Finding Taylor Polynomial Approximations of Functions - AP Calculus BC
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What is the $n^{th}$ degree Taylor polynomial for $f(x) = (1+x)^k$ centered at $0$?
What is the $n^{th}$ degree Taylor polynomial for $f(x) = (1+x)^k$ centered at $0$?
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$1 + kx + \frac{k(k-1)x^2}{2!} + \cdots + \frac{k(k-1)...(k-n+1)x^n}{n!}$. Binomial series with binomial coefficients.
$1 + kx + \frac{k(k-1)x^2}{2!} + \cdots + \frac{k(k-1)...(k-n+1)x^n}{n!}$. Binomial series with binomial coefficients.
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Determine the Taylor polynomial of degree 1 for $f(x) = \ln(x)$ at $x = 1$.
Determine the Taylor polynomial of degree 1 for $f(x) = \ln(x)$ at $x = 1$.
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$0 + (x-1)$. $f(1)=0$, $f'(1)=1$ gives linear approximation.
$0 + (x-1)$. $f(1)=0$, $f'(1)=1$ gives linear approximation.
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Identify the convergence condition for a Taylor series.
Identify the convergence condition for a Taylor series.
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Converges if $|x-a| < R$ (radius of convergence $R$). Series converges within its radius of convergence $R$.
Converges if $|x-a| < R$ (radius of convergence $R$). Series converges within its radius of convergence $R$.
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State the Taylor series for $\text{cos}(x)$ centered at $0$.
State the Taylor series for $\text{cos}(x)$ centered at $0$.
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$\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$. Alternating signs with even powers only.
$\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$. Alternating signs with even powers only.
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What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^{2x}$ at $x = 0$?
What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^{2x}$ at $x = 0$?
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$1 + 2x$. Replace $x$ with $2x$ in exponential series.
$1 + 2x$. Replace $x$ with $2x$ in exponential series.
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What is the $n^{th}$ degree Taylor polynomial of $f(x) = \text{ln}(1+x)$ centered at $0$?
What is the $n^{th}$ degree Taylor polynomial of $f(x) = \text{ln}(1+x)$ centered at $0$?
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$x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n-1}\frac{x^n}{n}$. Alternating signs with terms $\frac{(-1)^{n-1}x^n}{n}$.
$x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n-1}\frac{x^n}{n}$. Alternating signs with terms $\frac{(-1)^{n-1}x^n}{n}$.
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Find the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{ln}(1+x)$ centered at $0$.
Find the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{ln}(1+x)$ centered at $0$.
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$x - \frac{x^2}{2} + \frac{x^3}{3}$. First three terms of natural log series.
$x - \frac{x^2}{2} + \frac{x^3}{3}$. First three terms of natural log series.
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Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $0$.
Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $0$.
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$1 + x + \frac{x^2}{2}$. First three terms of exponential series.
$1 + x + \frac{x^2}{2}$. First three terms of exponential series.
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What is the Taylor polynomial approximation for $f(x) = \text{sin}(x)$ at $x = 0$ up to $x^2$?
What is the Taylor polynomial approximation for $f(x) = \text{sin}(x)$ at $x = 0$ up to $x^2$?
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$x$. Only first term survives since $x^2$ term has coefficient zero.
$x$. Only first term survives since $x^2$ term has coefficient zero.
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What is the remainder term $R_3(x)$ for Taylor polynomial of $f(x) = e^x$ at $x = 0$?
What is the remainder term $R_3(x)$ for Taylor polynomial of $f(x) = e^x$ at $x = 0$?
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$\frac{e^c x^4}{4!}$ for some $c \in (0,x)$. Next derivative term after cubic polynomial.
$\frac{e^c x^4}{4!}$ for some $c \in (0,x)$. Next derivative term after cubic polynomial.
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What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $1$?
What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $1$?
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$e + e(x-1)$. $e^1=e$ with slope $e$ at $x=1$.
$e + e(x-1)$. $e^1=e$ with slope $e$ at $x=1$.
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Calculate the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ at $x = \pi$.
Calculate the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ at $x = \pi$.
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$-1 + (x-\pi)^2$. $\cos(\pi)=-1$, derivatives give quadratic term.
$-1 + (x-\pi)^2$. $\cos(\pi)=-1$, derivatives give quadratic term.
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Find the Taylor polynomial of degree 2 for $f(x) = x^3$ at $a = 1$.
Find the Taylor polynomial of degree 2 for $f(x) = x^3$ at $a = 1$.
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$1 + 3(x-1) + 3(x-1)^2$. Taylor expansion of $x^3$ around $x=1$.
$1 + 3(x-1) + 3(x-1)^2$. Taylor expansion of $x^3$ around $x=1$.
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What is the $4^{th}$ degree Taylor polynomial for $f(x) = x^2 \text{e}^x$ at $x = 0$?
What is the $4^{th}$ degree Taylor polynomial for $f(x) = x^2 \text{e}^x$ at $x = 0$?
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$0 + 0 + x^2 + x^3 + \frac{x^4}{2}$. Product of $x^2$ and exponential series.
$0 + 0 + x^2 + x^3 + \frac{x^4}{2}$. Product of $x^2$ and exponential series.
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What is the radius of convergence for the Taylor series of $e^x$?
What is the radius of convergence for the Taylor series of $e^x$?
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Radius of convergence is $\infty$. Exponential function converges everywhere.
Radius of convergence is $\infty$. Exponential function converges everywhere.
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Find the Taylor polynomial of degree 2 for $f(x) = \text{e}^{-x}$ at $x = 0$.
Find the Taylor polynomial of degree 2 for $f(x) = \text{e}^{-x}$ at $x = 0$.
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$1 - x + \frac{x^2}{2}$. Replace $x$ with $-x$ in exponential series.
$1 - x + \frac{x^2}{2}$. Replace $x$ with $-x$ in exponential series.
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What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?
What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?
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$P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.
$P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.
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What is the Taylor series expansion of $e^x$ centered at $0$?
What is the Taylor series expansion of $e^x$ centered at $0$?
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$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$. Each term is $\frac{x^n}{n!}$ for the exponential function.
$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$. Each term is $\frac{x^n}{n!}$ for the exponential function.
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Find the first three non-zero terms of the Taylor series for $\text{arctan}(x)$ centered at $0$.
Find the first three non-zero terms of the Taylor series for $\text{arctan}(x)$ centered at $0$.
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$x - \frac{x^3}{3} + \frac{x^5}{5}$. Pattern follows $\frac{(-1)^n x^{2n+1}}{2n+1}$.
$x - \frac{x^3}{3} + \frac{x^5}{5}$. Pattern follows $\frac{(-1)^n x^{2n+1}}{2n+1}$.
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What is the Taylor series expansion for $\text{ln}(1-x)$ centered at $0$?
What is the Taylor series expansion for $\text{ln}(1-x)$ centered at $0$?
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$-x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots$. Replace $x$ with $-x$ in $\ln(1+x)$ series.
$-x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots$. Replace $x$ with $-x$ in $\ln(1+x)$ series.
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What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{sin}(2x)$ centered at $0$?
What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{sin}(2x)$ centered at $0$?
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$2x - \frac{8x^3}{3!}$. Replace $x$ with $2x$ in sine series.
$2x - \frac{8x^3}{3!}$. Replace $x$ with $2x$ in sine series.
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What is the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{sin}(x)$ centered at $\pi$?
What is the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{sin}(x)$ centered at $\pi$?
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$0 - (x-\pi)$. $\sin(\pi)=0$, $\sin'(\pi)=-1$ gives linear term.
$0 - (x-\pi)$. $\sin(\pi)=0$, $\sin'(\pi)=-1$ gives linear term.
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What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{arcsin}(x)$ at $x = 0$?
What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{arcsin}(x)$ at $x = 0$?
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$x + \frac{x^3}{6}$. Arcsine has pattern $\frac{(2n)!x^{2n+1}}{4^n(n!)^2(2n+1)}$.
$x + \frac{x^3}{6}$. Arcsine has pattern $\frac{(2n)!x^{2n+1}}{4^n(n!)^2(2n+1)}$.
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Find the $4^{th}$ degree Taylor polynomial for $f(x) = \text{ln}(x)$ centered at $1$.
Find the $4^{th}$ degree Taylor polynomial for $f(x) = \text{ln}(x)$ centered at $1$.
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$(x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4}$. Natural log series shifted to center at $x=1$.
$(x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4}$. Natural log series shifted to center at $x=1$.
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Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{cos}(3x)$ at $x = 0$.
Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{cos}(3x)$ at $x = 0$.
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$1 - \frac{9x^2}{2}$. Replace $x$ with $3x$ in cosine series.
$1 - \frac{9x^2}{2}$. Replace $x$ with $3x$ in cosine series.
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What is the $4^{th}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ centered at $0$?
What is the $4^{th}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ centered at $0$?
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$1 - \frac{x^2}{2} + \frac{x^4}{24}$. Even powers with alternating signs through degree 4.
$1 - \frac{x^2}{2} + \frac{x^4}{24}$. Even powers with alternating signs through degree 4.
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What is the formula for the remainder term $R_n(x)$ in Taylor polynomial?
What is the formula for the remainder term $R_n(x)$ in Taylor polynomial?
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$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$. Lagrange form with $c$ between $a$ and $x$.
$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$. Lagrange form with $c$ between $a$ and $x$.
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State the first four terms of the Taylor series for $\cosh(x)$ centered at $0$.
State the first four terms of the Taylor series for $\cosh(x)$ centered at $0$.
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$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!}$. Even powers only, like cosine but hyperbolic.
$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!}$. Even powers only, like cosine but hyperbolic.
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What is the Taylor series for $\text{sin}(x)$ centered at $0$?
What is the Taylor series for $\text{sin}(x)$ centered at $0$?
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$\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$. Alternating signs with odd powers only.
$\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$. Alternating signs with odd powers only.
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What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?
What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?
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$P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.
$P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.
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