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<title>Adventure 3 — Activity</title>
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<h1>✏️ Adventure 3 — Activity</h1>
<div class="subtitle">
Your goal: use geometry to compute <code>df(x)</code>, then find <code>df(x)/dx</code>.
Keep <code>dx</code> as a tiny number, and remember: terms with <code>dx</code> or <code>dx²</code> or <code>dx³</code> vanish as <code>dx → 0</code>.
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<p class="hint">
Tip: Think “new pieces only.” The area/volume of the original blue square/cube is already <code>x²</code> or <code>x³</code>.
<br>Only calculate what gets added when each side grows from <code>x</code> to <code>x + dx</code>.
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<h2>Part A — Growing a Square: <code>f(x)=x²</code></h2>
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<img src="images/adventure_3_image_1_student.png" alt="Growing square student image">
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<ol class="steps">
<li>Write the area of each new piece:
<div class="workline">Vertical strip area = <input class="blank sm" type="text" /></div>
<div class="workline">Horizontal strip area = <input class="blank sm" type="text" /></div>
<div class="workline">Corner square area = <input class="blank sm" type="text" /></div>
</li>
<li>Add them to find <code>df(x)</code>:
<div class="workline"><code>df(x) =</code> <input class="blank lg" type="text" /></div>
</li>
<li>Simplify:
<div class="workline"><code>df(x) =</code> <input class="blank lg" type="text" /></div>
</li>
<li>Divide by <code>dx</code>:
<div class="workline"><code>df(x)/dx =</code> <input class="blank lg" type="text" /></div>
</li>
<li>Now let <code>dx → 0</code>. Cross out any terms that vanish and write the derivative:
<div class="workline"><strong>Derivative:</strong> <code>d(x²)/dx =</code> <input class="blank sm" type="text" /></div>
</li>
</ol>
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<h2>Part B — Growing a Cube: <code>f(x)=x³</code></h2>
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<div class="imgwrap">
<img src="images/adventure_3_image_2_student.png" alt="Growing cube student image">
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<ol class="steps">
<li>Find the total volume of the new pieces that appear when the sides of a this cube grows from <code>x</code> to <code>x+dx</code>:
<ul>
<li>Total volume of large face "sheets" each of volume <code>x²·dx</code>? <input class="blank sm" type="text" /></li>
<li>Total volume of edge "bars" each of volume <code>x·dx²</code>? <input class="blank sm" type="text" /></li>
<li>Total volume of tiny corner cube of volume <code>dx³</code>? <input class="blank sm" type="text" /></li>
</ul>
</li>
<li>Add them to find <code>df(x)</code>:
<div class="workline"><code>df(x) =</code> <input class="blank lg" type="text" /></div>
</li>
<li>Factor out <code>dx</code> :
<div class="workline"><code>df(x) =</code> <input class="blank lg" type="text" /></div>
</li>
<li>Divide by <code>dx</code>:
<div class="workline"><code>df(x)/dx =</code> <input class="blank lg" type="text" /></div>
</li>
<li>Now let <code>dx → 0</code>. Cross out vanishing terms and write the derivative:
<div class="workline"><strong>Derivative:</strong> <code>d(x³)/dx =</code> <input class="blank sm" type="text" /></div>
</li>
</ol>
<p class="hint">
Check: your final answers should be <code>2x</code> and <code>3x²</code>.
</p>
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