12.1.1Can you name the mystery function?

Approximating with Polynomial Functions

In Chapter 10, you saw that polynomial functions could be created to mimic the behavior of non-polynomial functions near some shared point. We are now ready for a more definitive study of these special polynomial functions. You will learn how these polynomials are constructed as well as their accuracy in approximating the desired function.

12-1.

On you calculator, sketch a graph of the ninth-degree polynomial function $f ( x ) = x - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \frac { x ^ { 7 } } { 7 ! } + \frac { x ^ { 9 } } { 9 ! }$ over $–7 ≤ x ≤ 7$ and $–2 ≤ y ≤ 2$ .

Describe the graph of $y = f(x)$ . Which common function does this graph remind you of? Name this common function $g$ .
Graph $y = f(x)$ and $y = g(x)$ on the same set of axes. Over what interval does f closely approximate $g$ ?
Notice that there is a pattern among consecutive terms of $f$ . If $f$ had one more term, make a prediction about what the next term would be:
$f(x)+\text{____}\quad=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}+\text{____}$
Input this new $f$ into your calculator and compare with $g$ . What did you notice?
Use your calculator to explore what happens when $f$ has ten terms. Then compare it to $g$ .

12-2.

MYSTERY FUNCTION, Part One
Your diabolical teacher gives you the following puzzle:

Clues: $f$ is a third-degree polynomial.

$f(0)=-1,$ $f′(0)=5,$ $f′′(0)=-12,$ $f′′′(0)=18$

Your Challenge: Write a possible equation for $f$ .

Write the equation of mystery polynomial.
Describe your method.

12-3.

MYSTERY FUNCTION, Part Two

She is at it again! Your teacher is thinking of another mystery function. This time she wants you to estimate a specific value of a function.

Clues: $f(0)=2,$ $f′(0)=-2,$ $f′′(0)=4,$ $f′′′(0)=-12,$ $f(4)(0)=48$

Your Challenge: Estimate the value of $f(0.5)$ .

Notice that your teacher did not tell you if $f$ is (or is not) a polynomial. But that should not stop you! Write a polynomial, $h$ , that models the mystery function, $f$ . Then calculate $h(0.5)$ .
How close do you think $h\left(0.5\right)$ is to the actual value of $f(0.5)$ ? Explain your thinking.

12-4.

MYSTERY FUNCTION, Part Three

This time, your diabolical teacher challenges you to play the game in reverse.

Clues: The mystery function is a third-degree polynomial, $p_3(x)$ .
The graph of $y = p_3(x)$ looks a lot like the graph of $f(x) = e^{2x}$ near $x = 0$ .

Your Challenge: Write an equation for $p_3(x)$ .

If the graph of $y = p(x)$ looks like the graph of $y = f(x)$ near $x = 0$ , then $p$ and $f$ should have the same value, slope, and concavity at $x = 0$ . With your team, determine the values of:
$f\left(0\right) =\text{____}$ , $f^\prime\left(0\right) =\text{____}$ , $f^{\prime\prime}\left(0\right) =\text{____}$ , $f^{\prime\prime\prime}\left(0\right) =\text{____}$
Use the information from part (a) to write a polynomial that looks a lot like $f(x) = e^{2x}$ near $x = 0$ .

12-5.

MYSTERY FUNCTION, The Strategy

Let’s find a general way to build a polynomial, $p\left(x\right)$ , that is centered about $x = 0$ , that looks a lot like another function, $f\left(x\right)$ .

Justify why this first-degree polynomial has the general equation $p_1(x) = f(0) + f ′(0)x$ .
Write the general equation for $p_{2}(x)$ .
Write the general equation for $p_{3}(x)$ . Look for patterns!
Write the general equation for $p_{n}(x)$ , where $n$ is a positive integer.

12-6.

During each round of the Mystery Function Game, your diabolical teacher challenged you to write a polynomial function that looks a lot like another function that might or might not be a polynomial! A polynomial of this type is called a Taylor polynomial.

All three “mystery polynomials” in this lesson were “centered” about $x = 0$ . In future lessons, you will write Taylor polynomials that are centered about other values of $x$ .

When writing Taylor polynomials, it is important to decide how many terms to include and what degree the polynomial should be. Refer back to the Taylor polynomial you wrote in problem 12-4 for $f(x) = e^{2x}$ . How many terms were did it have? What degree was it?
Write an equation for $p_1(x)$ , the first-degree Taylor polynomial for $f(x) = e^{2x}$ centered at $x = 0$ . Use your calculator to graph $y = p_1(x)$ and $y = f(x)$ on the same set of axes. What does the graph of $y = p_1(x)$ represent in relation to $y = f(x)$ ?
Evaluate $f ′′\left(0\right)$ and determine if $p_1(x)$ gives an underestimate or an overestimate of $f(–0.01)$ . Justify your answer.

12-7.

Recall what you know about translations of functions. Homework Help ✎

What is the equation of the parabola that is identical in shape to $f(x) = x^2$ , but has its vertex at $(5, 2)$ ?
Write an equation of a cosine curve that has a maximum at $(2, –1)$ .

12-8.

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

$\displaystyle \sum _ { n = 1 } ^ { \infty } ( 1 - \frac { 3 } { n } ) ^ { 3 n }$

$\displaystyle \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { \operatorname { ln } ( n ) } { n }$

$\displaystyle \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { ln } ( k ) } { k ^ { 3 } }$

$\displaystyle \sum _ { k = 5 } ^ { \infty } k ! e ^ { - k }$

12-9.

A bug is climbing up a wall. When the bug starts out, it has the energy to climb up quickly, but as time passes it tires, and climbs more and more horizontally on the wall, which is less work than going up. The bug’s climb can be modeled by the derivative equations $\frac { d y } { d t } = t - \frac { t ^ { 2 } } { 10 }$ and $\frac { d x } { d t } = \operatorname { ln } ( t + 1 )$ , where $t$ is measured in minutes and distance is measured in feet. How far does the bug travel along the wall in the first $10$ minutes? Homework Help ✎

12-10.

Calculate the area of the region enclosed by the polar curve $r(θ) = 5 + \cos(6θ)$ for $0 ≤ θ ≤ 2π$ . Homework Help ✎

12-11.

A particle moves along the curve $y = \sin(x)$ at a constant speed of $5$ units per second. What is $x′(t)$ , the $x$ -component of the velocity vector, when $x = 2$ ? Homework Help ✎

Periodic curve with unscaled axes, passing through the origin, with point just past first positive maximum, highlighted & labeled as an order pair, 2, comma sine of 2.

12-12.

Write the equation of a third-degree polynomial function $f$ such that $f(0) = 7$ , $f ′(0) = -3$ , $f^{\prime\prime}(0) = 18$ , and $f^{\prime\prime\prime} (0) = -30$ . Homework Help ✎

12-13.

Multiple Choice: The interval of convergence of the power series $\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } x ^ { n } } { n ^ { 2 } }$ is: Homework Help ✎

$[–3, 3]$

$(–3, 3]$

$[–1, 1]$

$[ - \frac { 1 } { 3 } , \frac { 1 } { 3 } ]$

$( - \frac { 1 } { 3 } , \frac { 1 } { 3 } ]$

12-14.

Multiple Choice: The arc length of the cardioid given by $x(t) = 2\cos(t) - \cos(2t)$ and $y = 2\sin(t) - \sin(2t)$ over the interval $0 ≤ t ≤ 2π$ can be calculated by evaluating the integral: Homework Help ✎

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 - 8 \operatorname { cos } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 + 8 \operatorname { cos } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 - 8 \operatorname { sin } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 + 8 \operatorname { sin } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 5 - 4 \operatorname { cos } ( t ) } d t$

12-15.

Multiple Choice: The slope of the polar curve $r(θ) = \tan(θ)$ is given by: Homework Help ✎

$\sin(θ)(1 + \sec^2(θ))$

$\tan(θ)(1 + \sec^2(θ))$

$\sin(θ) \tan^2(θ)$

$\tan^3(θ)$

$\sec^2(θ)$

Calculus Third Edition