Displacement, Velocity & Acceleration: Where Calculus Enters

You'll learnhow the derivative reads velocity as the slope of position, and the integral runs it backward.

Velocity is the slope of position; acceleration is the slope of velocity; run it backwards and displacement is the area under the curve. Watch a secant collapse into a tangent, differentiate the lesson’s own polynomial, and catch the classic trap: the highest point on x(t) is where the motion is slowest.

8 STEPS · 6 QUICK CHECKS · v3

CALCULUS LAB

Before you start

What you're looking at

A position-time curve x(t) = −t³ + 6t² with two points marked: t = 4 s (biggest x) and t = 2 s (steepest).

The question

Along this motion, at what instant is the particle moving fastest?

Watch for

A secant line collapsing onto a tangent; the highest point on the curve is where motion is slowest.

What the symbols mean

Horizontal axis is time in seconds, vertical axis is position x in meters; the curve is x(t) = −t³ + 6t².

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