Math 161 Calculus I

Reading Outlines

You should answer these questions as you read the text.

Section 4.3

1. What does it mean to ``approximate'' a function with another function?
2. Why would you want to approximate a function with a polynomial?
3. What were the three conditions used in Example 2 to find a quadratic function which ``best'' approximates f (x) = $\sqrt{x}$? What were the two conditions used to find a linear function which ``best'' approximates f (x) = $\sqrt{x}$? What is the pattern?
4. Which graph (the constant, linear or quadratic) best approximated f (x) = $\sqrt{x}$?
5. What does the equation l (x) = f (x₀) + f'(x₀)(x - x₀) represent? What's x₀?
6. Explain the formula for q(x) in the definition on page 269.
7. How does this general formula for q(x) match the fromula for q(x) we found in Example 2? Does the general formula give the same q(x)?
8. What conditions would a third order (cubic polynomial) approximation of f near x₀ satisfy?
9. What is a Taylor polynomial? What is a Maclaurin polynomial?
10. Find P₁, P₂, P₃, P₄, the Maclaurin polynomials for f (x) = 1 + x + x² + x³ based at x = 0. Plot everything on one set of axes. How closely does each polynomial approximate f (.1)? What do you notice about P₃, P₄? What do you guess P₅ would be? Does this make sense in this case?