3.8: Linear Approximations and Taylor Polynomials

Tangent Line Approximations

A curve lies close to it's tangent line near the point of tangency.

This means we can use the tangent line to calculate approximate values near $x = a$.

If the tangent line is at $(a, f(a))$, then the point-slope form is \[y - f(a) = f'(a)(x - a)\]

If we call this line a function, it's called the linearization of $f$ at $a$:

The linearization of $f$ at $a$ is \[L(x) = f(a) + f'(a)(x-a)\]

Find the linearization of $f(x) = \sqrt{x + 3}$ at $a = 1$ and use it to approximate the numbers $\sqrt{3.98}$ and $\sqrt{4.05}$. Are these approximations overestimates or underestimates?

Find the linear approximation of $f(x) = \sin(x)$ at $x = 0$.

Taylor Polynomials

Our linear approximation is a first degree polynomial in which $L'(a) = f'(a)$.

You can think of $L(x)$ as the best linear approximation to $f$ at $a$ since the above says the rate of change of $f(x)$ is the same as $L(x)$ at $a$.

Let's try a degree two polynomial \[P(x) = A + B(x - a) + C(x - a)^2\] with the previous condition but also $P''(a) = f''(a)$.

The second-degree Taylor polynomial of $f$ centered at $a$ is \[P(x) = f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2\] and is usually denoted $T_2(x)$.

Find the second-degree Taylor polynomial $T_2(x)$ centered at $a = 0$ for $f(x) = \cos x$.

We can do this with a degree $n$ polynomial. If we force $P^{(k)}(x) = f^{(k)}(x)$, it turns out the coefficients we derived by hand for the second-degree polynomial have a nice form in general:

The polynomial \[T_n(x) = f(a) + f'(a)(x - a) + \dfrac{f''(a)}{2!}(x-a)^2 + \cdots + \dfrac{f^{(n)}(a)}{n!}(x-a)^n\] is called the $n$th-degree Taylor polynomial of $f$ centered at $a$.

Find the first three Taylor polynomials for $f(x) = \ln x$ at $a = 1$.

Find the first three Taylor polynomials for $f(x) = \sin x$ at $a = 0$.