Linear Approximation Calculator
Find tangent lines and linear approximations of functions with comprehensive error analysis. Our calculus calculator supports differential analysis, linearization, and detailed approximation studies for educational and professional use.
Last updated: December 15, 2024
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Use x as variable. Supported: +, -, *, /, ^, sin, cos, tan, ln, sqrt, exp
Calculus Analysis
Linear Approximation:
L(x) = 1.0000 + 2.0000(x - 1)
Tangent Line:
y = 2.0000x + -1.0000
Derivative f'(a):
2.0000
Actual f(x):
1.2100
Approximation L(x):
1.2000
Absolute Error:
0.010000
Relative Error:
0.826%
Analysis:
The linear approximation L(1.1) = 1.2000 compared to the actual value f(1.1) = 1.2100 gives an error of 0.0100 (0.83%). The approximation quality is excellent given the distance |x - a| = 0.1000.
Calculation Steps:
- Function: f(x) = x^2
- Point of tangency: a = 1
- f(1) = 1.0000
- f'(1) = 2.0000
- Linear approximation: L(x) = 1.0000 + 2.0000(x - 1)
- L(1.1) = 1.2000
- Actual: f(1.1) = 1.2100
- Error: |f(1.1) - L(1.1)| = 0.0100
Linear Approximation Properties:
- • Formula: L(x) = f(a) + f'(a)(x - a)
- • Accuracy: Best near point of tangency
- • Error: Grows approximately as (x - a)²
- • Applications: Approximations, differentials, Newton's method
Quick Example Result
For f(x) = x² at point a = 1, approximating x = 1.1:
L(x) = 1 + 2(x - 1)
Approximation: 1.2, Actual: 1.21, Error: 0.01 (0.83%)
How This Calculator Works
Our linear approximation calculator applies fundamental differential calculus principles to find tangent lines and approximate function values. The calculator uses the linearization formulaL(x) = f(a) + f'(a)(x - a) to provide accurate approximations with comprehensive error analysis.
Linear Approximation Formula
L(x) = f(a) + f'(a)(x - a)y = f(a) + f'(a)(x - a)|Error| ≤ |f''(c)|/2 × (x - a)²The linear approximation formula creates the best linear function that matches both the value and slope of the original function at point a. This tangent line provides excellent approximations for points near a, with error typically growing quadratically with distance.
Shows how the tangent line approximates the function near the point of tangency
Calculus Foundation
Linear approximation is based on the geometric interpretation of the derivative as the slope of the tangent line. When we know f(a) and f'(a), we can construct a linear function that matches the original function's value and instantaneous rate of change at point a. This concept is fundamental to differential calculus and has numerous practical applications.
- Best linear approximation to a function near a given point
- Foundation for Newton's method and other numerical algorithms
- Used in physics for small-angle approximations and linearization
- Essential for understanding differentials and related rates
Sources & References
- Calculus: Early Transcendentals - James Stewart (9th Edition)Comprehensive treatment of linear approximation and differential applications
- Mathematical Association of America - Calculus Education GuidelinesProfessional standards for teaching differential calculus concepts
- MIT OpenCourseWare - Single Variable CalculusEducational resources and applications of linear approximation
Need help with other calculus calculations? Check out our derivative calculator and tangent line calculator.
Get Custom Calculator for Your PlatformExample Analysis
Problem Setup:
- Function: f(x) = √x
- Known point: a = 1 (since √1 = 1)
- Target: x = 1.05
- Goal: Approximate √1.05
Solution Steps:
- f(1) = √1 = 1
- f'(x) = 1/(2√x), so f'(1) = 1/2 = 0.5
- L(x) = 1 + 0.5(x - 1)
- L(1.05) = 1 + 0.5(0.05) = 1.025
Result: √1.05 ≈ 1.025 (Actual: 1.02470, Error: 0.0003%)
The linear approximation gives an excellent estimate with minimal error. This demonstrates how linear approximation can provide quick, accurate estimates for values near known points. The small error (0.0003%) shows why this method is so useful in practical calculations where high precision isn't required.
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