Applications of Derivatives
Approximations and Errors
Grade 12
Question:
<p>Let S be a square with sides of length <span class="math inline">x</span>. If we approximate the change in size of the area of S by <span class="math inline">\frac{dA}{dx}\bigg|_{x=x_0} \cdot h</span>, when the sides are changed from <span class="math inline">x_0</span> to <span class="math inline">x_0 + h</span>, then the absolute value of the error in our approximation, is</p>
<p>(a) <span class="math inline">h^2</span></p>
<p>(b) <span class="math inline">2hx_0</span></p>
<p>(c) <span class="math inline">x_0^2</span></p>
<p>(d) <span class="math inline">h</span></p>
Step-by-Step Solution
Key Concept: The error in linear approximation is the difference between the actual change and the differential approximation, which comes from the higher-order terms.
<p><strong>Step 1:</strong> Area of square, <span class="math inline">A = x^2</span></p><p><strong>Step 2:</strong> Linear approximation of change in area: <span class="math inline">\Delta A \approx \frac{dA}{dx}\bigg|_{x=x_0} \cdot h = 2x_0 h</span></p><p><strong>Step 3:</strong> Actual change in area when side changes from <span class="math inline">x_0</span> to <span class="math inline">x_0 + h</span>: <span class="math inline">\Delta A_{actual} = (x_0 + h)^2 - x_0^2 = 2x_0h + h^2</span></p><p><strong>Step 4:</strong> Error = <span class="math inline">|\Delta A_{actual} - \Delta A_{approx}| = |2x_0h + h^2 - 2x_0h| = h^2</span></p><p>∴ Answer is (a).</p>
Correct Answer: A