Math / Calculus

QuestionFind the Taylor polynomial $T_{3}(x)$ for the function $f$ centered at the number $a$ . $\begin{array}{r} f(x)=x e^{-7 x}, \quad a=0 \\ T_{3}(x)=\square \end{array}$
Graph $f$ and $T_{3}$ in the same viewing rectangle.

Studdy Solution

STEP 1

1. We are finding the Taylor polynomial of degree 3 for the function $f(x) = x e^{-7x}$ .
2. The Taylor polynomial is centered at $a = 0$ .
3. The Taylor polynomial $T_3(x)$ approximates $f(x)$ around $x = 0$ .

STEP 2

1. Calculate the necessary derivatives of $f(x)$ .
2. Evaluate these derivatives at $x = a = 0$ .
3. Construct the Taylor polynomial $T_3(x)$ .
4. Graph the function $f(x)$ and the polynomial $T_3(x)$ .

STEP 3

Calculate the first derivative of $f(x) = x e^{-7x}$ .
$f'(x) = e^{-7x} - 7xe^{-7x}$

STEP 4

Calculate the second derivative of $f(x)$ .
$f''(x) = -14e^{-7x} + 49xe^{-7x}$

STEP 5

Calculate the third derivative of $f(x)$ .
$f'''(x) = 98e^{-7x} - 343xe^{-7x}$

STEP 6

Evaluate $f(x)$ and its derivatives at $x = 0$ .
$f(0) = 0 \cdot e^{0} = 0$
$f'(0) = e^{0} - 7 \cdot 0 \cdot e^{0} = 1$
$f''(0) = -14 \cdot e^{0} + 49 \cdot 0 \cdot e^{0} = -14$
$f'''(0) = 98 \cdot e^{0} - 343 \cdot 0 \cdot e^{0} = 98$

STEP 7

Construct the Taylor polynomial $T_3(x)$ using the formula:
\[ T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 $
Substituting the evaluated derivatives:
\[ T_3(x) = 0 + 1 \cdot x + \frac{-14}{2}x^2 + \frac{98}{6}x^3 $
Simplifying:
\[ T_3(x) = x - 7x^2 + \frac{49}{3}x^3 $

STEP 8

Graph $f(x) = x e^{-7x}$ and $T_3(x) = x - 7x^2 + \frac{49}{3}x^3$ in the same viewing rectangle.
The Taylor polynomial $T_3(x)$ is:
$\boxed{x - 7x^2 + \frac{49}{3}x^3}$

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QuestionFind the Taylor polynomial $T_{3}(x)$ for the function $f$ centered at the number $a$ . $\begin{array}{r} f(x)=x e^{-7 x}, \quad a=0 \\ T_{3}(x)=\square \end{array}$
Graph $f$ and $T_{3}$ in the same viewing rectangle.

Studdy Solution

STEP 1

1. We are finding the Taylor polynomial of degree 3 for the function $f(x) = x e^{-7x}$ .
2. The Taylor polynomial is centered at $a = 0$ .
3. The Taylor polynomial $T_3(x)$ approximates $f(x)$ around $x = 0$ .

STEP 2

1. Calculate the necessary derivatives of $f(x)$ .
2. Evaluate these derivatives at $x = a = 0$ .
3. Construct the Taylor polynomial $T_3(x)$ .
4. Graph the function $f(x)$ and the polynomial $T_3(x)$ .

STEP 3

Calculate the first derivative of $f(x) = x e^{-7x}$ .
$f'(x) = e^{-7x} - 7xe^{-7x}$

STEP 4

Calculate the second derivative of $f(x)$ .
$f''(x) = -14e^{-7x} + 49xe^{-7x}$

STEP 5

Calculate the third derivative of $f(x)$ .
$f'''(x) = 98e^{-7x} - 343xe^{-7x}$

STEP 6

Evaluate $f(x)$ and its derivatives at $x = 0$ .
$f(0) = 0 \cdot e^{0} = 0$
$f'(0) = e^{0} - 7 \cdot 0 \cdot e^{0} = 1$
$f''(0) = -14 \cdot e^{0} + 49 \cdot 0 \cdot e^{0} = -14$
$f'''(0) = 98 \cdot e^{0} - 343 \cdot 0 \cdot e^{0} = 98$

STEP 7

Construct the Taylor polynomial $T_3(x)$ using the formula:
\[ T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 $
Substituting the evaluated derivatives:
\[ T_3(x) = 0 + 1 \cdot x + \frac{-14}{2}x^2 + \frac{98}{6}x^3 $
Simplifying:
\[ T_3(x) = x - 7x^2 + \frac{49}{3}x^3 $

STEP 8

Graph $f(x) = x e^{-7x}$ and $T_3(x) = x - 7x^2 + \frac{49}{3}x^3$ in the same viewing rectangle.
The Taylor polynomial $T_3(x)$ is:
$\boxed{x - 7x^2 + \frac{49}{3}x^3}$

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QuestionFind the Taylor polynomial T3(x)T_{3}(x)T3​(x) for the function fff centered at the number aaa. f(x)=xe−7x,a=0T3(x)=□\begin{array}{r} f(x)=x e^{-7 x}, \quad a=0 \\ T_{3}(x)=\square \end{array}f(x)=xe−7x,a=0T3​(x)=□​Graph fff and T3T_{3}T3​ in the same viewing rectangle.

Studdy Solution

STEP 1

1. We are finding the Taylor polynomial of degree 3 for the function f(x)=xe−7x f(x) = x e^{-7x} f(x)=xe−7x.2. The Taylor polynomial is centered at a=0 a = 0 a=0.3. The Taylor polynomial T3(x) T_3(x) T3​(x) approximates f(x) f(x) f(x) around x=0 x = 0 x=0.

STEP 2

1. Calculate the necessary derivatives of f(x) f(x) f(x).2. Evaluate these derivatives at x=a=0 x = a = 0 x=a=0.3. Construct the Taylor polynomial T3(x) T_3(x) T3​(x).4. Graph the function f(x) f(x) f(x) and the polynomial T3(x) T_3(x) T3​(x).

STEP 3

Calculate the first derivative of f(x)=xe−7x f(x) = x e^{-7x} f(x)=xe−7x.f′(x)=e−7x−7xe−7x f'(x) = e^{-7x} - 7xe^{-7x} f′(x)=e−7x−7xe−7x

STEP 4

Calculate the second derivative of f(x) f(x) f(x).f′′(x)=−14e−7x+49xe−7x f''(x) = -14e^{-7x} + 49xe^{-7x} f′′(x)=−14e−7x+49xe−7x

STEP 5

Calculate the third derivative of f(x) f(x) f(x).f′′′(x)=98e−7x−343xe−7x f'''(x) = 98e^{-7x} - 343xe^{-7x} f′′′(x)=98e−7x−343xe−7x

STEP 6

STEP 7

STEP 8

Graph f(x)=xe−7x f(x) = x e^{-7x} f(x)=xe−7x and T3(x)=x−7x2+493x3 T_3(x) = x - 7x^2 + \frac{49}{3}x^3 T3​(x)=x−7x2+349​x3 in the same viewing rectangle. The Taylor polynomial T3(x) T_3(x) T3​(x) is:x−7x2+493x3 \boxed{x - 7x^2 + \frac{49}{3}x^3} x−7x2+349​x3​

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QuestionFind the Taylor polynomial $T_{3}(x)$ for the function $f$ centered at the number $a$ . $\begin{array}{r} f(x)=x e^{-7 x}, \quad a=0 \\ T_{3}(x)=\square \end{array}$
Graph $f$ and $T_{3}$ in the same viewing rectangle.

1. We are finding the Taylor polynomial of degree 3 for the function $f(x) = x e^{-7x}$ .
2. The Taylor polynomial is centered at $a = 0$ .
3. The Taylor polynomial $T_3(x)$ approximates $f(x)$ around $x = 0$ .

1. Calculate the necessary derivatives of $f(x)$ .
2. Evaluate these derivatives at $x = a = 0$ .
3. Construct the Taylor polynomial $T_3(x)$ .
4. Graph the function $f(x)$ and the polynomial $T_3(x)$ .

Calculate the first derivative of $f(x) = x e^{-7x}$ .
$f'(x) = e^{-7x} - 7xe^{-7x}$

Calculate the second derivative of $f(x)$ .
$f''(x) = -14e^{-7x} + 49xe^{-7x}$

Calculate the third derivative of $f(x)$ .
$f'''(x) = 98e^{-7x} - 343xe^{-7x}$

Graph $f(x) = x e^{-7x}$ and $T_3(x) = x - 7x^2 + \frac{49}{3}x^3$ in the same viewing rectangle.
The Taylor polynomial $T_3(x)$ is:
$\boxed{x - 7x^2 + \frac{49}{3}x^3}$