AP Calculus AB / Unit 4: Contextual Applications of Differentiation
4.6 Approximating Values of a Function Using Local Linearity and Linearization
Notes
Near $x=a$, a differentiable function looks like its tangent line:
$$L(x)=f(a)+f'(a)(x-a)$$
- If $f$ is concave up, the tangent line underestimates; concave down, it overestimates.
Practice
$f(2)=5$ and $f'(2)=3$. Estimate $f(2.1)$.
$5+3(0.1)$.
Use the tangent line to $\displaystyle \sqrt{x}$ at $x=9$ to estimate $\displaystyle \sqrt{9.6}$.
$\displaystyle 3+\frac{1}{6}(0.6)=3.1$.
If $f''>0$ near $a$, the tangent line approximation is:
A concave-up curve lies above its tangent lines.
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