Grade 12 · Free math lesson
Intro to Calculus
After this lesson, you will understand what calculus is and how to find the slope of a curve at any point.
Practice Intro to Calculus freeSkills in this unit
Pick a skill to try three free questions on just that.
- Limits from graphs and tables
- Evaluating limits algebraically
- Continuity
- Average and instantaneous rates of change
- The derivative and the power rule
- Product, quotient, and chain rules
- Tangent lines
- Derivatives and motion
- Antiderivatives and definite integrals
- Area under a curve
What Is Calculus?
Calculus is the math of change. It helps you measure how fast something is changing at any moment.
Example: A car speeds up from 0 to 60 mph. Calculus tells you exactly how fast it was accelerating at the 3-second mark.
Remember: Think of calculus as a zoom lens — it zooms into a curve to measure its exact steepness.
Slope of a Curve
A straight line has one fixed slope. A curve's slope changes at every point, and calculus finds it.
Example: On the curve y = x², the slope at x = 2 is 4, but at x = 3 the slope is 6. It keeps changing!
Remember: Picture a ball rolling down a hill — the slope gets steeper as it goes. That steepness is what we calculate.
Meet the Derivative
A derivative is the tool calculus uses to find the slope at any point. For y = x², the derivative is 2x.
Example: If y = x², the derivative is 2x. Plug in x = 3: slope = 2×3 = 6. That's the steepness at that exact point!
Remember: Derivative = slope at a point. Just multiply the exponent by the number, then drop the exponent by 1.
Worked example
Find the slope of y = x² at the point where x = 4.
- 1Step 1: Write the function — y = x².
- 2Step 2: Find the derivative — multiply the exponent (2) by x, drop the power by 1: derivative = 2x.
- 3Step 3: Plug in x = 4 — slope = 2 × 4 = 8.
Answer: The slope of y = x² at x = 4 is 8.
The #1 mistake to avoid
Students forget to use the derivative and just plug x into the original equation. Always find the derivative FIRST, then plug in your x value.
Key takeaways
- Calculus measures how fast things change at any exact moment.
- Curves have different slopes at different points — not one fixed slope.
- The derivative gives you the slope at any point: for y = x², derivative = 2x.
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