Single-variable calculus from first principles
Before calculus can do anything interesting, you need to be fluent with the functions it acts on. Two families carry most of the weight early on: lines and polynomials. The good news is you can read almost everything about them straight off their formula — no plotting required once you know what to look for.
A line is y = mx + b. The slope m is its steepness (rise over run); b is where it crosses the y-axis. Positive m tilts up, negative tilts down, zero is flat. That's the entire story of a line.
A candle burning down at a steady rate is a perfect straight line: its height drops by the same amount each hour, so the formula y = mx + b has a negative slope m (the burn rate) and intercept b (the starting height). A ball thrown into the air is different — its height rises, then falls, tracing a parabola, the U-shaped graph of a quadratic ax² + bx + c. One bends, the other stays straight, and the formula tells you which before you ever plot a point.