Differential Estimates of Change


In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.


The change in the volume V = (4/3)πr³ of a sphere when the radius changes from r₀ to r₀ + dr

Verified step by step guidance

1. Start by recalling the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \).
2. To find the differential \( dV \), differentiate the volume formula with respect to \( r \). This gives \( dV = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \cdot dr \).
3. Calculate the derivative: \( \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) = 4 \pi r^2 \).
4. Substitute the derivative back into the differential formula: \( dV = 4 \pi r^2 \cdot dr \).
5. This differential formula \( dV = 4 \pi r^2 \cdot dr \) estimates the change in volume when the radius changes from \( r_0 \) to \( r_0 + dr \).

Key Concepts

Differential Calculus

Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. In this context, it helps estimate how a small change in one variable, such as the radius of a sphere, affects another variable, like the volume. Understanding derivatives is crucial for applying differential formulas to approximate changes.

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius. This formula is essential for understanding how the volume changes with respect to the radius. Knowing this relationship allows us to apply calculus concepts to find the differential change in volume when the radius changes slightly.

Differential Formula

A differential formula is used to approximate the change in a function's value due to a small change in its input. For a function V(r), the differential dV is given by dV = V'(r)dr, where V'(r) is the derivative of V with respect to r. This formula helps estimate the change in volume of a sphere when its radius changes by a small amount dr.