Taylor Polynomials: Local Linear & Quadratic Approximations

1) The Taylor series for a function f centered at x = a is a power series of the form ∑ [f⁽ⁿ⁾(a)/n!] (x - a)ⁿ from n=0 to infinity. What is the coefficient of the term with (x - a)³?

F'''(a)/3

F'''(a)/3!

F(a)/6

3f'''(a)

The general term in a Taylor series centered at x = a is [f⁽ⁿ⁾(a)/n!] (x - a)ⁿ where n! means n factorial. For the term containing (x - a)³, the exponent n = 3, so the coefficient is f^(3)(a)/3!. Since 3! = 6, this is the same as f'''(a)/6, but the standard form uses 3! in the denominator.

Explanation

The general term in a Taylor series centered at x = a is [f⁽ⁿ⁾(a)/n!] (x - a)ⁿ where n! means n factorial. For the term containing (x - a)³, the exponent n = 3, so the coefficient is f^(3)(a)/3!. Since 3! = 6, this is the same as f'''(a)/6, but the standard form uses 3! in the denominator.

2) What is the Taylor series for eˣ centered at x = 0?

∑ xⁿ / n! from n=0 to ∞

∑ xⁿ from n=0 to ∞

∑ xⁿ / 2ⁿ from n=0 to ∞

∑ (-1)ⁿ xⁿ / n! from n=0 to ∞

For eˣ, all derivatives are eˣ, and at x = 0 they equal 1. The coefficient of xⁿ is f⁽ⁿ⁾(0)/n! = 1/n!, so the series is 1 + x + x²/2! + x³/3! + ... which is written as ∑ xⁿ / n! from n=0 to ∞.

Explanation

For eˣ, all derivatives are eˣ, and at x = 0 they equal 1. The coefficient of xⁿ is f⁽ⁿ⁾(0)/n! = 1/n!, so the series is 1 + x + x²/2! + x³/3! + ... which is written as ∑ xⁿ / n! from n=0 to ∞.

3) What is the third-degree Taylor polynomial for f(x) = ln(x) centered at x = 1?

(x-1) - (x-1)²/2 + (x-1)³/3

(x-1) - (x-1)²/2 - (x-1)³/3

X - 1 - (x-1)² + (x-1)³

(x-1) + (x-1)²/2 + (x-1)³/3

For ln(x) centered at a=1: f(1)=0, f'(x)=1/x so f'(1)=1, f''(x)=-1/x² so f''(1)=-1, f'''(x)=2/x³ so f'''(1)=2, f^(4)(x)=-6/x⁴ so f^(4)(1)=-6. Coefficients: f(1)/0! = 0, f'(1)/1! = 1, f''(1)/2! = -1/2, f'''(1)/3! = 2/6 = 1/3. So T3(x) = (x-1) - (x-1)²/2 + (x-1)³/3.

Explanation

For ln(x) centered at a=1: f(1)=0, f'(x)=1/x so f'(1)=1, f''(x)=-1/x² so f''(1)=-1, f'''(x)=2/x³ so f'''(1)=2, f^(4)(x)=-6/x⁴ so f^(4)(1)=-6. Coefficients: f(1)/0! = 0, f'(1)/1! = 1, f''(1)/2! = -1/2, f'''(1)/3! = 2/6 = 1/3. So T3(x) = (x-1) - (x-1)²/2 + (x-1)³/3.

4) What is the Taylor polynomial of degree 2 for f(x) = sin(x) centered at x = π/4?

√2/2 + (√2/2)(x - π/4) + (√2/4)(x - π/4)²

1/2 + (√3/2)(x - π/4) - (1/4)(x - π/4)²

√2/2 + (√2/2)(x - π/4)

√2/2 + (√2/2)(x - π/4) - (√2/4)(x - π/4)²

T2(x) = f(a) + f'(a)(x-a) + f''(a)/2!(x-a)². Here a = π/4, f(π/4) = √2/2, f'(x) = cos(x) so f'(π/4) = √2/2, f''(x) = -sin(x) so f''(π/4) = -√2/2. Thus T2(x) = √2/2 + (√2/2)(x - π/4) + (-√2/2)/2 (x - π/4)² = √2/2 + (√2/2)(x - π/4) - (√2/4)(x - π/4)².

Explanation

T2(x) = f(a) + f'(a)(x-a) + f''(a)/2!(x-a)². Here a = π/4, f(π/4) = √2/2, f'(x) = cos(x) so f'(π/4) = √2/2, f''(x) = -sin(x) so f''(π/4) = -√2/2. Thus T2(x) = √2/2 + (√2/2)(x - π/4) + (-√2/2)/2 (x - π/4)² = √2/2 + (√2/2)(x - π/4) - (√2/4)(x - π/4)².

5) What is the first term (n=0) in any Taylor series centered at x = a?

F(a)

F'(a)

F''(a)/2!

0

The n=0 term is f^(0)(a)/0! (x-a)⁰= f(a)/1 * 1 = f(a). This is true for any function and any center a.

Explanation

The n=0 term is f^(0)(a)/0! (x-a)⁰= f(a)/1 * 1 = f(a). This is true for any function and any center a.

6) Why do higher-degree Taylor polynomials generally give better approximations near the center point?

Because they include more terms from the series

Because they match more derivatives at the center

Because they are linear functions

Because they are always exact

A degree-n Taylor polynomial matches the function and its first n derivatives at the center point x = a. The more derivatives that match, the closer the graph of the polynomial stays to the graph of the original function near that point.

Explanation

A degree-n Taylor polynomial matches the function and its first n derivatives at the center point x = a. The more derivatives that match, the closer the graph of the polynomial stays to the graph of the original function near that point.

7) What is the relationship between a Taylor polynomial and a Taylor series?

A Taylor polynomial is a partial sum of the Taylor series

A Taylor series is a partial sum of the Taylor polynomial

They are unrelated concepts

A Taylor polynomial is the limit of the Taylor series

A Taylor series is an infinite series representation of a function. A Taylor polynomial is obtained by truncating this infinite series after a finite number of terms. If we take the first n+1 terms of a Taylor series, we get the nth degree Taylor polynomial. In other words, a Taylor polynomial is exactly a partial sum of the infinite Taylor series. Option B reverses the relationship. Option C is false as they are directly related. Option D is incorrect because the limit of partial sums gives the infinite series itself, not just a polynomial.

Explanation

A Taylor series is an infinite series representation of a function. A Taylor polynomial is obtained by truncating this infinite series after a finite number of terms. If we take the first n+1 terms of a Taylor series, we get the nth degree Taylor polynomial. In other words, a Taylor polynomial is exactly a partial sum of the infinite Taylor series. Option B reverses the relationship. Option C is false as they are directly related. Option D is incorrect because the limit of partial sums gives the infinite series itself, not just a polynomial.

8) Using T3(x) (third-degree Taylor polynomial) for eˣ centered at 0 to approximate e^1, what do we get?

1 + 1 + 1/2 + 1/6 ≈ 2.666...

1 + 1 + 1 + 1 = 4

1 + 1 + 1/2 = 2.5

2.718...

T3(x) = 1 + x + x²/2 + x³/6. At x=1: 1 + 1 + 1/2 + 1/6 = 1 + 1 + 0.5 + 0.1667 ≈ 2.6667.

Explanation

T3(x) = 1 + x + x²/2 + x³/6. At x=1: 1 + 1 + 1/2 + 1/6 = 1 + 1 + 0.5 + 0.1667 ≈ 2.6667.

9) When using a Taylor polynomial to approximate a function value, the error is usually smallest:

Near the center point

Far from the center point

At the endpoints of the interval

Only exactly at the center

Taylor polynomials are designed to match the function and its derivatives at the center. The farther you move away from the center, the more the higher derivatives cause the curves to diverge, so the approximation is best close to the center.

Explanation

Taylor polynomials are designed to match the function and its derivatives at the center. The farther you move away from the center, the more the higher derivatives cause the curves to diverge, so the approximation is best close to the center.