Midterm 2 information/Learning goals

Week 5

1. Recognize that a given word problem is an optimization problem. (Something to this effect)
2. Set up an optimization problem by finding the objective and constraint functions.
3. Find an optimal value and show that it is optimal using appropriate methods.

Week 6

1. Find the slope of the line of best fit, that passes through the origin, to a set of data by minimizing the sum of the squared residuals.
2. Identify which line in a given set of lines best fits a given set of data.

Week 7

1. Calculate the derivative of functions of the form $p(q(x))$, where $p(x)$ and $q(x)$ are either polynomials or rational functions.
2. Calculate the derivative of polynomial functions defined implicitly.
3. Recognize that a given word problem is a related rates problem, translate the word problem into a mathematical problem, solve this mathematical problem and relate the answer back to the original word problem.

Week 8

1. Use the properties of exponential and logarithmic functions to simplify, or rearrange expressions involving exponential and logarithmic functions.
   • In particular, transform an expression involving general exponential and logarithmic functions to one involving the natural exponential and logarithm, $e$ and $\ln$.
2. Calculate the derivative of $y=a^{f(x)}$, where $a$ is a constant and $f(x)$ is a function of $x$.
3. Calculate the derivative of $y=\log_a(f(x))$, where $a$ is a constant and $f(x)$ is a function of $x$.
4. Convert $(x,y)$ data to $(\ln(x),\ln(y))$ data.
5. Find a line of best fit to $(\ln(x), \ln(y))$ data and transform this line to a power function in the $x-y$ plane.

Week 9

1. Verify that a given function satisfies a given differential equation.
2. Find a function that satisfies a given elementary differential equation.
3. Find a function that describes the change in a quantity given information about that quantity's doubling or characteristic time.
4. Solve for the constants in a general solution to a differential equation using the information given in an initial value problem.
5. Solve for the doubling times and half lives in exponential growth/decay problems.