Taylor Polynomials I

Part 4: Summary

The first four questions refer to an unspecified function f that can be differentiated infinitely many times. The notation p_n refers to the n-th degree Taylor polynomial of f.

1. What property do f and p₀ have in common?

2. What properties do f and p₁ have in common? What is another name for the graph of p₁?

3. What properties do f and p₂ have in common?

4. What properties do f and p_n have in common?

5. Would you expect the arctangent function to have a seventh-degree Taylor polynomial? Why or why not?

6. Would you expect the square root function to have a seventh-degree Taylor polynomial? Why or why not?