Calculating Volume Error Using Differentials: Solved Calculus Examples

📚 Quick Study Guide

• 📏 Differentials provide a way to estimate the change in a function (like volume) based on a small change in its input (like radius).
• 📐 The formula for the differential of a volume $V$ with respect to a variable $x$ is $dV = \frac{dV}{dx} dx$, where $dx$ represents the error in $x$.
• 🧮 Common volume formulas:
  • Sphere: $V = \frac{4}{3}\pi r^3$
  • Cube: $V = s^3$
  • Cylinder: $V = \pi r^2 h$
• 💡 To find the maximum error, use the absolute value of $dV$.
• 📈 Relative error is calculated as $\frac{dV}{V}$, often expressed as a percentage.

🧪 Practice Quiz

1. What is the differential $dV$ used to estimate?
   1. (A) The exact change in volume
   2. (B) The approximate change in volume
   3. (C) The initial volume
   4. (D) The rate of change of volume


2. If the radius of a sphere is measured with a possible error of 2%, what is the approximate maximum percentage error in the calculated volume?
   1. (A) 2%
   2. (B) 4%
   3. (C) 6%
   4. (D) 8%


3. The side of a cube is measured to be 10 cm with a possible error of $\pm 0.1$ cm. Use differentials to estimate the maximum error in the calculated volume.
   1. (A) 1 $cm^3$
   2. (B) 3 $cm^3$
   3. (C) 30 $cm^3$
   4. (D) 100 $cm^3$


4. The radius of a cylinder is measured to be 5 cm, and the height is measured to be 10 cm. Both measurements have a possible error of $\pm 0.2$ cm. Estimate the maximum error in the calculated volume.
   1. (A) $5\pi$ $cm^3$
   2. (B) $10\pi$ $cm^3$
   3. (C) $15\pi$ $cm^3$
   4. (D) $20\pi$ $cm^3$


5. If $V = \frac{4}{3}\pi r^3$, find $dV$ when $r = 3$ and $dr = 0.1$.
   1. (A) $1.2\pi$
   2. (B) $3.6\pi$
   3. (C) $4.8\pi$
   4. (D) $7.2\pi$


6. A sphere's volume is calculated using a radius of 4 cm, but the actual radius is 4.1 cm. Use differentials to approximate the error in the volume calculation.
   1. (A) $6.4\pi$ $cm^3$
   2. (B) $8.0\pi$ $cm^3$
   3. (C) $9.6\pi$ $cm^3$
   4. (D) $19.2\pi$ $cm^3$


7. The side of a square is measured as 5 cm with a possible error of 0.05 cm. What is the approximate percentage error in the area calculation?
   1. (A) 1%
   2. (B) 2%
   3. (C) 3%
   4. (D) 4%