Arc Lengths of Curves: Parametric Equations - AP Calculus BC
Card 1 of 30
Calculate $\frac{dy}{dt}$ for $y(t) = \cot(t)$.
Calculate $\frac{dy}{dt}$ for $y(t) = \cot(t)$.
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$ -\csc^2(t) $. Derivative of $\cot(t)$ is $ -\csc^2(t) $.
$ -\csc^2(t) $. Derivative of $\cot(t)$ is $ -\csc^2(t) $.
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What is the interval of integration for a curve from $t=a$ to $t=b$?
What is the interval of integration for a curve from $t=a$ to $t=b$?
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$[a, b]$. Standard interval notation for parameter bounds.
$[a, b]$. Standard interval notation for parameter bounds.
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State the formula for $\frac{dx}{dt}$ when $x(t) = a^t$.
State the formula for $\frac{dx}{dt}$ when $x(t) = a^t$.
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$a^t \ln(a)$. Derivative of exponential $a^t$ using logarithmic differentiation.
$a^t \ln(a)$. Derivative of exponential $a^t$ using logarithmic differentiation.
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Find the integral for $x(t)=\sin(t)$, $y(t)=\cos(t)$ from $t=0$ to $t=\pi$.
Find the integral for $x(t)=\sin(t)$, $y(t)=\cos(t)$ from $t=0$ to $t=\pi$.
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$\int_{0}^{\pi} \sqrt{\cos^2(t) + \sin^2(t)} , dt$. Using Pythagorean identity $\cos^2(t)+\sin^2(t)=1$ simplifies integrand to 1.
$\int_{0}^{\pi} \sqrt{\cos^2(t) + \sin^2(t)} , dt$. Using Pythagorean identity $\cos^2(t)+\sin^2(t)=1$ simplifies integrand to 1.
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What integral represents the arc length of $x(t)=t$, $y(t)=t^2$ from $t=0$ to $t=2$?
What integral represents the arc length of $x(t)=t$, $y(t)=t^2$ from $t=0$ to $t=2$?
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$\int_{0}^{2} \sqrt{1 + (2t)^2} , dt$. Arc length formula with $\frac{dx}{dt}=1$ and $\frac{dy}{dt}=2t$.
$\int_{0}^{2} \sqrt{1 + (2t)^2} , dt$. Arc length formula with $\frac{dx}{dt}=1$ and $\frac{dy}{dt}=2t$.
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Find $\frac{dy}{dt}$ if $y(t) = \cos(t)$.
Find $\frac{dy}{dt}$ if $y(t) = \cos(t)$.
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$-\sin(t)$. Derivative of $\cos(t)$ is $-\sin(t)$.
$-\sin(t)$. Derivative of $\cos(t)$ is $-\sin(t)$.
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State the Pythagorean identity used in the arc length formula.
State the Pythagorean identity used in the arc length formula.
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$\cos^2\theta + \sin^2\theta = 1$. Fundamental trigonometric identity used to simplify expressions.
$\cos^2\theta + \sin^2\theta = 1$. Fundamental trigonometric identity used to simplify expressions.
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State the formula for $\frac{dy}{dt}$ when $y(t) = \log_b(t)$.
State the formula for $\frac{dy}{dt}$ when $y(t) = \log_b(t)$.
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$\frac{1}{t \ln(b)}$. Derivative of logarithm base $b$ using change of base formula.
$\frac{1}{t \ln(b)}$. Derivative of logarithm base $b$ using change of base formula.
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What is $\frac{dx}{dt}$ for $x(t) = \sec(t)$?
What is $\frac{dx}{dt}$ for $x(t) = \sec(t)$?
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$\sec(t)\tan(t)$. Derivative of $\sec(t)$ using quotient rule.
$\sec(t)\tan(t)$. Derivative of $\sec(t)$ using quotient rule.
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What is the derivative of $x(t) = a\sin(t)$ with respect to $t$?
What is the derivative of $x(t) = a\sin(t)$ with respect to $t$?
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$a\cos(t)$. Derivative of $a\sin(t)$ using constant multiple rule.
$a\cos(t)$. Derivative of $a\sin(t)$ using constant multiple rule.
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Find the integral for $x(t)=\ln(t)$, $y(t)=e^t$ from $t=1$ to $t=2$.
Find the integral for $x(t)=\ln(t)$, $y(t)=e^t$ from $t=1$ to $t=2$.
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$\int_{1}^{2} \sqrt{(\frac{1}{t})^2 + (e^t)^2} , dt$. Setup with $\frac{dx}{dt}=\frac{1}{t}$ and $\frac{dy}{dt}=e^t$.
$\int_{1}^{2} \sqrt{(\frac{1}{t})^2 + (e^t)^2} , dt$. Setup with $\frac{dx}{dt}=\frac{1}{t}$ and $\frac{dy}{dt}=e^t$.
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What does $\frac{dy}{dt}$ represent in parametric equations?
What does $\frac{dy}{dt}$ represent in parametric equations?
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The rate of change of $y$ with respect to $t$. Measures how $y$ changes as parameter $t$ varies.
The rate of change of $y$ with respect to $t$. Measures how $y$ changes as parameter $t$ varies.
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What is the result of $\frac{dy}{dt}$ for $y(t) = \ln(t)$?
What is the result of $\frac{dy}{dt}$ for $y(t) = \ln(t)$?
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$\frac{1}{t}$. Derivative of natural logarithm $\ln(t)$ is $\frac{1}{t}$.
$\frac{1}{t}$. Derivative of natural logarithm $\ln(t)$ is $\frac{1}{t}$.
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What is $\frac{dy}{dt}$ for $y(t) = \csc(t)$?
What is $\frac{dy}{dt}$ for $y(t) = \csc(t)$?
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$-\csc(t)\cot(t)$. Derivative of $\csc(t)$ using quotient rule.
$-\csc(t)\cot(t)$. Derivative of $\csc(t)$ using quotient rule.
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What is the derivative of $x(t) = \ln(t)$?
What is the derivative of $x(t) = \ln(t)$?
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$\frac{1}{t}$. Derivative of natural logarithm function.
$\frac{1}{t}$. Derivative of natural logarithm function.
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What is the value of $\frac{dy}{dt}$ for $y(t) = \sqrt{t}$?
What is the value of $\frac{dy}{dt}$ for $y(t) = \sqrt{t}$?
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$\frac{1}{2\sqrt{t}}$. Derivative of $\sqrt{t} = t^{1/2}$ using power rule.
$\frac{1}{2\sqrt{t}}$. Derivative of $\sqrt{t} = t^{1/2}$ using power rule.
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What does $\frac{dx}{dt}$ represent in parametric equations?
What does $\frac{dx}{dt}$ represent in parametric equations?
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The rate of change of $x$ with respect to $t$. Measures how $x$ changes as parameter $t$ varies.
The rate of change of $x$ with respect to $t$. Measures how $x$ changes as parameter $t$ varies.
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What is the primary use of parametric equations in calculus?
What is the primary use of parametric equations in calculus?
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To describe curves in terms of a parameter. Parametric form allows complex curves not expressible as functions.
To describe curves in terms of a parameter. Parametric form allows complex curves not expressible as functions.
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What is the derivative of $y(t) = a\cos(t)$ with respect to $t$?
What is the derivative of $y(t) = a\cos(t)$ with respect to $t$?
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$-a\sin(t)$. Derivative of $a\cos(t)$ using constant multiple rule.
$-a\sin(t)$. Derivative of $a\cos(t)$ using constant multiple rule.
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Find $\frac{dx}{dt}$ if $x(t) = \sin(t)$.
Find $\frac{dx}{dt}$ if $x(t) = \sin(t)$.
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$\cos(t)$. Derivative of $\sin(t)$ is $\cos(t)$.
$\cos(t)$. Derivative of $\sin(t)$ is $\cos(t)$.
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What is the result of $\frac{dx}{dt}$ for $x(t) = e^t$?
What is the result of $\frac{dx}{dt}$ for $x(t) = e^t$?
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$e^t$. Derivative of exponential function $e^t$ is itself.
$e^t$. Derivative of exponential function $e^t$ is itself.
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State the formula for the arc length of a parametric curve.
State the formula for the arc length of a parametric curve.
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$L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} , dt$. Standard formula using derivatives and Pythagorean theorem.
$L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} , dt$. Standard formula using derivatives and Pythagorean theorem.
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Calculate the arc length for $x(t)=t$, $y(t)=t$ from $t=0$ to $t=1$.
Calculate the arc length for $x(t)=t$, $y(t)=t$ from $t=0$ to $t=1$.
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$\sqrt{2}$. Both derivatives equal 1, so $\sqrt{1^2+1^2}=\sqrt{2}$ over unit interval.
$\sqrt{2}$. Both derivatives equal 1, so $\sqrt{1^2+1^2}=\sqrt{2}$ over unit interval.
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For $x(t)=\cos(t)$, $y(t)=\sin(t)$, what is $\frac{dy}{dt}$?
For $x(t)=\cos(t)$, $y(t)=\sin(t)$, what is $\frac{dy}{dt}$?
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$\cos(t)$. Derivative of $\sin(t)$ is $\cos(t)$.
$\cos(t)$. Derivative of $\sin(t)$ is $\cos(t)$.
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Find $\frac{dx}{dt}$ for $x(t) = 3t^2 + 2t$.
Find $\frac{dx}{dt}$ for $x(t) = 3t^2 + 2t$.
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$6t + 2$. Power rule: derivative of $3t^2$ is $6t$, derivative of $2t$ is $2$.
$6t + 2$. Power rule: derivative of $3t^2$ is $6t$, derivative of $2t$ is $2$.
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What symbol represents arc length in parametric equations?
What symbol represents arc length in parametric equations?
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$L$. Standard notation for arc length measurement.
$L$. Standard notation for arc length measurement.
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Identify the parameter in the equations $x(t)$ and $y(t)$.
Identify the parameter in the equations $x(t)$ and $y(t)$.
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The parameter is $t$. The independent variable that defines both $x$ and $y$.
The parameter is $t$. The independent variable that defines both $x$ and $y$.
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What is the value of $\frac{dx}{dt}$ for $x(t) = t^4$?
What is the value of $\frac{dx}{dt}$ for $x(t) = t^4$?
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$4t^3$. Power rule: derivative of $t^4$ is $4t^3$.
$4t^3$. Power rule: derivative of $t^4$ is $4t^3$.
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What is the derivative of $y(t) = e^t$?
What is the derivative of $y(t) = e^t$?
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$e^t$. Derivative of exponential function with base $e$.
$e^t$. Derivative of exponential function with base $e$.
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Find $\frac{dy}{dt}$ for $y(t) = 4t^3 - t$.
Find $\frac{dy}{dt}$ for $y(t) = 4t^3 - t$.
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$12t^2 - 1$. Power rule: derivative of $4t^3$ is $12t^2$, derivative of $-t$ is $-1$.
$12t^2 - 1$. Power rule: derivative of $4t^3$ is $12t^2$, derivative of $-t$ is $-1$.
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