Write a differential formula that estimates the given change...

Write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area $S=2 \pi r h$ of a right circular cylinder when the height changes from $h_{0}$ to $h_{0}+d h$ and the radius does not change

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Key Concepts

Differentials

Differentials refer to the small changes in variables and are used in calculus to approximate how a function changes when its inputs change slightly. They provide the tool for linear approximations, expressing the change in a function as the derivative evaluated at a certain point multiplied by the small change in the variable.

Linear Approximation

Linear approximation is the method of estimating the value of a function near a given point by using the tangent line. This technique uses the derivative, which represents the rate of change, to predict how a small change in the input (such as height) will affect the output (such as surface area).

Lateral Surface Area of a Cylinder

The lateral surface area of a cylinder is the area of the curved surface excluding the bases, and is calculated using the formula S = 2?rh, where r is the radius and h is the height. Understanding this geometric formula is crucial when applying differential calculus to estimate changes in surface area with respect to changes in one of the dimensions.

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