SYSTEMATIC MATHEMATICS

Single-Variable Calculus I & II

How can an exact speed be found at one instant? How can infinitely many thin pieces give a finite area? Sixty-five visible chapters answer those questions without compressing several major methods into one lesson. Begin with graphs, numerical tables, and the plain idea of getting as close as desired. Turn that idea into a precise limit, use it to build the derivative as local rate and the integral as accumulated change, and then prove why these two operations undo one another. Only after those meanings are secure do rules, hyperbolic functions, applications, weighted physical accumulation, parametric and polar tangent geometry, integration methods, infinite series, error-controlled approximations, and exponential and logistic differential models appear. This course does not claim to cover multivariable or vector calculus, nor a full differential-equations sequence; those belong to the wider curriculum.

Before this course: Completed Precalculus, including function domains and composition, algebraic and trigonometric identities, radians, exponential and logarithmic functions, sequences, informal limits, and complete model validation.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A limit controls outputs near a point

Objective: Why does the definition exclude x=a even when f(a) is defined?

The statement lim(x→a)f(x)=L concerns values f(x) for inputs close to a but different from a. It does not initially say that f(a) exists or equals L. A graph suggests approach, a table samples it, and algebra may expose it, but the definition asks for a uniform guarantee: every sufficiently close permitted input must force the output into any requested tolerance around L. This separates evidence from proof.

In the ε–δ definition, ε names the desired output accuracy and δ names an input closeness that is sufficient to achieve it. The order matters: an opponent chooses any ε>0, then the proof constructs δ>0 before the input x is tested. The punctured condition 0<|x−a|<δ deliberately omits a. A successful proof shows how output error is bounded by input error without selecting only favorable sample points.

Picture turning an input dial toward a without quite clicking onto a. The limit asks where the output needle can be forced to stay once the input is close enough. It is not a guess from a few readings: after someone names any output error they will tolerate, you must name an input distance that makes every permitted reading obey it.

For f(x)=3x−2, lim(x→a)f(x)=3a−2 for every real a.

Let ε>0 be given. The output error is |(3x−2)−(3a−2)|=3|x−a|.

Choose δ=ε/3. If 0<|x−a|<δ, multiplying by 3 gives 3|x−a|<3δ=ε.

Therefore |f(x)−(3a−2)|<ε for every qualifying x, exactly matching the quantified definition.

Prove directly that lim(x→2)(5x+1)=11, and state the δ that works for a requested ε.

  1. Compute the error: |(5x+1)−11|=|5x−10|=5|x−2|.
  2. To make this smaller than ε, it is enough to require |x−2|<ε/5, so choose δ=ε/5.
  3. Then every x with 0<|x−2|<δ satisfies |(5x+1)−11|<5δ=ε.

Result: δ=ε/5 proves the limit for every ε>0.