Learning Objectives for Quiz 6

Define a root of a function \(f\).
Explain why we might need to find roots of a function \(f\) numerically, rather than algebraically.
Give an example of a function where we cannot, in general, find its roots algebraically.
Use uniroot() to find a root of function \(f\).
State the special case of the intermediate value theorem involving roots, and sketch a picture of why the intermediate value must be true under the hypotheses of the intermediate value theorem.
Give a (rough) description of how the Bisection Method finds a root of a function.
Use uniroot() to find where a function \(f\) takes a non-zero value.
Use uniroot() to find the quantiles of a continuous random variable \(X\) given its cumulative distribution function \(F\).

Explain why statisticians care about optimization.
State the 3 main methods in Base R used for numerical optimization.
Use optimize() to find local minima and / or maxima of a function \(f\) of a scalar variable \(x\).
Give a (rough) description of how successive parabolic interpolation can be used to find local optima of a function \(f\).
Use Vectorize() to vectorize a function in terms of one or more of its arguments.
Use curve() to find a reasonable starting value for the interval argument of optimize().

Define numerical quadrature.
Explain why one might need to use numerical quadrature rather than use an indefinite integral and the Fundamental Theorem of Calculus to find the area under a curve.
Use integrate() to evaluate a definite integral.
- Both proper and improper definite integrals.
Find the cumulative distribution function \(F\) of a continuous random variable \(X\) using its probability density function \(f\) and integrate().

Define a symbolic derivative.
Use Deriv() to compute the symbolic derivative of a function \(f\) up to some order.
Define the \(r\)-th moment of a random variable \(X\).
Use a given moment generating function \(M_{X}(t)\) of a random variable \(X\) to find its \(r\)-th moment using Deriv().