Math with Mrs. Sarah

(Finney, AP Calculus BC)

1. Construct a 4th order Taylor polynomial at x = 0 for f(x) = √(1 + x²)

2. Construct the fifth-order Taylor polynomial and the Taylor series at x = 0 for f(x) = e^(1−x)

3. Given f(x) = ln(1−x), use the table of Maclaurin series

a) to construct the first 3 non-zero terms

b) to construct the general term

c) to construct the interval of convergence

4. Given f(x) = x²cosx, use the table of Maclaurin series

a) to construct the first 3 non-zero terms

b) to construct the general term

c) to construct the interval of convergence

5. Find the Taylor series generated by f(x) = 1/(x+1) at x = 2.

6. Find the Taylor polynomial of order 3 generated by f(x) = cosx at x = π/4

7. Let f(t) = 2/(1−t²) and G(x) = ∫₀ˣ f(t) dt

a) Find the first four terms and the general term for the Maclaurin series generated by f.

b) Find the first four nonzero terms and the Maclaurin series for G.

8. Show that if f″(x) exists over the domain of f and the graph of f(x) has an inflection point at x = a, then the linearization of f at x = a is also the second-order Taylor polynomial of f at x = a.

9. To what number does x − x³/3 + x⁵/5 − … + (−1)ⁿ x^(2n+1)/(2n+1) + … converge when x = 1? when x = −1?

1. Construct a 4th order Taylor polynomial at x = 0 for f(x) = sqrt(1 + x^2)
2. Construct the fifth-order Taylor polynomial and the Taylor series at x = 0 for f(x) = e^(1-x)
3. Given f(x) = ln(1-x), use the table of Maclaurin series
   a) to construct the first 3 non-zero terms
   b) to construct the general term
   c) to construct the interval of convergence
4. Given f(x) = x^2 * cos(x), use the table of Maclaurin series
   a) to construct the first 3 non-zero terms
   b) to construct the general term
   c) to construct the interval of convergence
5. Find the Taylor series generated by f(x) = 1/(x+1) at x = 2.
6. Find the Taylor polynomial of order 3 generated by f(x) = cos(x) at x = pi/4.
7. Let f(t) = 2/(1-t^2) and G(x) = integral from 0 to x of f(t) dt.
   a) Find the first four terms and the general term for the Maclaurin series generated by f.
   b) Find the first four nonzero terms and the Maclaurin series for G.
8. Show that if f”(x) exists over the domain of f and the graph of f(x) has an inflection point at x = a, then the linearization of f at x = a is also the second-order Taylor polynomial of f at x = a.
9. To what number does x – x^3/3 + x^5/5 – … + (-1)^n * x^(2n+1)/(2n+1) + … converge when x = 1? when x = -1?