Unit 10 of 10

The Bridge to Calculus

  • 4 lessons
  • Parts 217 to 224

The last unit previews the central ideas of calculus with the tools of precalculus. It looks at sequences that settle toward a value and at function values that approach a number as the input gets close to a point, using tables and graphs.

It then shrinks secant lines to approach an instantaneous rate of change, and adds up rectangles to estimate the area under a curve. These are the starting points of limits, derivatives and integrals in the calculus course.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Limits

  1. Sequence Behavior and Limiting Values

    What it means for a sequence to settle on a limiting value, how convergent, divergent and oscillating sequences differ, and why a table of terms suggests a limit but never proves one.

    Theory part 217Practice part 218

  2. Approaching a Limit Numerically and Graphically

    How to estimate the limit of a function at a point from tables and graphs, why the value at the point does not matter, how the two sides can disagree, and why evidence is not proof.

    Theory part 219Practice part 220

Rates and Areas

  1. Difference Quotients and the Instantaneous-Rate Idea

    How to write and simplify the difference quotient with h kept nonzero, how shrinking secant lines lead to a rate at a single instant when that limit exists, and why a corner has no single tangent slope.

    Theory part 221Practice part 222

  2. Area Approximation with Rectangles

    How to estimate the area under a nonnegative curve with left and right rectangles, when those sums are guaranteed lower and upper bounds, and how finer partitions close in on the exact area.

    Theory part 223Practice part 224