Lesson test: Taylor series

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Taylor series

1.[3p]
Estimate $e^{0.5}$ with the first four terms of its Maclaurin series, to four decimal places.

CorrectNot quite: 1.6458

2.[3p]
Estimate $\cos (0.3)$ with the first three non-zero terms, to five decimal places.

CorrectNot quite: 0.95534

3.[2p]
What is the coefficient of $(x - a)^{k}$ in the Taylor polynomial of $f$ at $a$ ?

$f^{(k)} (a) / k!$

Correct

$f^{(k)} (a)$

The answer is: $f^{(k)} (a) / k!$

$f (a) / k$

The answer is: $f^{(k)} (a) / k!$

$f' (a)^{k} / k!$

The answer is: $f^{(k)} (a) / k!$

4.[3p]
Two terms give $\sin (0.5) \approx 0.4791667$ . What is the Lagrange bound on the error, to five decimal places?

CorrectNot quite: 0.00026

5.[2p]
The Mean Value Theorem is the case $n = 0$ of Taylor's theorem with remainder.

True

Correct

False

The answer is: True

6.[3p]
Replacing $\sin θ$ by $θ$ at 10 degrees gives what percentage error, relative to the true sine, to two decimal places?

CorrectNot quite: 0.51

7.[3p]
The series for $1 / (1 + x^{2})$ converges only for $| x | < 1$ , though the function is smooth on the whole real line. Why?

The function has singularities at $\pm i$ , and a power series cannot reach past the nearest one

Correct

The function is not differentiable at $x = 1$

The answer is: The function has singularities at $\pm i$ , and a power series cannot reach past the nearest one

The series was computed incorrectly

The answer is: The function has singularities at $\pm i$ , and a power series cannot reach past the nearest one

Every power series has radius of convergence 1

The answer is: The function has singularities at $\pm i$ , and a power series cannot reach past the nearest one

8.[3p]
Which of these Maclaurin series converge to their function for every real $x$ ?

Select all that apply

$e^{x}$

Correct

$\sin x$

Correct

$\cos x$

Correct

$1 / (1 - x)$

The answer is: $e^{x}$ , $\sin x$ , $\cos x$

9.[3p]
Match each function to the start of its Maclaurin series.

$e^{x}$
$\sin x$
$\cos x$
$1 / (1 - x)$

$1 - x^{2} / 2 + \dots$
$1 + x + x^{2} + \dots$
$x - x^{3} / 6 + \dots$
$1 + x + x^{2} / 2 + \dots$

Show the answer

$e^{x}$ : $1 + x + x^{2} / 2 + \dots$ $\sin x$ : $x - x^{3} / 6 + \dots$ $\cos x$ : $1 - x^{2} / 2 + \dots$ $1 / (1 - x)$ : $1 + x + x^{2} + \dots$

1.[3p] Estimate e0.5 with the first four terms of its Maclaurin series, to four decimal places.

2.[3p] Estimate cos(0.3) with the first three non-zero terms, to five decimal places.

3.[2p] What is the coefficient of (x−a)k in the Taylor polynomial of f at a?

4.[3p] Two terms give sin(0.5)≈0.4791667. What is the Lagrange bound on the error, to five decimal places?

5.[2p] The Mean Value Theorem is the case n=0 of Taylor's theorem with remainder.

6.[3p] Replacing sinθ by θ at 10 degrees gives what percentage error, relative to the true sine, to two decimal places?

7.[3p] The series for 1/(1+x2) converges only for |x|<1, though the function is smooth on the whole real line. Why?

8.[3p] Which of these Maclaurin series converge to their function for every real x?

9.[3p] Match each function to the start of its Maclaurin series.

1.[3p]
Estimate $e^{0.5}$ with the first four terms of its Maclaurin series, to four decimal places.

2.[3p]
Estimate $\cos (0.3)$ with the first three non-zero terms, to five decimal places.

3.[2p]
What is the coefficient of $(x - a)^{k}$ in the Taylor polynomial of $f$ at $a$ ?

4.[3p]
Two terms give $\sin (0.5) \approx 0.4791667$ . What is the Lagrange bound on the error, to five decimal places?

5.[2p]
The Mean Value Theorem is the case $n = 0$ of Taylor's theorem with remainder.

6.[3p]
Replacing $\sin θ$ by $θ$ at 10 degrees gives what percentage error, relative to the true sine, to two decimal places?

7.[3p]
The series for $1 / (1 + x^{2})$ converges only for $| x | < 1$ , though the function is smooth on the whole real line. Why?

8.[3p]
Which of these Maclaurin series converge to their function for every real $x$ ?

9.[3p]
Match each function to the start of its Maclaurin series.