Differentials
Definition
Differentials provide an approximation of how a function changes as its input changes. They are used to approximate small changes in functions using the derivative.
5 Must Know Facts For Your Next Test
- Differentials can be expressed as $dy = f'(x)dx$, where $dy$ is the differential change in $y$ and $dx$ is the differential change in $x$.
- The differential $dx$ is often treated as an independent variable representing a small change in $x$.
- Differentials are crucial for linear approximations, which estimate the value of a function near a given point using its tangent line.
- In practical applications, differentials help estimate errors and changes in measurements.
- The notation for differentials aligns with Leibniz's notation for derivatives, making them easier to understand within the context of calculus.
Review Questions
- What is the formula for expressing differentials?
- How do differentials assist in linear approximations?
- Why are differentials important for estimating errors?
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Related terms
Derivative: A measure of how a function changes as its input changes; represented as $f'(x)$ or $\frac{dy}{dx}$.
Linear Approximation: An estimation method using the tangent line to approximate the value of a function near a given point.
Tangent Line: A straight line that touches a curve at one point and has the same slope as the curve at that point.
