AMATH 301 Summer 2026 Final Exam Knowledge Checklist Flashcards

TypeError

An error message indicating that an operation or function is applied to an object of an inappropriate type, which can involve int, float, or str types.

Floating-point arithmetic error

Small errors introduced during numerical computations that require the application of a tolerance check when comparing values.

Discretization

The process of representing a continuous function $f(x)$ for $x \times [a, b]$ using discrete parameters including $a$, $b$, $N$, and $\triangle x$.

np.linspace

A NumPy function used to discretize a continuous function into an array of evenly spaced points.

Matplotlib Figure vs. Axes

The distinction in plotting where the figure is the overall window while the axes are the specific subplots used for data representation, typically created using $fig, ax = plt.subplots()$.

Modularity

The improvement in code structure and reusability achieved by implementing Python functions.

Newton's Method

An iterative root-finding algorithm derived from the root of the tangent line at $x_k$, defined by the formula $x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$.

Bisection Method

A root-finding algorithm that repeatedly divides an interval in half and selects the subinterval containing the root.

First-order forward difference

A finite difference scheme for approximating a derivative defined by $f'(x) \times \frac{f(x+h)-f(x)}{h} + O(h)$, which is first-order accurate.

First-order backward difference

A finite difference scheme for approximating a derivative defined by $f'(x) \times \frac{f(x)-f(x-h)}{h} + O(h)$, which is first-order accurate.

Second-order centered difference

A finite difference scheme for approximating a derivative defined by $f'(x) \times \frac{f(x+h)-f(x-h)}{2h} + O(h^2)$, derived from calculating the secant line between two points.

Truncation Error

The error introduced by approximating a continuous mathematical process with a finite one, distinct from round-off or coding errors.

Big-O Notation

A mathematical notation used to describe the order of accuracy of numerical schemes, such as finite difference methods.

Loglog plots

Plots where both axes use logarithmic scales, used to reveal the relationship between variables through the slope, specifically the order of accuracy.

Trapezoidal Rule

A numerical integration rule derived from averaging the left-hand and right-hand rectangle rules