Building Taylor Polynomials: Matching Derivatives & Approximating Function Values

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| Questions: 15 | Updated: Dec 15, 2025
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1) What is the Taylor series for cos(x) centered at x = 0?

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)!.

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About This Quiz
Building Taylor Polynomials: Matching Derivatives & Approximating Function Values - Quiz

Taylor series provide a powerful bridge between functions and polynomials, allowing us to approximate complicated expressions using derivatives evaluated at a single point. In this quiz, you'll dive deeper into how Taylor series are constructed, how specific coefficients arise, and how centered expansions change the form of familiar functions like... see moreexe^xex, ln⁡xln xlnx, sin⁡xsin xsinx, and cos⁡xcos xcosx. You’ll practice building Taylor polynomials of various degrees, interpreting general terms, analyzing convergence, and applying remainder estimates such as the Lagrange bound. You’ll also explore when a Taylor series perfectly equals its function and how alternating series give precise control over error. Whether you're calculating values, identifying correct expansions, or understanding why polynomial approximations behave so well near the center, this quiz will strengthen your mastery of Taylor theory and its applications. see less

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

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

Explanation

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

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

Explanation

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

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

Explanation

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

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

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 ∞.

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

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.

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

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.

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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:

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.

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

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.

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

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).

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

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

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 + ...

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

Explanation

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

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

Explanation

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

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What is the Taylor series for cos(x) centered at x = 0?
Find the second-degree Taylor polynomial for f(x) = √x centered...
Using T4(x) for eˣ at x=0.5, what is the approximation?
The Taylor series for f(x) = 1/(1-x) centered at x=0 is:
What is the first-degree Taylor polynomial (linear approximation) for...
What is the general term for the Taylor series of eˣ centered at x =...
The radius of convergence of the Taylor series for sin(x) centered at...
A degree-n Taylor polynomial centered at a matches which properties of...
If we use the alternating series for sin(x) and stop after the x⁵...
The Lagrange form of the remainder for a Taylor polynomial of degree n...
For which function does the Taylor series centered at x=0 equal the...
The Taylor series for ln(x) centered at x=1 has coefficient of...
The Taylor series centered at x=0 for 1/(1+x)² is:
What is the coefficient of (x-3)⁴ in the Taylor series for eˣ...
The Taylor series for arctan(x) centered at 0 is:
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