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Linear approximation uses the tangent line at a point \(a\) to estimate a function nearby: \(L(x) = f(a) + f'(a)(x – a)\). For example, \(\sqrt{4.1} \approx 2 + \frac14(0.1) = 2.025\). The real value is about 2.0248.
How to Use the Linear Approximation Calculator
- Type the function, such as
sqrt(x). - Enter the center \(a\). Pick a number close to your target where the function is easy to evaluate, like \(a = 4\) for \(\sqrt{4.1}\).
- Enter the \(x\)-value you want to estimate, then press Solve.
- You get \(L(x)\), the estimate, the exact value, and the error. The graph shows the curve and its tangent line.
The Formula, Step by Step
- Find \(f(a)\).
- Find the derivative \(f'(x)\), then \(f'(a)\). (Use the derivative calculator if you need the steps.)
- Write \(L(x) = f(a) + f'(a)(x – a)\).
- Plug in your \(x\)-value.
Linear approximation is also called linearization or the tangent line approximation. In differential notation it is \(\Delta y \approx dy = f'(a)\,\Delta x\).
Worked Examples
Example 1: Estimate \(\sqrt{4.1}\)
\(f(x) = \sqrt{x}\), \(a = 4\). Then \(f(4) = 2\) and \(f'(x) = \frac{1}{2\sqrt{x}}\), so \(f'(4) = \frac14\).
\(L(x) = 2 + \frac14(x – 4)\), so \(L(4.1) = 2.025\). The exact value is about 2.02485, an error of about 0.00015.
Example 2: Estimate \(\sin(0.2)\)
\(a = 0\): \(\sin 0 = 0\) and \(\cos 0 = 1\), so \(L(x) = x\). The estimate is 0.2. The exact value is about 0.19867. This is the famous “small-angle approximation” \(\sin x \approx x\).
Example 3: Estimate \(\ln(1.05)\)
\(a = 1\): \(\ln 1 = 0\) and \(\frac{d}{dx}\ln x = \frac1x\), so \(f'(1) = 1\). \(L(1.05) = 0.05\). The exact value is about 0.04879.
Example 4: Estimate \(\sqrt[3]{8.2}\)
\(a = 8\): \(f(8) = 2\) and \(f'(x) = \frac{1}{3}x^{-2/3}\), so \(f'(8) = \frac{1}{12}\). \(L(8.2) = 2 + \frac{0.2}{12} \approx 2.01667\). The exact value is about 2.01653.
When Is the Estimate Good?
- It is best when \(x\) is close to \(a\).
- If the graph curves up (concave up), the tangent line sits below it, so \(L(x)\) is an underestimate. If it curves down, \(L(x)\) is an overestimate. Check the curve with the second derivative calculator.
Learn the Math Behind the Linear Approximation Calculator
Linear approximation uses the tangent line to estimate values near a known point. It is the idea behind Euler’s method.
Frequently Asked Questions
Why use linear approximation if calculators exist?
It shows how a small change in input changes the output. Engineers and scientists use it to estimate errors and to simplify hard equations near a known point.
How do I choose the point a?
Choose a value close to your target where \(f(a)\) and \(f'(a)\) are easy to find, like a perfect square for square roots or 0 for trig functions.
Is linear approximation the same as a Taylor polynomial?
It is the first-degree Taylor polynomial. Higher-degree Taylor polynomials add more terms for a better estimate.