Finding Linearization of Functions and Estimating Values Using Derivatives

Understanding Linearization

• 💡 Linearization (L of X) is defined as the equation of the tangent line to a function at a specific point.
• 🔑 The formula for linearization is presented as f(a) + f'(a) * (x - a), which is analogous to the point-slope form of a line.

Calculating Linearization Steps

• 🎯 First, evaluate the function at point 'a' (f(a)). For f(x) = x³, at a=4, f(4) = 4³ = 64.
• 🔬 Next, find the derivative of the function (f'(x)). The derivative of x³ is 3x².
• 📈 Then, evaluate the derivative at point 'a' (f'(a)). For f'(x) = 3x², at a=4, f'(4) = 3 * 4² = 48.
• ✍️ Substitute these values into the linearization formula: L(x) = 64 + 48 * (x - 4).

Estimating Function Values

• 🚀 The linearization function L(x) can be used to estimate function values near the point 'a'.
• 📊 To estimate 4.01³, plug x = 4.01 into the linearization function: L(4.01) = 64 + 48 * (4.01 - 4) = 64 + 48 * 0.01 = 64.48.
• ✅ This estimation (64.48) is very close to the exact value of 4.01³ (64.481201), demonstrating the utility of linearization for approximations without a calculator.