Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

15. Power Series

Taylor Series & Taylor Polynomials

Calculus

15. Power Series

Taylor Series & Taylor Polynomials

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Multiple Choice

Approximate $l n 1.5$ to four decimal places using the third-degree Taylor polynomial for $f (x) = \ln x$ .

$0.0417$

$0.4167$

$0.125$

$0.6667$

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Verified step by step guidance

Step 1: Recall the general formula for the Taylor polynomial of a function f(x) centered at a point a. The nth-degree Taylor polynomial is given by: T_n(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... up to the nth derivative.

Step 2: Identify the function f(x) = ln(x) and the center a = 1. Compute the derivatives of f(x) at x = 1. For f(x) = ln(x): f'(x) = 1/x, f''(x) = -1/x^2, and f'''(x) = 2/x^3. Evaluate these derivatives at x = 1: f(1) = ln(1) = 0, f'(1) = 1, f''(1) = -1, and f'''(1) = 2.

Step 3: Write the third-degree Taylor polynomial for f(x) = ln(x) centered at a = 1 using the derivatives computed: T_3(x) = f(1) + f'(1)(x-1) + f''(1)(x-1)^2/2! + f'''(1)(x-1)^3/3!. Substitute the values: T_3(x) = 0 + 1(x-1) - (x-1)^2/2 + 2(x-1)^3/6.

Step 4: Simplify the polynomial: T_3(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3. This is the third-degree Taylor polynomial for ln(x) centered at x = 1.

Step 5: Approximate ln(1.5) by substituting x = 1.5 into the Taylor polynomial T_3(x). Compute each term: (1.5-1), (1.5-1)^2/2, and (1.5-1)^3/3. Add these terms together to get the approximation for ln(1.5) to four decimal places.