How do individual terms affect the shape of the approximation?

Conditions

• The terms are evaluated in the context of the partial sum polynomial.
• $x$ is a real variable.

Reasoning, step by step

1. Analyze the constant term ($1$): it provides a y-intercept of $1$.
2. Analyze the quadratic term ($-\frac{x^2}{2!}$): it acts as a negative parabola, curving the graph downwards near the origin.
3. Analyze the quartic term ($+\frac{x^4}{4!}$): for larger $x$, this positive term dominates the quadratic, forcing the graph to turn upwards.
4. Analyze the sextic term ($-\frac{x^6}{6!}$): for even larger $x$, this negative term takes over, pulling the graph sharply downwards again.
5. Synthesize these effects to see how the combination of terms creates the wavy shape that mimics the periodic cosine function.

Example

The video uses purple annotations to highlight how the constant '1' sets the intercept, the '-x^$2/2$!' term pulls the graph down, the '+$x^4/4$!' term pushes it up, and the '-x^$6/6$!' term forces it down again.

Common misconceptions

• Thinking that each term only affects the graph locally without influencing the overall end behavior.
• Assuming that higher-degree terms are always negligible and do not change the shape of the polynomial.

Watch the explanation

Explore next

• Why does the blue curve diverge from the red curve?
• What is the Maclaurin series for the cosine function?
• How do partial sums approximate the function defined by the series?