Unit 7 of 8

Geometry, Models, and Finite Changes

  • 4 lessons
  • Parts 117 to 124

Applied calculus problems start with geometry and a model. This unit collects the area, volume and surface formulas with their units, uses similar triangles to relate changing lengths, and turns a description in words into a function with clearly named variables and parameters.

It ends with difference quotients, the average rates of change over shrinking intervals that the derivative is built from, computed and simplified before any limit is taken.

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.

Geometry and Models

  1. Geometry Formulas and Units

    How to pick the right lengths from a picture, use the standard formulas for circles, triangles, rings and cylinders, and keep units consistent so that lengths, areas and volumes come out right.

    Theory part 117Practice part 118

  2. Similar Triangles and Changing Dimensions

    How matching angles prove two triangles similar, how to pair up corresponding sides, and how one fixed ratio links the radius and depth of water in a cone or the height of a tree and its shadow.

    Theory part 119Practice part 120

  3. Variables, Parameters, and Building a Model

    How to name the quantities in a situation, use a constraint to reduce them to one input, and write a formula with units and an honest domain.

    Theory part 121Practice part 122

Toward the Derivative

  1. Difference Quotients Before Limits

    How to build the difference quotient of a function, substitute a + h correctly, and simplify it by honest cancellation while keeping h nonzero and both inputs in the domain.

    Theory part 123Practice part 124