Math / Calculus

QuestionExercise 1. Determine limited development in the neighborhood of $x_{0}=0$ to the indicated order $n$ for the following functions
1. $\frac{1+x+x^{2}}{1+x+x^{3}} \quad(n=3){ }^{I}$
3. $\frac{\arctan (2 x)}{-1+\sin x} \quad(n=3)$ ,
2. $\sqrt{\frac{1+x}{2-x^{2}}} \quad(n=2)$
4. $\frac{\operatorname{sh}^{2} x}{\ln (\cos x)} \quad(n=2)$ ,
5. $e^{\sqrt{2+\cos x}} \quad(n=2)$
Exercise 2.
1. Determine the limited development in the neighborhood of $x_{0}$ to the indicated order $n$ of the following functions : (a) $f(x)=\cos x, x_{0}=\frac{\pi}{4}, n=3$ . (b) $\frac{\ln (x)}{x^{2}}\left(x_{0}=1, n=3\right)$ .
2. Determine Determine the limited development in the neighborhood of $+\infty$ to the indicated order $n$ of the following functions : (a) $f(x)=\frac{\sqrt{x+2}}{\sqrt{x}}, n=3$ (b) $\ln \left(x+\sqrt{1+x^{2}}\right)-\ln x(n=4)$

Studdy Solution

STEP 1

1. We are tasked with finding the Taylor series expansion (limited development) of given functions around a specified point $x_0$ to a certain order $n$ .
2. We will use standard Taylor series expansions and algebraic manipulation to achieve the desired results.
3. The functions are assumed to be differentiable up to the required order in the neighborhood of the specified point.

STEP 2

1. Exercise 1: Find the Taylor series expansion for each function around $x_0 = 0$ to the specified order $n$ . - Function 1: $\frac{1+x+x^{2}}{1+x+x^{3}}$ with $n=3$ - Function 2: $\frac{\arctan (2 x)}{-1+\sin x}$ with $n=3$ - Function 3: $\sqrt{\frac{1+x}{2-x^{2}}}$ with $n=2$ - Function 4: $\frac{\operatorname{sh}^{2} x}{\ln (\cos x)}$ with $n=2$ - Function 5: $e^{\sqrt{2+\cos x}}$ with $n=2$
2. Exercise 2: Find the Taylor series expansion for each function around the specified point $x_0$ or at $+\infty$ to the specified order $n$ . - Part 1: - Function (a): $f(x)=\cos x$ , $x_{0}=\frac{\pi}{4}$ , $n=3$ - Function (b): $\frac{\ln (x)}{x^{2}}$ , $x_{0}=1$ , $n=3$ - Part 2: - Function (a): $f(x)=\frac{\sqrt{x+2}}{\sqrt{x}}$ , $n=3$ - Function (b): $\ln \left(x+\sqrt{1+x^{2}}\right)-\ln x$ , $n=4$
Exercise 1:

STEP 3

For Function 1: $\frac{1+x+x^{2}}{1+x+x^{3}}$ , expand both the numerator and denominator using Taylor series around $x_0 = 0$ .

STEP 4

Simplify the expression by dividing the series expansions of the numerator and denominator and truncate to order $n=3$ .

STEP 5

For Function 2: $\frac{\arctan (2 x)}{-1+\sin x}$ , use the Taylor series for $\arctan(2x)$ and $\sin x$ around $x_0 = 0$ .

STEP 6

Simplify and truncate the expression to order $n=3$ .

STEP 7

For Function 3: $\sqrt{\frac{1+x}{2-x^{2}}}$ , use the binomial series expansion for both the numerator and denominator.

STEP 8

Simplify and truncate the expression to order $n=2$ .

STEP 9

For Function 4: $\frac{\operatorname{sh}^{2} x}{\ln (\cos x)}$ , use the series expansion for $\operatorname{sh} x$ and $\ln(\cos x)$ .

STEP 10

Simplify and truncate the expression to order $n=2$ .

STEP 11

For Function 5: $e^{\sqrt{2+\cos x}}$ , use the series expansion for $\cos x$ and the exponential function.

STEP 12

Simplify and truncate the expression to order $n=2$ .
Exercise 2:

STEP 13

For Part 1, Function (a): $f(x)=\cos x$ , expand around $x_0 = \frac{\pi}{4}$ using Taylor series to order $n=3$ .

STEP 14

Simplify the expression.

STEP 15

For Part 1, Function (b): $\frac{\ln (x)}{x^{2}}$ , expand around $x_0 = 1$ using Taylor series to order $n=3$ .

STEP 16

Simplify the expression.

STEP 17

For Part 2, Function (a): $f(x)=\frac{\sqrt{x+2}}{\sqrt{x}}$ , expand around $+\infty$ using asymptotic expansion to order $n=3$ .

STEP 18

Simplify the expression.

STEP 19

For Part 2, Function (b): $\ln \left(x+\sqrt{1+x^{2}}\right)-\ln x$ , expand around $+\infty$ using asymptotic expansion to order $n=4$ .

STEP 20

Simplify the expression.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

STEP 3

For Function 1: 1+x+x21+x+x3\frac{1+x+x^{2}}{1+x+x^{3}}1+x+x31+x+x2​, expand both the numerator and denominator using Taylor series around x0=0 x_0 = 0 x0​=0.

STEP 4

Simplify the expression by dividing the series expansions of the numerator and denominator and truncate to order n=3 n=3 n=3.

STEP 5

For Function 2: arctan⁡(2x)−1+sin⁡x\frac{\arctan (2 x)}{-1+\sin x}−1+sinxarctan(2x)​, use the Taylor series for arctan⁡(2x)\arctan(2x)arctan(2x) and sin⁡x\sin xsinx around x0=0 x_0 = 0 x0​=0.

STEP 6

Simplify and truncate the expression to order n=3 n=3 n=3.

STEP 7

For Function 3: 1+x2−x2\sqrt{\frac{1+x}{2-x^{2}}}2−x21+x​​, use the binomial series expansion for both the numerator and denominator.

STEP 8

Simplify and truncate the expression to order n=2 n=2 n=2.

STEP 9

For Function 4: sh⁡2xln⁡(cos⁡x)\frac{\operatorname{sh}^{2} x}{\ln (\cos x)}ln(cosx)sh2x​, use the series expansion for sh⁡x\operatorname{sh} xshx and ln⁡(cos⁡x)\ln(\cos x)ln(cosx).

STEP 10

Simplify and truncate the expression to order n=2 n=2 n=2.

STEP 11

For Function 5: e2+cos⁡xe^{\sqrt{2+\cos x}}e2+cosx​, use the series expansion for cos⁡x\cos xcosx and the exponential function.

STEP 12

Simplify and truncate the expression to order n=2 n=2 n=2. Exercise 2:

STEP 13

For Part 1, Function (a): f(x)=cos⁡xf(x)=\cos xf(x)=cosx, expand around x0=π4 x_0 = \frac{\pi}{4} x0​=4π​ using Taylor series to order n=3 n=3 n=3.

STEP 14

Simplify the expression.

STEP 15

For Part 1, Function (b): ln⁡(x)x2\frac{\ln (x)}{x^{2}}x2ln(x)​, expand around x0=1 x_0 = 1 x0​=1 using Taylor series to order n=3 n=3 n=3.

STEP 16

Simplify the expression.

STEP 17

For Part 2, Function (a): f(x)=x+2xf(x)=\frac{\sqrt{x+2}}{\sqrt{x}}f(x)=x​x+2​​, expand around +∞ +\infty +∞ using asymptotic expansion to order n=3 n=3 n=3.

STEP 18

Simplify the expression.

STEP 19

For Part 2, Function (b): ln⁡(x+1+x2)−ln⁡x\ln \left(x+\sqrt{1+x^{2}}\right)-\ln xln(x+1+x2​)−lnx, expand around +∞ +\infty +∞ using asymptotic expansion to order n=4 n=4 n=4.

STEP 20

Simplify the expression.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

For Function 1: $\frac{1+x+x^{2}}{1+x+x^{3}}$ , expand both the numerator and denominator using Taylor series around $x_0 = 0$ .

Simplify the expression by dividing the series expansions of the numerator and denominator and truncate to order $n=3$ .

For Function 2: $\frac{\arctan (2 x)}{-1+\sin x}$ , use the Taylor series for $\arctan(2x)$ and $\sin x$ around $x_0 = 0$ .

Simplify and truncate the expression to order $n=3$ .

For Function 3: $\sqrt{\frac{1+x}{2-x^{2}}}$ , use the binomial series expansion for both the numerator and denominator.

Simplify and truncate the expression to order $n=2$ .

For Function 4: $\frac{\operatorname{sh}^{2} x}{\ln (\cos x)}$ , use the series expansion for $\operatorname{sh} x$ and $\ln(\cos x)$ .

Simplify and truncate the expression to order $n=2$ .

For Function 5: $e^{\sqrt{2+\cos x}}$ , use the series expansion for $\cos x$ and the exponential function.

Simplify and truncate the expression to order $n=2$ .
Exercise 2:

For Part 1, Function (a): $f(x)=\cos x$ , expand around $x_0 = \frac{\pi}{4}$ using Taylor series to order $n=3$ .

For Part 1, Function (b): $\frac{\ln (x)}{x^{2}}$ , expand around $x_0 = 1$ using Taylor series to order $n=3$ .

For Part 2, Function (a): $f(x)=\frac{\sqrt{x+2}}{\sqrt{x}}$ , expand around $+\infty$ using asymptotic expansion to order $n=3$ .

For Part 2, Function (b): $\ln \left(x+\sqrt{1+x^{2}}\right)-\ln x$ , expand around $+\infty$ using asymptotic expansion to order $n=4$ .