12.1.3Can I change the center?

Taylor Polynomials About $x=c$

12-28.

For $f(x)=2(x-1)^2-5(x-1)+7$ , calculate the values of $f(1)$ , $f ^\prime(1)$ , and $f ^{\prime\prime}(1)$ . Then compare these values with $g(x) = 2x^2 – 5x + 1$ .

12-29.

MYSTERY FUNCTION, Part Four

Clues: $f$ is a third-degree polynomial function.
$f(2)=-1,$ $f′(2)=5,$ $f′′(2)=-12,$ $f′′′(2)=18$

Your Challenge: Write the equation of $f(x)$ .

Notice that the values given in this puzzle are centered at $x = 2$ (not $x = 0$ ). It may help to write the polynomial in the form $f(x)=a(x-k)^3+b(x-k)^2+c(x-k)+d$ .
Describe your strategy. Be prepared to share your strategy with the class.

12-30.

Your next challenge is to write the equation of a fourth-degree Taylor polynomial that can approximate $f(x) = \ln(x)$ centered at $x = 1$ .

Explain why the first-degree polynomial can be written as $p_1(x) = f(1) + f ′(1)(x - 1) = x - 1$ .
Next, write an equation for, $p_2(x)$ , the second-degree Taylor polynomial that approximates $f(x) = \ln(x)$ at $x = 1$ .
Similarly write the equations of third-degree and fourth-degree Taylor polynomials that approximate f at $x = 1$ .
Sketch $f(x) = \ln(x)$ and $y = p_4(x)$ using a suitable domain and range. State the domain and range you chose. How well does $p_{4}$ approximate $f$ ?

12-31.

Let $f(x) =\sqrt { x }$ .

Write the equation of a fifth-degree Taylor polynomial, $p_5(x)$ , centered at the point $(4,2)$ , to approximate $f$ .
Use $p_5(x)$ to approximate $\sqrt { 5 }$ , then calculate the error of your approximation.
Describe how you could determine the value of $f ^{\prime\prime}(4)$ using the equation for $p_5(x)$ .

12-32.

Write the equation of a second-degree polynomial to approximate $f(x) = \sin(x)$ near $x =\frac { \pi } { 2 }$ . Homework Help ✎

12-33.

Write an equation for $\frac { d y } { d x }$ given each equation below. (Hint for part (c): Use natural log to rewrite this equation.) Homework Help ✎

$y = \int _ { 3 } ^ { x } \sqrt { 1 + e ^ { u } } d u$

$\left\{ \begin{array} { l } { x ( t ) = \operatorname { cos } ( 2 t ) } \\ { y ( t ) = \operatorname { cos } ^ { - 1 } ( 2 t ) } \end{array} \right.$

$y = x^{x}$

$y = x \csc(\ln(x))$

12-34.

Write the equation of the line tangent to the curve $x^{2}y - xy^{2} = 6$ at the point in the first quadrant where $y = 1$ . Homework Help ✎

12-35.

The diagram at right shows a slope field. Homework Help ✎

Explain why $\frac { d y } { d x }$ must be a function of both $x$ and $y$ .
Sketch a solution curve that passes through the point $(0, 3)$ .
The slope field is for the differential equation $\frac { d y } { d x }= -2xy$ . Write the equation of your solution curve.

12-36.

A ball is thrown from a window $25$ meters above the ground with an initial velocity of $40$ m/sec and an angle of inclination of $\frac { \pi } { 6 }$ . Let the origin be the point on the (level) ground below the window. Homework Help ✎

Assume the acceleration due to gravity is $–10$ m/sec². Write equations for $x(t)$ and $y\left(t\right)$ .
Determine the angle at which the ball hits the ground.

12-37.

Multiple Choice: To calculate the area of the region in the first quadrant outside the circle $r = 4\cos(θ)$ and inside the lemniscate $r^2 = 8\sin(2θ)$ , use the integral: Homework Help ✎

$\frac { 1 } { 2 } \int _ { \pi / 4 } ^ { \pi / 2 } ( 16 \operatorname { cos } ^ { 2 } ( \theta ) - 8 \operatorname { sin } ^ { 2 } ( 2 \theta ) ) d \theta$

$\frac { 1 } { 2 } \int _ { \pi / 4 } ^ { \pi / 2 } ( 8 \operatorname { sin } ^ { 2 } ( 2 \theta ) - 16 \operatorname { cos } ^ { 2 } ( \theta ) ) d \theta$

$\frac { 1 } { 2 } \int _ { 0 } ^ { \pi / 4 } ( 16 \operatorname { cos } ^ { 2 } ( \theta ) - 8 \operatorname { sin } ^ { 2 } ( 2 \theta ) ) d \theta$

$\frac { 1 } { 2 } \int _ { 0 } ^ { \pi / 4 } ( 8 \operatorname { sin } ^ { 2 } ( 2 \theta ) - 16 \operatorname { cos } ^ { 2 } ( \theta ) ) d \theta$

$\int _ { 0 } ^ { \pi / 4 } ( 4 \operatorname { cos } ( \theta ) - 2 \sqrt { 2 } \operatorname { sin } ( 2 \theta ) ) d \theta$

12-38.

Multiple Choice: The function $f$ is continuous on the closed interval $[3, 9]$ and has values given in the table below. Homework Help ✎

$x$	$3$	$4$	$7$	$9$
$f\left(x\right)$	$50$	$40$	$20$	$30$

Using the subintervals $[3, 4]$ , $[4, 7]$ , and $[7, 9]$ , what is the trapezoidal approximation of $\int _ { 3 } ^ { 9 } f ( x ) d x$ ?

$160$

$185$

$200$

$210$

$300$

Calculus Third Edition

Taylor Polynomials