2.10.1: Resources and Key Concepts
- Page ID
- 192973
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Prerequisite Topics
The following is a list of prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.
- Arithmetic
- Percents: Used when calculating percentage error from relative error.
Videos
- Linear approximation
- Differentials
Key Concepts
Definitions
- Linear Approximation (or Tangent Line Approximation): For a function \(f\) differentiable at \(a\), the linear approximation of \(f\) at \(a\) is the linear function \(L(x) = f(a) + f^{\prime}(a)(x-a)\). For \(x\) near \(a\), \(f(x) \approx L(x)\).
- Linearization: The function \(L(x)\) is also known as the linearization of \(f\) at \(x=a\).
- Differentials:
- Let \(y=f(x)\) be a differentiable function. The differential \(dx\) is an independent variable that can be assigned any nonzero real number.
- The differential \(dy\) is defined as \(dy = f^{\prime}(x) \, dx\).
- Measurement Error (\(dx\) or \(\Delta x\)): If the exact value of a measured quantity is \(a\) and the measured value is \(a+dx\), then \(dx\) is the measurement error.
- Propagated Error (\(\Delta y\)): The error that occurs in a calculated quantity \(f(x)\) due to a measurement error \(dx\) in the input \(x\). It is given by \(\Delta y = f(a+dx) - f(a)\).
- Approximation of Propagated Error using Differentials (\(dy\)): \(\Delta y \approx dy = f^{\prime}(a) \, dx\).
- Relative Error: If \(\Delta q\) is the absolute error in a quantity whose actual value is \(q\), the relative error is \(\frac{\Delta q}{q}\). (Can be approximated by \(\frac{dq}{q}\)).
- Percentage Error: The relative error expressed as a percentage.
Common Mistakes
- Incorrect Evaluation Point for Linear Approximation: When approximating \(f(x)\) using \(L(x) = f(a) + f^{\prime}(a)(x-a)\) for \(x\) near \(a\), if \(x = a + \text{decimal part}\), students might incorrectly evaluate \(f\) and \(L\) at \(x\) instead of evaluating \(f(a)\) and \(f^{\prime}(a)\) and then \(L\) at the "decimal part" relative to \(a\). For example, to approximate \((1.01)^3\) using \(f(x)=(1+x)^3\) linearized at \(a=0\), one evaluates \(L(0.01)\), not \(L(1.01)\) directly if \(L(x)\) was defined as \(f(0)+f^{\prime}(0)x\).
- Confusing \(dy\) and \(\Delta y\): While \(dy\) approximates \(\Delta y\), they are not identical. \(dy\) is the change along the tangent line, while \(\Delta y\) is the actual change in the function value.
- Leibniz Notation as a Fraction: Treating \(\frac{dy}{dx}\) as a true fraction that can be "pulled apart" by multiplying by \(dx\) to get \(dy = f^{\prime}(x) \, dx\) is a notational convenience that works, but the underlying logic is based on \(\frac{dy}{dx}\) as an operator.