What are the first and second Taylor polynomial approximations to $\cos (x)$ at $a=0$ shown in the graph?

Conditions

• Function is $\cos (x)$
• Center is $a=0$

Reasoning, step by step

1. Identify the first-degree approximation on the graph, which is the yellow horizontal line labeled 1.
2. Identify the second-degree approximation on the graph, which is the purple parabola labeled $1-x^2/2$.
3. Observe that both approximations are centered at $x=0$ and hug the blue $\cos (x)$ curve closely near the origin.
4. Note that the quadratic approximation captures the downward curvature of the cosine function at its peak.

Example

The graph shows $\cos (x)$ in blue, a yellow horizontal line labeled 1, and a purple parabola labeled $1-x^2/2$. The speaker identifies these as the first and second Taylor polynomial approximations of cosine about $a=0$.

Common misconceptions

• Assuming the first approximation is a slanted tangent line, whereas for $\cos (x)$ at $x=0$, the tangent line is perfectly horizontal.
• Believing that the quadratic approximation $1-x^2/2$ is a global replacement for $\cos (x)$, when the video explicitly shows it diverging far from $x=0$.

Watch the explanation

Explore next

• Why are Taylor polynomial approximations considered local rather than global?
• What happens to the Taylor approximations of $\cos (x)$ when the center changes from 0 to pi?