✅ Adventure 2 — Solutions
Use this page after you try the activity. It shows the tangent picture, the rise/run idea, and the limit form using
h.
Tip: On the tangent segment, from t=2 to t=4 the run is 2 and the rise is 12.
Slope: rise/run = 12/2 = 6.
2.2 — Table of Values
For h(t)=t^2, the completed table is:
| t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| h(t) | 0 | 1 | 4 | 9 | 16 | 25 |
Difference Quotient at t = 3
f'(3) = lim_(dt->0) ( f(3+dt) - f(3) ) / dt
With f(t)=t^2:
( (3+dt)^2 - 9 ) / dt = (9 + 6dt + dt^2 - 9)/dt = 6 + dt,
so f'(3)=6.
General Rule
f'(t) = lim_(dt->0) ( f(t+dt) - f(t) ) / dt
( (t+dt)^2 - t^2 ) / dt = (2t*dt + dt^2)/dt = 2t + dt,
so f'(t)=2t.