12.1.5Is there a shortcut for writing a Taylor series?

Write the first four terms and the general term of the Maclaurin series for $y = \cos(x)$, $y = \sin(x)$, and $y = e^{x}$. Refer back to previous problems to help you do this efficiently.

The Maclaurin series that you wrote for $y = \cos(x)$ in problem 12-48 can help you write other Maclaurin series quickly.

1. Consider the equations $y = \cos(2x)$ and $y = \cos(x)$. Use substitution to write a Maclaurin series for $f(x) = \cos(2x)$.

2. Likewise, use substitution to write a Maclaurin series for $f(x) = \cos(x^2)$.

Use substitution to write the first four terms and the general term of the Maclaurin series for each of the functions below.

1. $f(x) = \cos(-x)$ 

2. $g ( x ) = e ^ { \sqrt { x } }$ 

3. $h(x) = \sin(x^2)$ 

Without knowing the antiderivative of $y = e ^ { x ^ { 2 } }$, the definite integral $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ is difficult to evaluate without a calculator. You can, however, approximate the value of the integral by integrating a Taylor polynomial for $y = e ^ { x ^ { 2 } }$.

1. Use the first three non-zero terms of the Maclaurin series for $y = e ^ { x ^ { 2 } }$ to write a fourth-degree Taylor polynomial, $p_{4}(x)$, centered at $x = 0$.

2. Approximate the value of $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ by integrating the polynomial from part (a) on the interval $[0, 1]$.

3. Use a calculator to evaluate $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ and compare this value to your answer from part (b). What is the error of your approximation?

Use the Maclaurin series for $\sin(x)$ to demonstrate that $\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ) } { x } = 1$.

For $x$ near $0$, use the Maclaurin series for $\cos\left(x\right)$ to demonstrate that $\lim\limits_ { x \rightarrow 0 } \frac { 1 - \operatorname { cos } ( x ) } { x } = 0$.

Use a Taylor series to evaluate $\int \frac { \operatorname { cos } ( x ^ { 2 } ) - 1 } { x ^ { 2 } } d x$.

Compare the Maclaurin series for $f(x) = \sin(x)$ and $g(x) = \cos(x)$. Homework Help ✎

1. Use what you know about $f$ and $g$ to find a way that will help you remember which the expanded forms of each function.

2. Since $\frac{d}{dx}\sin(x)=\cos(x)$, Khalel thinks the polynomial approximations of these functions should work the same way. Test his theory.

Write the equation of the third-degree Taylor polynomial, $p_3(x)$, about $x = 1$ for $f(x) =\sqrt { x }$. Then use substitution to write a sixth-degree Taylor polynomial for $f ( x ) = \sqrt { x ^ { 2 } + 1 }$. Homework Help ✎

Calculate the volume of the solid formed by revolving the region bounded by the $y$-axis, the graph of $f ( x ) = 4 \sqrt { x ^ { 3 / 2 } + 1 }$, and the line $y = 12$ about the $y$-axis. Homework Help ✎

If the solid in problem 12-58 is a storage tank and liquid is being added at a rate of $8$ cubic units per minute. How fast is the level of the liquid in the tank rising when the level is $4$ units above the lowest point of the tank? Homework Help ✎

Multiple Choice: If $\frac{dy}{dx}=\log_2(y^2+1)$ , then $\frac { d ^ { 2 } y } { d x ^ { x } }=$  Homework Help ✎

Multiple Choice: Use your calculator to compute the error of the sixth-degree Taylor polynomial approximation of $y = e ^ { - x ^ { 2 } }$ centered at $x = 0.5$. Homework Help ✎