Taylor Polynomial Approximations Quiz

1) What is the Taylor series for cos(x) centered at x = 0?

∑ (-1)ⁿ x^{2n} / (2n)! from n=0 to ∞

∑ x^{2n} / (2n)! from n=0 to ∞

∑ (-1)ⁿ xⁿ / n! from n=0 to ∞

1 + x²/2! + x⁴/4! + ...

Only even powers appear for cos(x). The derivatives cycle cos → -sin → -cos → sin → cos with alternating signs. The standard form is 1 - x²/2! + x⁴/4! - x⁶/6! + ... written compactly as ∑ (-1)ⁿ x^{2n} / (2n)!.

Explanation

Only even powers appear for cos(x). The derivatives cycle cos → -sin → -cos → sin → cos with alternating signs. The standard form is 1 - x²/2! + x⁴/4! - x⁶/6! + ... written compactly as ∑ (-1)ⁿ x^{2n} / (2n)!.

2) Find the second-degree Taylor polynomial for f(x) = √x centered at x = 4.

2 + (x-4)/4 - (x-4)²/64

2 + (x-4)/4 - (x-4)²/32

2 + (x-4)/8 - (x-4)²/64

2 + (1/4)(x-4) + (1/32)(x-4)²

f(x) = x^{1/2}, f(4)=2,

f'(x)= (½)x^{-1/2} → f'(4)=1/4,

f''(x)= (-1/4)x^{-3/2} → f''(4)= -1/32.

Then T2(x) = 2 + (1/4)(x-4) + (-1/32)/2 (x-4)². This gives option A.

Explanation

f(x) = x^{1/2}, f(4)=2,

f'(x)= (½)x^{-1/2} → f'(4)=1/4,

f''(x)= (-1/4)x^{-3/2} → f''(4)= -1/32.

Then T2(x) = 2 + (1/4)(x-4) + (-1/32)/2 (x-4)². This gives option A.

3) Using T4(x) for eˣ at x=0.5, what is the approximation?

1 + 0.5 + 0.25/2 + 0.125/6 + 0.0625/24 ≈ 1.6487

1 + 0.5 + 0.125 + 0.03125 + 0.0078125 ≈ 1.664

1.648721

1 + 0.5 + (0.5)²/2 + (0.5)³/6 + (0.5)⁴/24

T4(x) = 1 + x + x²/2 + x³/6 + x⁴/24. Plugging x=0.5 gives exactly that expression.

Explanation

T4(x) = 1 + x + x²/2 + x³/6 + x⁴/24. Plugging x=0.5 gives exactly that expression.

4) The Taylor series for f(x) = 1/(1-x) centered at x=0 is:

∑ xⁿ from n=0 to ∞

1 + x + x² + x³ + ...

Both A and B

Only valid for |x| < 1

This is the geometric series with ratio x. It equals 1 + x + x² + x³ + ... = ∑ xⁿ from n=0 to ∞ for |x| < 1.

Explanation

This is the geometric series with ratio x. It equals 1 + x + x² + x³ + ... = ∑ xⁿ from n=0 to ∞ for |x| < 1.

5) What is the first-degree Taylor polynomial (linear approximation) for f(x) = tan(x) at x = 0?

X

1 + x

X + x³/3

1

f(0)=0, f'(x)=sec²(x) → f'(0)=1, so T1(x) = f(0) + f'(0)x = 0 + 1·x = x.

Explanation

f(0)=0, f'(x)=sec²(x) → f'(0)=1, so T1(x) = f(0) + f'(0)x = 0 + 1·x = x.

6) What is the general term for the Taylor series of eˣ centered at x = 2?

(x-2)ⁿ / n!

E² (x-2)ⁿ / n!

E^{x} (x-2)ⁿ / n!

(x-2)ⁿ / (n! e²)

Every derivative of eˣ is eˣ. At x=2, f⁽ⁿ⁾(2) = e² for all n. So coefficient is e² / n!, and the series is ∑ e² (x-2)ⁿ / n! from n=0 to ∞.

Explanation

Every derivative of eˣ is eˣ. At x=2, f⁽ⁿ⁾(2) = e² for all n. So coefficient is e² / n!, and the series is ∑ e² (x-2)ⁿ / n! from n=0 to ∞.

7) The radius of convergence of the Taylor series for sin(x) centered at any point is:

Infinite

1

π

0

The derivatives of sin(x) are bounded (always ±sin or ±cos), so the ratio test |a_{n+1}/aₙ| → 0 as n → ∞ regardless of center, giving infinite radius.

Explanation

The derivatives of sin(x) are bounded (always ±sin or ±cos), so the ratio test |a_{n+1}/aₙ| → 0 as n → ∞ regardless of center, giving infinite radius.

8) A degree-n Taylor polynomial centered at a matches which properties of the original function at x = a?

The value and the first n derivatives

Only the value

Only the first derivative

The value and all derivatives up to order n-1

By construction, Tₙ(a) = f(a), Tₙ'(a) = f'(a), ..., Tₙ^{(n)}(a) = f^{(n)}(a), and higher derivatives of the polynomial are zero.

Explanation

By construction, Tₙ(a) = f(a), Tₙ'(a) = f'(a), ..., Tₙ^{(n)}(a) = f^{(n)}(a), and higher derivatives of the polynomial are zero.

9) If we use the alternating series for sin(x) and stop after the x⁵ term, the error is at most the absolute value of the next term:

|x⁷/7!|

Less than |x⁷/7!|

Exactly x⁷/7!

For an alternating series that satisfies the alternating series test conditions (which sin(x) does), the error when truncating is less than the magnitude of the first omitted term and has the same sign as that term.

Explanation

For an alternating series that satisfies the alternating series test conditions (which sin(x) does), the error when truncating is less than the magnitude of the first omitted term and has the same sign as that term.

10) The Lagrange form of the remainder for a Taylor polynomial of degree n gives the error as:

F^{(n+1)}(c)/(n+1)! (x-a)ⁿ⁺¹

The next term in the series

Zero

The difference between the function and the polynomial

Taylor's theorem with remainder states that the exact error is exactly that expression for some c in the interval between a and x.

Explanation

Taylor's theorem with remainder states that the exact error is exactly that expression for some c in the interval between a and x.

11) For which function does the Taylor series centered at x=0 equal the function everywhere?

Eˣ

Sin(x)

Cos(x)

All of the above

eˣ, sin(x), and cos(x) are equal to their Maclaurin series for all real x (infinite radius of convergence and the remainder goes to zero).

Explanation

eˣ, sin(x), and cos(x) are equal to their Maclaurin series for all real x (infinite radius of convergence and the remainder goes to zero).

12) The Taylor series for ln(x) centered at x=1 has coefficient of (x-1)⁵:

-1/5

1/5

5(-1)⁵/5

6(-1)⁶/5

The general term for n ≥ 1 is (-1)ⁿ⁺¹ (x-1)ⁿ / n. For n=5: (-1)^{6} / 5 = +1/5.

Explanation

The general term for n ≥ 1 is (-1)ⁿ⁺¹ (x-1)ⁿ / n. For n=5: (-1)^{6} / 5 = +1/5.

13) The Taylor series centered at x=0 for 1/(1+x)² is:

∑ (-1)ⁿ (n+1) xⁿ

1 - 2x + 3x² - 4x³ + ...

Both A and B

Converges only for |x| < 1

Differentiating the geometric series 1/(1+x) = ∑ (-1)ⁿ xⁿ, then multiplying by -1 gives the series for 1/(1+x)² with coefficients -(n+1)(-1)ⁿ = (-1)ⁿ⁺¹ (n+1), or commonly written 1 - 2x + 3x² - 4x³ + ...

Explanation

Differentiating the geometric series 1/(1+x) = ∑ (-1)ⁿ xⁿ, then multiplying by -1 gives the series for 1/(1+x)² with coefficients -(n+1)(-1)ⁿ = (-1)ⁿ⁺¹ (n+1), or commonly written 1 - 2x + 3x² - 4x³ + ...

14) What is the coefficient of (x-3)⁴ in the Taylor series for eˣ centered at x=3?

E³ / 24

1/24

E³ / 4!

E³

All derivatives of eˣ are eˣ, so f^(4)(3) = e³, coefficient = e³ / 4! = e³ / 24.

Explanation

All derivatives of eˣ are eˣ, so f^(4)(3) = e³, coefficient = e³ / 4! = e³ / 24.

15) The Taylor series for arctan(x) centered at 0 is:

∑ (-1)ⁿ x^{2n+1}/(2n+1)

X - x³/3 + x⁵/5 - x⁷/7 + ...

Valid for |x| ≤ 1

All of the above

This is the standard alternating series for arctan(x), converges for |x| ≤ 1 (at endpoints by Abel theorem).

Explanation

This is the standard alternating series for arctan(x), converges for |x| ≤ 1 (at endpoints by Abel theorem).

Building Taylor Polynomials: Matching Derivatives & Approximating Function Values

1) What is the Taylor series for cos(x) centered at x = 0?

2)

What first name or nickname would you like us to use?

2) Find the second-degree Taylor polynomial for f(x) = √x centered at x = 4.

3) Using T4(x) for eˣ at x=0.5, what is the approximation?

4) The Taylor series for f(x) = 1/(1-x) centered at x=0 is:

5) What is the first-degree Taylor polynomial (linear approximation) for f(x) = tan(x) at x = 0?

6) What is the general term for the Taylor series of eˣ centered at x = 2?

7) The radius of convergence of the Taylor series for sin(x) centered at any point is:

8) A degree-n Taylor polynomial centered at a matches which properties of the original function at x = a?

9) If we use the alternating series for sin(x) and stop after the x⁵ term, the error is at most the absolute value of the next term:

10) The Lagrange form of the remainder for a Taylor polynomial of degree n gives the error as:

11) For which function does the Taylor series centered at x=0 equal the function everywhere?

12) The Taylor series for ln(x) centered at x=1 has coefficient of (x-1)⁵:

13) The Taylor series centered at x=0 for 1/(1+x)² is:

14) What is the coefficient of (x-3)⁴ in the Taylor series for eˣ centered at x=3?

15) The Taylor series for arctan(x) centered at 0 is: