🍎 Adventure 2 — Student Activity (Slope Only)

Complete the table, study the curve, and use the tangent line to understand slope at one instant.

2.1 — The Curve and the Tangent

Curve: h(t) = t^2

The marked point is (3, 9). The tangent line is drawn only from t = 2 to t = 4.

Complete the table for h(t)=t^2.

t	0	1	2	3	4	5
h(t)

Step 1: Write the slope near t=3:

dh(3)/dt = (h(3 + Δt) − h(3)) / Δt =

Substitute h(t)=t^2 and simplify until you get “6 + Δt”:

Step 2: As Δt → 0, the slope at t = 3 is:

Slope at t=3 =

Repeat for an unknown t:

(h(t + Δt) − h(t)) / Δt =

As Δt → 0, dh/dt =

This is the “secret of slope” for h(t)=t^2.