1. Given a function and two points in its domain, compute the average rate of change of the function between those points. I.e. determine the slope of the secant line between two points on a function’s graph.
2. Given either a formulaic or graphical presentation of a function \(f\) and some point \(x_0\) in its domain, determine whether or not the limits \[ \lim_{x \to x_0^{-}} f(x) \qquad \lim_{x \to x_0} f(x) \qquad \lim_{x \to x_0^{+}} f(x) \] exist, and if they exist numerically approximate their value or determine their exact value if possible.
3. Given either a formulaic or graphical or narrative description of a function \(f,\) determine whether or not the limits \[ \lim_{x \to -\infty} f(x) \quad\qquad \lim_{x \to \infty} f(x) \] exist, and if they exist numerically approximate their value or determine their exact value if possible.
4. Given either a formulaic or graphical or narrative description of a function, determine whether or not the function is continuous on its domain.
5. Given a formula for a function expressed as a sum, difference, product, quotient, or composite of algebraic and trigonometric functions, write a formula for its derivative.
6. Determine a function’s derivative from the definition: \[ \frac{\mathrm{d}}{\mathrm{d}x}f(x) = \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}\,, \]
7. Given a function and a point in its domain around which the function is differentiable, compute the instantaneous rate of change of the function at that point.
8. Given a function and a point in its domain around which the function is differentiable, determine an equation of the line tangent to the graph of the function at that point.
9. Given a multivariable equation consisting of sums, products, quotients, or composites of algebraic and trigonometric functions, write down a implicit formula for the derivative of any dependent variable in that equation with respect to any variable on which it depends.
10. For any function expressible piecewise in terms of sums, products, quotients, or composites of algebraic and trigonometric functions, determine its extreme values and inflection points and any intervals on which it is increasing/decreasing or accelerating/decelerating, and from this information accurately sketch the graph of the function.
11. Use Newton’s method to approximate the value to any precision of the root of any function that is differentiable around that root.
12. Express the value of a definite integral as a limit of a Riemann sum, and approximate the value to any precision of that definite integral using the Riemann sum.
13. Determine an antiderivative of any sufficiently simple function expressible in terms of sums, products, quotients, or composites of algebraic and trigonometric functions, and use that antiderivative to calculate the exact value of the definite integral of the function according to the Fundamental Theorem of Calculus.
14. Given two sufficiently simple functions expressible in terms of sums, products, quotients, or composites of algebraic and trigonometric functions whose graphs intersect at points with algebraic coordinates, determine the area of the region bound by those graphs.
15. Compute the volume of any solid whose cross-sections along some axis have an area describable as an integrable function. (e.g. solids of revolution).
16. Compute the work being done as some linear, potentially variable, force is being applied to a potentially variable mass.