✏️ Adventure 3 — Activity

Your goal: use geometry to compute df(x), then find df(x)/dx. Keep dx as a tiny number, and remember: terms with dx or dx² or dx³ vanish as dx → 0.

Tip: Think “new pieces only.” The area/volume of the original blue square/cube is already x² or x³.
Only calculate what gets added when each side grows from x to x + dx.

Part A — Growing a Square: f(x)=x²

Growing square student image
  1. Write the area of each new piece:
    Vertical strip area =
    Horizontal strip area =
    Corner square area =
  2. Add them to find df(x):
    df(x) =
  3. Simplify:
    df(x) =
  4. Divide by dx:
    df(x)/dx =
  5. Now let dx → 0. Cross out any terms that vanish and write the derivative:
    Derivative: d(x²)/dx =

Part B — Growing a Cube: f(x)=x³

Growing cube student image
  1. Find the total volume of the new pieces that appear when the sides of a this cube grows from x to x+dx:
    • Total volume of large face "sheets" each of volume x²·dx?
    • Total volume of edge "bars" each of volume x·dx²?
    • Total volume of tiny corner cube of volume dx³?
  2. Add them to find df(x):
    df(x) =
  3. Factor out dx :
    df(x) =
  4. Divide by dx:
    df(x)/dx =
  5. Now let dx → 0. Cross out vanishing terms and write the derivative:
    Derivative: d(x³)/dx =

Check: your final answers should be 2x and 3x².