Calculus / Unit 5: Applications of Derivatives I
5.5 Linear Approximation
Notes
Near $x = a$, a function is close to its tangent line. The linearization is $L(x) = f(a) + f'(a)(x - a)$.
- Use it to estimate values that are hard to compute by hand.
- Example: $\displaystyle f(x) = \sqrt{x}$ at $a = 4$ has $\displaystyle L(x) = 2 + \frac{1}{4}(x - 4)$. So $\displaystyle \sqrt{4.2} \approx 2 + \frac{1}{4}(0.2) = 2.05$.
- The approximation is best close to $a$.
Practice
Using $\displaystyle f(x) = \sqrt{x}$ near $a = 4$, estimate $\displaystyle \sqrt{4.2}$.
$\displaystyle L(4.2) = 2 + \frac{1}{4}(0.2) = 2.05$.
Use the linearization of $f(x) = x^2$ at $a = 3$ to estimate $(3.1)^2$.
$L(x) = 9 + 6(x - 3)$, so $L(3.1) = 9.6$. (The exact value is 9.61.)
Use the linearization of $e^x$ at $a = 0$ to estimate $e^{0.1}$.
$L(x) = 1 + x$, so $L(0.1) = 1.1$.
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