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4.2 Linear Approximations and Differentials

3 min read•june 24, 2024

Linear approximations and are powerful tools in calculus for estimating function values. They use tangent lines to approximate curves near specific points, providing a simpler way to understand complex functions.

These techniques are crucial for solving real-world problems where exact calculations are impractical. By understanding linear approximations and differentials, you'll gain insights into function behavior and improve your problem-solving skills in calculus.

Linear Approximations and Differentials

Linear approximation at a point

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  • Estimates the value of a function near a specific point using a linear function (the tangent line to the curve at that point)
  • Tangent line provides a good approximation of the original function near the point of tangency (local approximation)
  • Based on the idea that a differentiable function can be closely approximated by a linear function near a given point due to the proximity of the tangent line and the curve in that vicinity

Construction of function linearization

  • of a function f(x)f(x) at a point aa given by the formula: L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)
    • f(a)f(a) value of the function at the point of linearization
    • f′(a)f'(a) derivative of the function at the point of linearization
    • (x−a)(x - a) difference between the input value and the point of linearization
  • L(x)L(x) represents the equation of the tangent line to the curve of f(x)f(x) at the point (a,f(a))(a, f(a))
  • Linearization used to estimate the value of f(x)f(x) for xx near aa
    • Example: f(x)=sin⁡(x)f(x) = \sin(x), estimate sin⁡(0.1)\sin(0.1) using linearization at a=0a = 0
      1. L(x)=sin⁡(0)+cos⁡(0)(x−0)=xL(x) = \sin(0) + \cos(0)(x - 0) = x
      2. L(0.1)=0.1L(0.1) = 0.1, good approximation of sin⁡(0.1)≈0.0998\sin(0.1) \approx 0.0998
  • The of the linearization represents the instantaneous rate of change of the function at the point of linearization

Graphical representation of differentials

  • Differentials estimate the change in a function's value based on a small change in its input
  • Function y=f(x)y = f(x), differential dydy represents a small change in yy corresponding to a small change dxdx in xx
    • dydy approximated by the product of the derivative f′(x)f'(x) and dxdx: dy≈f′(x) dxdy \approx f'(x) \, dx
  • Graphically, dydy represented as the vertical change between the tangent line and the original function curve at a given point
    • Tangent line () used to estimate the change in the function's value
    • Actual change is the vertical distance between the original curve and the point on the curve corresponding to the new input value

Error analysis in differential approximations

  • Important to understand the accuracy of differential approximations
  • Absolute error: difference between actual change in function's value and estimated change using the differential
    • Absolute error = |Actual change - Estimated change|
  • : ratio of absolute error to actual change
    • Relative error = (Absolute error) / (Actual change)
  • : relative error expressed as a percentage
    • Percentage error = (Relative error) × 100%
  • Interpreting errors helps understand the accuracy of the differential approximation
    • Smaller relative or percentage error indicates a more accurate approximation
    • Acceptable level of error depends on context and desired precision

Continuity and Differentiability

  • is a prerequisite for differentiability
  • A function is differentiable at a point if it is continuous at that point and has a well-defined derivative
  • Differentiability implies that the function can be approximated by a linear function (tangent line) near the point of interest

Key Terms to Review (12)

Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Differential form: A differential form is an expression involving differentials that can be used to approximate changes in a function. It allows for the linear approximation of how functions change with respect to their variables.
Differentials: 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.
Linear approximation: Linear approximation is a method of estimating the value of a function near a given point using the tangent line at that point. It leverages the fact that the tangent line closely resembles the function in the vicinity of the point.
Linearization: Linearization is the process of approximating a function near a given point using the tangent line at that point. It provides a simpler linear function that closely matches the original function in the vicinity of the point.
Percentage error: Percentage error measures the accuracy of an approximate value compared to an exact value. It is expressed as a percentage and calculated using the formula $\left( \frac{\text{approximate value} - \text{exact value}}{\text{exact value}} \right) \times 100\%$.
Point-slope equation: The point-slope equation of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. It is useful for writing the equation of a line when you know one point and the slope.
Propagated error: Propagated error is the measure of how uncertainties in variables affect the uncertainty in a function derived from those variables. It is often calculated using partial derivatives to approximate the total differential of the function.
Relative error: Relative error is a measure of the uncertainty in a measurement compared to the size of the measurement itself. It is typically expressed as a percentage or fraction.
Slope: Slope is a measure of the steepness or incline of a line or curve, representing the rate of change between two points. It is a fundamental concept in calculus that underpins the understanding of functions, rates of change, and the shape of graphs.
Tangent line approximation: A tangent line approximation uses the tangent line at a point to estimate the value of a function near that point. It is based on linearization and is useful for making quick, approximate calculations.
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4.3 Maxima and Minima