Computer Science › Foundations of Computer Science
Functions as mappings with a domain and a range, then the families the rest of the discipline uses constantly: affine, quadratic, polynomial, exponential, logarithmic, and trigonometric, with composition and inverses. This lesson absorbs the precalculus the discipline assumes.
What it means for f(x) to approach a value, first through tables and graphs and then through the epsilon-delta definition, with one-sided limits and the cases where a limit does not exist.
The algebraic laws that let limits be computed instead of guessed, the squeeze theorem, and how factoring and rationalizing resolve 0/0 while dividing by the highest power resolves infinity over infinity.
Limits as x grows without bound, horizontal and vertical asymptotes, and the dominance hierarchy of logarithms, polynomials, and exponentials, which is the calculus underneath the formal statement of Big-O.
Continuity at a point and on an interval, the kinds of discontinuity, and the Intermediate Value Theorem, the existence guarantee behind bisection search.
The derivative as the limit of a difference quotient: instantaneous rate of change, slope of the tangent line, best local linear approximation, and the functions where it fails to exist.
The power, product, quotient, and chain rules, derived from the limit definition, with the chain rule read as a composition of local linear approximations.
Why e is the base whose exponential is its own derivative, the derivatives of logarithms and trigonometric functions, and implicit and logarithmic differentiation for expressions the basic rules do not reach.
Rolle's theorem and the Mean Value Theorem, and what they justify: a positive derivative means increasing, and a zero derivative on an interval means constant.
Critical points, the first and second derivative tests, concavity and inflection points, and how to turn a word problem into an optimization on a closed interval.
L'Hôpital's Rule for 0/0 and infinity over infinity, when its hypotheses fail, and how to rewrite the other indeterminate forms so it applies, including comparing growth rates.
Reversing differentiation: the family of antiderivatives, the constant of integration, and how an initial condition picks a single solution.
Area as the limit of Riemann sums, the left, right, and midpoint rules, the formal definition of the definite integral, and its basic properties.
The two parts of the Fundamental Theorem of Calculus, which tie accumulation to rates of change and reduce a definite integral to evaluating an antiderivative.
Substitution as the chain rule run backwards and integration by parts as the product rule run backwards, with how to choose the substitution and the parts.
Setting up integrals for the area between curves, volumes by slicing and by revolution, and the average value of a function over an interval.
Integrals over unbounded intervals or of unbounded functions as limits, convergence and divergence, the p-test, and why a probability density has to integrate to 1 over the whole line.
Sequences as functions on the natural numbers, the formal definition of convergence, and the monotone convergence theorem for bounded sequences.
Arithmetic and geometric progressions, closed forms for their terms and finite sums, and where they appear in computing: loop counts, the amortized cost of a doubling array, and compound growth.
An infinite series as the limit of its partial sums, the geometric series, telescoping sums, the divergence test, and why the harmonic series diverges.
The comparison, integral, ratio, root, and alternating series tests, absolute versus conditional convergence, and how to choose a test for a given series.
Power series as functions, the radius and interval of convergence, and term-by-term differentiation and integration.
Taylor and Maclaurin series, the remainder and its error bound, and how many terms a target accuracy requires, which is the base of numerical approximation.
Functions of two or more variables, their domains, graphs, and level curves, and limits and continuity when a point can be approached along many paths.
Partial derivatives, the tangent plane as the linear approximation, and the multivariable chain rule for compositions through intermediate variables.
Deepens the existing gradients lesson without repeating it: the directional derivative, why the gradient points in the direction of steepest ascent, and why it is perpendicular to level curves.
Critical points of functions of two variables, the second-derivative test through the Hessian, saddle points, and absolute extrema on a closed region.
Polar coordinates, conversion to and from Cartesian form, curves such as circles, cardioids, and roses, and area in polar form.
Double integrals over rectangles and general regions, iterated integrals and Fubini's theorem, changing the order of integration, and the switch to polar coordinates that evaluates the Gaussian integral.
Triple integrals in Cartesian, cylindrical, and spherical coordinates, with volume and mass as the applications and the Jacobian determinant as the scaling factor of a change of variables.
What a differential equation is, order and solutions, initial-value problems, slope fields, and exponential growth and decay as the first model.
Separation of variables, the logistic equation, Newton's law of cooling, and checking a solution by substituting it back.
A decision guide for the whole discipline: given a problem, choose between limits, derivatives, integrals, series, and multivariable tools, and recognize the trap each one sets.
Part of the Computer Science curriculum.
Functions as mappings with a domain and a range, then the families the rest of the discipline uses constantly: affine, quadratic, polynomial, exponential, logarithmic, and trigonometric, with composition and inverses. This lesson absorbs the precalculus the discipline assumes.
What it means for f(x) to approach a value, first through tables and graphs and then through the epsilon-delta definition, with one-sided limits and the cases where a limit does not exist.
The algebraic laws that let limits be computed instead of guessed, the squeeze theorem, and how factoring and rationalizing resolve 0/0 while dividing by the highest power resolves infinity over infinity.
Limits as x grows without bound, horizontal and vertical asymptotes, and the dominance hierarchy of logarithms, polynomials, and exponentials, which is the calculus underneath the formal statement of Big-O.
Continuity at a point and on an interval, the kinds of discontinuity, and the Intermediate Value Theorem, the existence guarantee behind bisection search.
The derivative as the limit of a difference quotient: instantaneous rate of change, slope of the tangent line, best local linear approximation, and the functions where it fails to exist.
The power, product, quotient, and chain rules, derived from the limit definition, with the chain rule read as a composition of local linear approximations.
Why e is the base whose exponential is its own derivative, the derivatives of logarithms and trigonometric functions, and implicit and logarithmic differentiation for expressions the basic rules do not reach.
Rolle's theorem and the Mean Value Theorem, and what they justify: a positive derivative means increasing, and a zero derivative on an interval means constant.
Critical points, the first and second derivative tests, concavity and inflection points, and how to turn a word problem into an optimization on a closed interval.
L'Hôpital's Rule for 0/0 and infinity over infinity, when its hypotheses fail, and how to rewrite the other indeterminate forms so it applies, including comparing growth rates.
Reversing differentiation: the family of antiderivatives, the constant of integration, and how an initial condition picks a single solution.
Area as the limit of Riemann sums, the left, right, and midpoint rules, the formal definition of the definite integral, and its basic properties.
The two parts of the Fundamental Theorem of Calculus, which tie accumulation to rates of change and reduce a definite integral to evaluating an antiderivative.
Substitution as the chain rule run backwards and integration by parts as the product rule run backwards, with how to choose the substitution and the parts.
Setting up integrals for the area between curves, volumes by slicing and by revolution, and the average value of a function over an interval.
Integrals over unbounded intervals or of unbounded functions as limits, convergence and divergence, the p-test, and why a probability density has to integrate to 1 over the whole line.
Sequences as functions on the natural numbers, the formal definition of convergence, and the monotone convergence theorem for bounded sequences.
Arithmetic and geometric progressions, closed forms for their terms and finite sums, and where they appear in computing: loop counts, the amortized cost of a doubling array, and compound growth.
An infinite series as the limit of its partial sums, the geometric series, telescoping sums, the divergence test, and why the harmonic series diverges.
The comparison, integral, ratio, root, and alternating series tests, absolute versus conditional convergence, and how to choose a test for a given series.
Power series as functions, the radius and interval of convergence, and term-by-term differentiation and integration.
Taylor and Maclaurin series, the remainder and its error bound, and how many terms a target accuracy requires, which is the base of numerical approximation.
Functions of two or more variables, their domains, graphs, and level curves, and limits and continuity when a point can be approached along many paths.
Partial derivatives, the tangent plane as the linear approximation, and the multivariable chain rule for compositions through intermediate variables.
Deepens the existing gradients lesson without repeating it: the directional derivative, why the gradient points in the direction of steepest ascent, and why it is perpendicular to level curves.
Critical points of functions of two variables, the second-derivative test through the Hessian, saddle points, and absolute extrema on a closed region.
Polar coordinates, conversion to and from Cartesian form, curves such as circles, cardioids, and roses, and area in polar form.
Double integrals over rectangles and general regions, iterated integrals and Fubini's theorem, changing the order of integration, and the switch to polar coordinates that evaluates the Gaussian integral.
Triple integrals in Cartesian, cylindrical, and spherical coordinates, with volume and mass as the applications and the Jacobian determinant as the scaling factor of a change of variables.
What a differential equation is, order and solutions, initial-value problems, slope fields, and exponential growth and decay as the first model.
Separation of variables, the logistic equation, Newton's law of cooling, and checking a solution by substituting it back.
A decision guide for the whole discipline: given a problem, choose between limits, derivatives, integrals, series, and multivariable tools, and recognize the trap each one sets.