped harmonic oscillators and beating oscillators. Fourier fit of periodic functions and periodic data with Sines and Cosines. Tan[x]. Morphing one function into another. Lesson 5: Functions folks use for calculation in science and enginering: Power functions xk. - Science and math experience Dominance in the global scale: The dominant power function in a polynomial is the highest power involved. In the global scale, exponential growth dominates power growth. Using power functions to calculate accurate values of other functions. Geometric sums and drug dosing. Experiments in factoring. Area, pictures and algebra. Using the fact a2-b2 = (a-b ) (a+b) to advantage. Young Carl Gauss, the Luke Skywalker of math. More morphing. Lesson 7 Measurements - Science and math experience Scaling and proportions. Pythagorean theorem. Monte Carlo sampling estimates of area and volume measurements. The bell-shaped curve and normal probability estimates via Monte Carlo. IQ percentiles. Area and volume measurements resulting from stretching along the axes. Light bulb failures via Monte Carlo sampling. Lesson 8: Trips on the unit circle, Sines and Cosines and radians - Science and math experience {Cos[t],Sin[t]} always plots out on the unit circle. To get to the position of {Cos[t],Sin[t]} on the unit circle, you start at {1,0} and take a trip of length t on the unit circle. A trip on the unit circle of length t corresponds to t radians. Sines and Cosines in automotive mechanics: Wheel bolt circle diameter. Parametric plotting with Sines and Cosines: Elliptical orbits of planets and asteroids. Lesson 9: Rotaition and reflections Using rotations to get the formulas Cos[s + t] = Cos[s] Cos[t] - Sin[s] Sin[t] and Sin[s + t] = Sin[s] Cos[t] + Sin[t] Cos[s]. When you flip {x,y} over the line through {0,0} and {Cos[s],Sin[s]}about {0,0}, you get {x Cos[2 s] + y Sin[2 s], x Cos[2 s] -y Sin[2 s]} . This results from a rotation, a flip over the x-axis, followed by another rotation. Using flips to do ray tracing: Bouncing light rays off curves. Lesson 10: Solving equation - Science and math experience Equations are different from formulas. When you put f[x] = x2 - 2 x - 4 , you are giving a formula for f[x] which works no matter what x is. When you go after the particular x's that make f[x] = 0, you are solving the equation f[x] = 0. The quadratic formula helps you solve this equation. Coming up with the deepest dip or highest crest of a quadratic and using it to get the quadratic formula without the traditional "completing the square." To solve for more than one variable, you generally need at least as many equations as variables. Lesson 11: Imaginary and complex numbers (will not appear in the preliminary version) Lesson 12: The arithmetic-geometric mean inequality and how to sue it to solve many max-min problems (will not appear in the preliminary version) |
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