Solving Related Rates Problems - AP Calculus AB
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What is the formula for the area of a circle?
What is the formula for the area of a circle?
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$A = \pi r^2$. Basic circle area formula.
$A = \pi r^2$. Basic circle area formula.
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What is the formula for the area of a rectangle?
What is the formula for the area of a rectangle?
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$A = l w$. Length times width for rectangular area.
$A = l w$. Length times width for rectangular area.
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Differentiate $A = 4\pi r^2$ with respect to time $t$.
Differentiate $A = 4\pi r^2$ with respect to time $t$.
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$\frac{dA}{dt} = 8\pi r \frac{dr}{dt}$. Chain rule applied to sphere surface area.
$\frac{dA}{dt} = 8\pi r \frac{dr}{dt}$. Chain rule applied to sphere surface area.
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What is the derivative of the Pythagorean Theorem with respect to time $t$?
What is the derivative of the Pythagorean Theorem with respect to time $t$?
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$2a \frac{da}{dt} + 2b \frac{db}{dt} = 2c \frac{dc}{dt}$. Implicit differentiation of Pythagorean theorem.
$2a \frac{da}{dt} + 2b \frac{db}{dt} = 2c \frac{dc}{dt}$. Implicit differentiation of Pythagorean theorem.
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What is the formula for the surface area of a cube?
What is the formula for the surface area of a cube?
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$A = 6s^2$. Six square faces with side length $s$.
$A = 6s^2$. Six square faces with side length $s$.
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Differentiate $PV = nRT$ with respect to time $t$.
Differentiate $PV = nRT$ with respect to time $t$.
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$P \frac{dV}{dt} + V \frac{dP}{dt} = 0$. Product rule assuming temperature constant.
$P \frac{dV}{dt} + V \frac{dP}{dt} = 0$. Product rule assuming temperature constant.
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Differentiate $x^2 + y^2 = l^2$ with respect to time $t$.
Differentiate $x^2 + y^2 = l^2$ with respect to time $t$.
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$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$. Ladder length $l$ remains constant, so derivative is zero.
$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$. Ladder length $l$ remains constant, so derivative is zero.
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What is the formula for the area of a sector of a circle?
What is the formula for the area of a sector of a circle?
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$A = \frac{1}{2} r^2 \theta$. Half radius squared times central angle.
$A = \frac{1}{2} r^2 \theta$. Half radius squared times central angle.
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Which principle is used in related rates to connect different rates of change?
Which principle is used in related rates to connect different rates of change?
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Chain Rule. Links rates through composite function differentiation.
Chain Rule. Links rates through composite function differentiation.
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What is the first step in using implicit differentiation?
What is the first step in using implicit differentiation?
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Differentiate both sides with respect to $t$. Apply $\frac{d}{dt}$ to the entire equation.
Differentiate both sides with respect to $t$. Apply $\frac{d}{dt}$ to the entire equation.
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What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
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$A = \frac{1}{2} b h$. Standard triangle area with base $b$ and height $h$.
$A = \frac{1}{2} b h$. Standard triangle area with base $b$ and height $h$.
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State the formula for the volume of a cylinder.
State the formula for the volume of a cylinder.
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$V = \pi r^2 h$. Standard cylinder volume with radius $r$ and height $h$.
$V = \pi r^2 h$. Standard cylinder volume with radius $r$ and height $h$.
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How do you differentiate $C = 2\pi r$ with respect to time $t$?
How do you differentiate $C = 2\pi r$ with respect to time $t$?
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$\frac{dC}{dt} = 2\pi \frac{dr}{dt}$. Linear relationship gives constant coefficient.
$\frac{dC}{dt} = 2\pi \frac{dr}{dt}$. Linear relationship gives constant coefficient.
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What is the formula for the volume of a cone?
What is the formula for the volume of a cone?
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$V = \frac{1}{3} \pi r^2 h$. One-third of cylinder volume formula.
$V = \frac{1}{3} \pi r^2 h$. One-third of cylinder volume formula.
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Differentiate $P = 2l + 2w$ with respect to time $t$.
Differentiate $P = 2l + 2w$ with respect to time $t$.
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$\frac{dP}{dt} = 2 \frac{dl}{dt} + 2 \frac{dw}{dt}$. Linear combination of length and width rates.
$\frac{dP}{dt} = 2 \frac{dl}{dt} + 2 \frac{dw}{dt}$. Linear combination of length and width rates.
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Differentiate $A = \frac{1}{2} b h$ with respect to time $t$.
Differentiate $A = \frac{1}{2} b h$ with respect to time $t$.
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$\frac{dA}{dt} = \frac{1}{2} \left( b \frac{dh}{dt} + h \frac{db}{dt} \right)$. Product rule applied to triangle area.
$\frac{dA}{dt} = \frac{1}{2} \left( b \frac{dh}{dt} + h \frac{db}{dt} \right)$. Product rule applied to triangle area.
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What is the relationship between linear and angular speed?
What is the relationship between linear and angular speed?
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$v = r \omega$. Linear velocity equals radius times angular velocity.
$v = r \omega$. Linear velocity equals radius times angular velocity.
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What is the formula for the surface area of a sphere?
What is the formula for the surface area of a sphere?
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$A = 4\pi r^2$. Four times $\pi$ times radius squared.
$A = 4\pi r^2$. Four times $\pi$ times radius squared.
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Differentiate $A = l w$ with respect to time $t$.
Differentiate $A = l w$ with respect to time $t$.
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$\frac{dA}{dt} = l \frac{dw}{dt} + w \frac{dl}{dt}$. Product rule for rectangle area differentiation.
$\frac{dA}{dt} = l \frac{dw}{dt} + w \frac{dl}{dt}$. Product rule for rectangle area differentiation.
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What is the derivative of $V = \frac{4}{3} \pi r^3$ with respect to time $t$?
What is the derivative of $V = \frac{4}{3} \pi r^3$ with respect to time $t$?
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$\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$. Chain rule applied to sphere volume formula.
$\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$. Chain rule applied to sphere volume formula.
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How do you differentiate $v = r \omega$ with respect to time $t$?
How do you differentiate $v = r \omega$ with respect to time $t$?
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$\frac{dv}{dt} = \omega \frac{dr}{dt} + r \frac{d\omega}{dt}$. Product rule applied to velocity relationship.
$\frac{dv}{dt} = \omega \frac{dr}{dt} + r \frac{d\omega}{dt}$. Product rule applied to velocity relationship.
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How do you find the relationship between given and unknown rates?
How do you find the relationship between given and unknown rates?
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Use a geometric or physical relationship. Connect variables through mathematical or physical laws.
Use a geometric or physical relationship. Connect variables through mathematical or physical laws.
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What is the definition of a related rates problem?
What is the definition of a related rates problem?
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A problem involving rates of change of related variables. Variables change simultaneously at connected rates.
A problem involving rates of change of related variables. Variables change simultaneously at connected rates.
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Differentiate $D^2 = x^2 + y^2$ with respect to time $t$.
Differentiate $D^2 = x^2 + y^2$ with respect to time $t$.
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$2D \frac{dD}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$. Chain rule for distance formula differentiation.
$2D \frac{dD}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$. Chain rule for distance formula differentiation.
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What is the Pythagorean Theorem?
What is the Pythagorean Theorem?
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$a^2 + b^2 = c^2$. Fundamental right triangle relationship.
$a^2 + b^2 = c^2$. Fundamental right triangle relationship.
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Identify the formula for differentiating $A = \pi r^2$ with respect to $t$.
Identify the formula for differentiating $A = \pi r^2$ with respect to $t$.
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$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Chain rule applied to circle area.
$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Chain rule applied to circle area.
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What is the formula for the volume of a rectangular prism?
What is the formula for the volume of a rectangular prism?
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$V = lwh$. Length times width times height.
$V = lwh$. Length times width times height.
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What is the formula for the perimeter of a rectangle?
What is the formula for the perimeter of a rectangle?
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$P = 2l + 2w$. Sum of all four sides of rectangle.
$P = 2l + 2w$. Sum of all four sides of rectangle.
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What is the relationship between the circumference and radius of a circle?
What is the relationship between the circumference and radius of a circle?
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$C = 2\pi r$. Circumference equals $2\pi$ times radius.
$C = 2\pi r$. Circumference equals $2\pi$ times radius.
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Differentiate $A = \frac{1}{2} r^2 \theta$ with respect to time $t$.
Differentiate $A = \frac{1}{2} r^2 \theta$ with respect to time $t$.
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$\frac{dA}{dt} = r \theta \frac{dr}{dt} + \frac{1}{2} r^2 \frac{d\theta}{dt}$. Product rule for sector area differentiation.
$\frac{dA}{dt} = r \theta \frac{dr}{dt} + \frac{1}{2} r^2 \frac{d\theta}{dt}$. Product rule for sector area differentiation.
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