Chapter 1: Calculus I — Limits, Derivatives, and Integration

Build the core Calculus I ideas: limits, continuity, derivatives, applications of derivatives, integrals, and accumulation.

What You Will Practice

Calculus I studies change and accumulation. Limits help describe what a function approaches. Derivatives describe instantaneous rate of change. Integrals describe accumulated change or area.

Limits from Tables
Continuity Concept
Power Rule Derivatives
Tangent Slope
Basic Integrals
Area and Accumulation

Mini Lesson

1. Limits

A limit describes what value a function approaches as the input gets closer to a certain number.

lim f(x) as x → a

Plain meaning: What is the function getting close to?

2. Continuity

A function is continuous at a point if the graph does not break, jump, or have a hole there.

Plain meaning: You can draw the graph through that point without lifting your pencil.

3. Derivatives

A derivative gives the instantaneous rate of change of a function.

If f(x) = x², then f′(x) = 2x

Plain meaning: The derivative tells the slope of the tangent line.

4. Power Rule

The power rule is one of the first derivative rules students learn.

d/dx(xⁿ) = nxⁿ⁻¹

Example: d/dx(x³) = 3x²

5. Integrals

An integral represents accumulation. It can also represent area under a curve.

∫ x dx = 1/2 x² + C

Plain meaning: Integration rebuilds totals from rates.

6. Definite Integrals and Area

A definite integral gives accumulated change over an interval.

Area under y = c from x = a to x = b is c(b - a)

Example: Area under y = 4 from x = 0 to x = 3 is 12.

Interactive Calculus I Practice

Choose a topic and practice with instant feedback. Type simple answers such as 4, continuous, 2x, 3x^2, or 1/2x^2+C.

Typing tip: Use x^2 for x², +C for the integration constant, and type concept answers like continuous or not continuous.
Big idea: Calculus I is the foundation. Limits prepare the ground, derivatives explain change, and integrals explain accumulation.

Mastery Check

Before moving to Book 3 Chapter 2, students should be able to recognize these ideas.

Limits

I can estimate what value a function approaches from a table.

Continuity

I can identify whether a function has a break, jump, or hole.

Derivatives

I can use the power rule for simple derivatives.

Tangent Slope

I know the derivative gives the slope of the tangent line.

Integrals

I can recognize basic antiderivatives and area under simple curves.

Go to Chapter 2 Back to Book 3