12.3.1How close am I to the curve?

 ACCURACY AND APPROXIMATIONS

1. Sketch $f(x) = \cos(x)$ and $y = p_4(x)$, its fourth-degree Taylor polynomial centered at $0$.

2. If $p_4(x)$ can be used to approximate $f(0.5)$ and $f(0.6)$, do you think that $p_4(0.5)$ is a better approximation of $f(0.5)$ than $p_4(0.6)$ is of $f(0.6)$? Explain.

3. If $p_6(x)$ is the sixth-degree Taylor polynomial centered at $0$ for $f(x) = \cos(x)$, which will give a better approximation of $f(0.5)$:  $p_6(0.5)$ or $p_4(0.5)$? Explain your reasoning.

4. Summarize the ideas of parts (b) and (c). What two factors influence the accuracy of a polynomial’s approximation?

 Taylor polynomials are useful for approximating the value of a function, but there will always be an error. Even small errors could lead to catastrophic results for scientists and engineers. So, before evaluating a Taylor polynomial, it is important to have a sense of the margin of error for a given value of $x$. If that margin of error is too large, you might want to add more terms to the polynomial.

1. Since the Taylor series, $p(x)$, has infinitely many terms, $p(1.5)$ will accurately calculate the value of $\ln(1.5)$. However, it is not practical (or humanly possible) to evaluate infinitely many terms! Approximating the value of $\ln(1.5)$ using a Taylor polynomial, with a finite number of terms, will be much more efficient.
   
   Make a prediction: What is the minimum number of terms that a Taylor polynomial centered at $x = 1$ needs to have in order to predict the value of $\ln(1.5)$ within two decimal places accuracy?

2. In order to answer this question, make a table of data. Let $n$ represent the degree of the Taylor polynomial for $\ln(x)$ centered at $x = 1$, and let $p_{n}(1.5)$ represent the polynomial approximation.

   $n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
   $p_n(1.5)$

3. Examine your table. As $n$ increases, what value does $p_n(1.5)$ seem to be converging to? What do you predict the actual value of $\ln(1.5)$ to be?    

4. To check the accuracy of your prediction, use your calculator to obtain a decimal value of $\ln(1.5)$. In order to approximate $\ln(1.5)$ within two decimal places accuracy, what degree Taylor polynomial centered at $x = 1$ should you use for $f(x) = \ln(x)$?

BOUNDING THE ERROR for Alternating Taylors Series

In problem 12-88, you created a table of data to determine the lowest degree polynomial in which a Taylor polynomial centered at $x = 1$ could be used to approximate $\ln(1.5)$ within two decimal place accuracy (or with an error less than $0.01$).

In the case of $\ln(x)$, each of these Taylor polynomials has terms that alternate. In cases like this, there is an efficient method to answer (with accuracy) whether the error will be less than a certain value (such as $0.01$). We will explore this method.

1. Explain why the Taylor series for $\ln(x)$ centered at $x = 1$ can be considered an alternating series.

2. Consider the sixth-degree polynomial such that $p_6(x)=$$(x-1)-\frac{(x-1)^2}{2}+$$\frac{(x-1)^3}{3}-\frac{(x-1)^4}{4}+$$\frac{(x-1)^5}{5}-\frac{(x-1)^6}{6}$.

   Express each term of the alternating series $p_6(1.5)$ as a fraction, and look for a pattern among the consecutive terms.

   term	$(x - 1)$	$- \frac { ( x - 1 ) ^ { 2 } } { 2 }$	$+ \frac { ( x - 1 ) ^ { 3 } } { 3 }$	$- \frac { ( x - 1 ) ^ { 4 } } { 4 }$	$+ \frac { ( x - 1 ) ^ { 5 } } { 5 }$	$- \frac { ( x - 1 ) ^ { 6 } } { 6 }$
   term value if $x = 1.5$

3. Refer to the table you created in part (b) of problem 12-88. When using the third-degree Taylor polynomial, $p_3(1.5)$, you approximated that $\ln(1.5) ≈ 0.417$. Use the table above to explain why that approximation must be less than $\frac { 1 } { 64 }$ of the actual value of $\ln(1.5)$.

4. If you know the Taylor series that corresponds with an $n$^th-degree Taylor polynomial, then the error when evaluating that polynomial at $x = a$ can be calculated by evaluating the next (non-zero) term of the Taylor series at $x = a$. However, this only works if the terms of the corresponding Taylor series decrease and alternate.

   1. Explain why it is essential that the terms of the corresponding Taylor series decrease.  

   2. Explain why it is essential that the terms of the corresponding Taylor series alternate.

5. Determine the maximum error when using $p_5(1.5)$, the fifth-degree Taylor polynomial centered at $x = 1$, to approximate $\ln(1.5)$. Justify your answer.

Let $f(x) = \sin(x)$.

1. Use $p_3(x)$, the third-degree Taylor polynomial for $f(x)$ centered at $x = 0$, to approximate the value of $\sin(2)$.

2. What is the maximum error of your approximation? Justify your answer.

3. Use your calculator to compute the actual error. Confirm that it is less than the maximum error you computed in part (b).

Write the equation of the fourth-degree Taylor polynomial centered at $x = 0$ for $f(x) = e^{–2x}$. Use it to approximate $f(0.5) = e^{–1 }=\frac { 1 } { e }$, and calculate a bound for the error in this approximation. Then approximate $f ^ { \prime \prime \prime } ( 0.5 )$ using your polynomial approximation. Homework Help ✎

A projectile is launched from the ground at a $45^\circ$ angle. Its height in feet after $t$ seconds is given by $y(t) = 96t - 16t^2$. Its horizontal displacement in feet is given by $x(t) = 96t$. Homework Help ✎

1. Write the velocity vector as a function of $t$.

2. Calculate the magnitude of the velocity vector at $t = 0$ and at the moment when the projectile hits the ground. Make a conjecture about the speed with which projectiles returns to earth.

Rewrite the polar equation $r ( \theta ) = \frac { 4 } { \sqrt { \operatorname { cos } ( 2 \theta ) } }$ in rectangular form. State the domain of $r(θ)$. Describe the graph and explain how the domain relates to the graph. Homework Help ✎

Consider the infinite series below. For each series, decide if it converges conditionally, converges absolutely, or diverges and justify your conclusion. State the tests you used. Homework Help ✎

Multiple Choice: A point moves in the plane according to the set of parametric equations $x(t) = 4 + \cos(π t)$ and $y ( t ) = \sqrt { t }$. What is the slope of the line tangent to the particle’s path at $t = 4$? Homework Help ✎

Multiple Choice: What is the solution to the differential equation $y^2 · y′ - \cos(x) = 0$ with initial condition $y(0) = 3$? Homework Help ✎