Homework 1
SHE 8.7: 1, 3, 13, 17, 23, 24 (you will probably need to use a calculator of some kind)
SHE 9.1: 1, 3, 4, 7, 11, 25, 27, 37
SHE 9.2: 2, 3, 13, 14, 24
Important Concepts for the Week
1. Integral Approximations - Be able to use the left hand, right hand, midpoint, trapezoidal, and Simpson's rules to approximate an integral, and understand where those rules come from. Be able to estimate the error for an approximation using the trapezoidal and Simpson's rules for a given "n". Conversely, given an acceptable error, be able to find an "n" for which you can use these rules to approximate an integral within this error.
2. Differential Equations - Given any differential equation and a function, be able to tell if that function is a solution to that differential equation. Be able to tell if a differential equation is linear or separable, and if so, be able to solve it using the appropriate technique. Be able to interpret a basic differential equation involving velocity and acceleration.
Why We Care
1. Integral Approximations - Often in practice, an integral does not have a "closed form" antiderivative that we can use to find the exact value, and we need to use an approximation to figure out the value of the integral. Additionally, computers and calculators can use methods like this to evaluate integrals and spit back a number.
2. Differential Equations - Every science deals with functions that change over time: population growth, heat dispersion, velocity, acceleration, the spread of a virus, the decay of atoms. When you have an equation that involves the rate of change of a function, you naturally get a differential equation. Thus they are fundamentally important to science and engineering fields.