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9.1 Defining and Differentiating Parametric Equations

3 min readjanuary 23, 2023

Sumi Vora

Sumi Vora

Jed Quiaoit

Jed Quiaoit

Sumi Vora

Sumi Vora

Jed Quiaoit

Jed Quiaoit

🪐 Unit 9 of AP Calculus BC deals with three major topics:

  1. Parametric equations
  2. Polar coordinates - a two-dimensional coordinate system dealing with a line’s distance from the origin (rr) and the angle said line makes with the positive x-axis (θθ).
  3. Vector-valued functions - functions that returns a vector after taking one or more variables.

We’ll dive deeper into the second and third topics in future sections; for now, we’ll focus on parametric functions as they actually tell us a lot of information about real-world phenomena like projectile and circular motion.

💭 What is a Parametric Function?

Parametric functions are a set of related functions where x and y are independent from each other, but they are connected using the dummy variable t, which represents time. When we use the Cartesian graph, we assume that we are moving along the x-axis in only one direction at a constant rate. However, parametric equations give us more freedom to manipulate horizontal motion. 🗺️

A parametric equation would look something like this:

x(t)=t21,y(t)=3tx(t)=t^2-1, y(t)=3t

In this equation, your x-coordinate would be determined by t21t² - 1 and your y-coordinate would be determined by 3t3t. So, when t = 1, you would plot the point (0, 3). In a parametric equation, t isn’t actually on the graph; we just use t as our constant so that our points are independent from one another.

There are several methods for calculating derivatives of real-valued functions, such as the limit definition, the power rule, the product rule, and the quotient rule. These methods can be extended to parametric functions, which are functions that depend on both a real variable and one or more parameters.

One way to extend these methods to parametric functions is to treat the parameters as constants and use the usual rules for differentiation. For example, if a parametric function is given by f(x,p)=px2f(x,p) = px², where x is the real variable and p is the parameter, then the derivative with respect to x can be calculated using the power rule as df/dx=2pxdf/dx = 2p*x.

However, our method for computing derivatives will actually be much simpler than the method above. Excited? 😄


✏️ Differentiating Parametric Equations

Like we discussed earlier, a parametric function is still graphed in 2D on an xy-plane, so if we wanted to find the slope of the tangent line, we would still need to find dy/dx. 🔍

When dealing with parametric functions, where both x and y are expressed in terms of a third variable t, you can find the slope of the tangent line by taking the derivative of y with respect to t (dy/dt) and dividing it by the derivative of x with respect to t (dx/dt):

dydtdxdt=dydx\frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx}

This ratio simplifies to dy/dx, which gives you the slope of the tangent line. This is analogous to the traditional approach of finding the slope of a curve in terms of x and y. Note than dx/dt cannot be zero!

🧠 Understanding Differentiation of Parametric Equations

Still confused about the theory? Let's go into further detail on what this idea really means physically and mathematically. 👇

This idea is known as the "parametric derivative" in calculus, often used to find the instantaneous rate of change of a parametric curve at a specific point. This method can only be used for parametric equations, where the curve is defined using a set of parametric equations in terms of a parameter, such as t.

To find the slope of the tangent line at a point on the curve, we first find the derivative of the x-coordinate function with respect to the parameter and the derivative of the y-coordinate function with respect to the parameter. These derivatives are denoted as dx/dt and dy/dt respectively. ⛰️

Then, at a specific point on the curve (x, y), the slope of the tangent line is found by taking the ratio of the derivative of the y-coordinate function with respect to the parameter (dy/dt) to the derivative of the x-coordinate function with respect to the parameter (dx/dt). This is the equation dy/dx = dy/dt / dx/dt we saw earlier! It’s important to note that the above equation is only valid if dx/dt is not equal to zero at the point of interest, as a vertical tangent line would not have a well-defined slope.

Now, this explanation still seems abstract when only talking about the theory. Why don’t we work this out with some examples? 😁


📝 Practice Differentiating Parametric Equations

Here are some concrete examples!

🥇 Differentiating Parametric Equations: Example 1

Find the slope of the tangent line of the parametrically defined equation at t = 3.

x(t)=t22t,y(t)=t2+1x(t)=t^2-2t, y(t)=t^2+1

We first need to find dy/dt (based on y(t)) and dx/dt (based on x(t)):

dxdt=2t2\frac{dx}{dt}=2t-2
dydt=2t\frac{dy}{dt}=2t

To get our final ratio dy/dx:

dydx=dydtdxdt=2t2t2=tt1\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{2t}{2t-2}=\frac{t}{t-1}

Normally, we’d stop here but we have a given parameter value (aka t = 3). Plug this into t:

dydxt=3=331=32\frac{dy}{dx}|_{t=3}=\frac{3}{3-1}=\frac{3}{2}

Therefore, the slope of the tangent line at t = 1 is 32\frac32!

🥈 Differentiating Parametric Equations: Example 2

Find the slope of the tangent line of the parametrically defined equation at t = -1.

x(t)=ln(t),y(t)=3t4+2t5+3t8x(t)=ln(t), y(t)=3t^4+2t^5+3t-8

You know the drill: find dy/dt and dx/dt by deriving x(t) and y(t), respectively:

dxdt=1t\frac{dx}{dt}=\frac{1}{t}
dydt=12t3+10t4+3\frac{dy}{dt}=12t^3+10t^4+3

To get our final ratio dy/dx:

dydx=dydtdxdt=12t3+10t4+31t=12t4+10t5+3t\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{12t^3+10t^4+3}{\frac{1}{t}}=12t^4+10t^5+3t

Plug t = -1 into our result:

dydxt=1=12(1)4+10(1)5+3(1)=1\frac{dy}{dx}|_{t=-1}=12(-1)^4+10(-1)^5+3(-1)=-1

Therefore, the slope of the tangent line at t = -1 is equal to -1!


💫 Closing

Feeling better? Like most of AP Calc, this unit becomes more and more navigable with practice and repeated exposure! It’s highly encouraged that you brush up on your derivatives and tangent line calculations from the differentiation-focused sections in earlier units. 🗒️

If you have limited time, the key takeaway for this section is the following:

As usual, good luck! 💯

Untitled

Image Courtesy of CalcWorkshop.

Key Terms to Review (16)

Cartesian Graph

: A Cartesian graph, also known as an xy-plane or coordinate plane, is a two-dimensional grid system used to plot points and graph functions. It consists of two perpendicular number lines called the x-axis (horizontal) and y-axis (vertical), which intersect at the origin (0, 0).

Derivative

: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.

df/dx

: df/dx represents the derivative of a function f with respect to x. It measures the rate at which the function is changing at any given point and provides information about its slope or steepness.

Dummy Variable

: A dummy variable is a placeholder variable used in mathematical equations or statistical analysis to represent different categories or conditions. It does not have any real meaning or value, but it helps organize and analyze data.

dy/dt

: The notation dy/dt represents the derivative of a function y with respect to the independent variable t. It measures the rate at which y is changing with respect to t.

Horizontal Motion

: Horizontal motion refers to the movement of an object in a straight line parallel to the ground, without any vertical displacement.

Limit Definition

: The limit definition provides a precise mathematical way to describe what happens as x approaches some value. It determines the behavior and value that a function approaches as its input gets arbitrarily close to a certain number.

Parameters

: Parameters are variables or constants that are used to define or describe mathematical objects or relationships. They allow us to customize or adjust certain aspects of these objects or relationships.

Parametric Curve

: A parametric curve is a set of equations that describe the coordinates of points on a curve in terms of one or more parameters. These equations allow us to represent curves that cannot be easily expressed as functions.

Parametric Equations

: Parametric equations are a set of equations that express the coordinates of points on a curve or surface in terms of one or more parameters. They allow us to represent complex shapes and motions by breaking them down into simpler components.

Parametric Function

: A parametric function describes a set of equations that define coordinates as functions of an independent parameter (usually time). Instead of expressing y directly in terms of x, both x and y are expressed separately as functions with respect to another variable.

Power Rule

: The power rule is a calculus rule used to find the derivative of a function that is raised to a constant power. It states that if f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is equal to n*x^(n-1).

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Quotient Rule

: The quotient rule is a formula used to find the derivative of a quotient or division between two functions. It states that the derivative of a fraction is equal to (the denominator times the derivative of numerator) minus (the numerator times the derivative of denominator), all divided by (the square root of denominator squared).

Real-valued functions

: Real-valued functions are mathematical functions that take real numbers as inputs and produce real numbers as outputs.

Tangent Line

: A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.

9.1 Defining and Differentiating Parametric Equations

3 min readjanuary 23, 2023

Sumi Vora

Sumi Vora

Jed Quiaoit

Jed Quiaoit

Sumi Vora

Sumi Vora

Jed Quiaoit

Jed Quiaoit

🪐 Unit 9 of AP Calculus BC deals with three major topics:

  1. Parametric equations
  2. Polar coordinates - a two-dimensional coordinate system dealing with a line’s distance from the origin (rr) and the angle said line makes with the positive x-axis (θθ).
  3. Vector-valued functions - functions that returns a vector after taking one or more variables.

We’ll dive deeper into the second and third topics in future sections; for now, we’ll focus on parametric functions as they actually tell us a lot of information about real-world phenomena like projectile and circular motion.

💭 What is a Parametric Function?

Parametric functions are a set of related functions where x and y are independent from each other, but they are connected using the dummy variable t, which represents time. When we use the Cartesian graph, we assume that we are moving along the x-axis in only one direction at a constant rate. However, parametric equations give us more freedom to manipulate horizontal motion. 🗺️

A parametric equation would look something like this:

x(t)=t21,y(t)=3tx(t)=t^2-1, y(t)=3t

In this equation, your x-coordinate would be determined by t21t² - 1 and your y-coordinate would be determined by 3t3t. So, when t = 1, you would plot the point (0, 3). In a parametric equation, t isn’t actually on the graph; we just use t as our constant so that our points are independent from one another.

There are several methods for calculating derivatives of real-valued functions, such as the limit definition, the power rule, the product rule, and the quotient rule. These methods can be extended to parametric functions, which are functions that depend on both a real variable and one or more parameters.

One way to extend these methods to parametric functions is to treat the parameters as constants and use the usual rules for differentiation. For example, if a parametric function is given by f(x,p)=px2f(x,p) = px², where x is the real variable and p is the parameter, then the derivative with respect to x can be calculated using the power rule as df/dx=2pxdf/dx = 2p*x.

However, our method for computing derivatives will actually be much simpler than the method above. Excited? 😄


✏️ Differentiating Parametric Equations

Like we discussed earlier, a parametric function is still graphed in 2D on an xy-plane, so if we wanted to find the slope of the tangent line, we would still need to find dy/dx. 🔍

When dealing with parametric functions, where both x and y are expressed in terms of a third variable t, you can find the slope of the tangent line by taking the derivative of y with respect to t (dy/dt) and dividing it by the derivative of x with respect to t (dx/dt):

dydtdxdt=dydx\frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx}

This ratio simplifies to dy/dx, which gives you the slope of the tangent line. This is analogous to the traditional approach of finding the slope of a curve in terms of x and y. Note than dx/dt cannot be zero!

🧠 Understanding Differentiation of Parametric Equations

Still confused about the theory? Let's go into further detail on what this idea really means physically and mathematically. 👇

This idea is known as the "parametric derivative" in calculus, often used to find the instantaneous rate of change of a parametric curve at a specific point. This method can only be used for parametric equations, where the curve is defined using a set of parametric equations in terms of a parameter, such as t.

To find the slope of the tangent line at a point on the curve, we first find the derivative of the x-coordinate function with respect to the parameter and the derivative of the y-coordinate function with respect to the parameter. These derivatives are denoted as dx/dt and dy/dt respectively. ⛰️

Then, at a specific point on the curve (x, y), the slope of the tangent line is found by taking the ratio of the derivative of the y-coordinate function with respect to the parameter (dy/dt) to the derivative of the x-coordinate function with respect to the parameter (dx/dt). This is the equation dy/dx = dy/dt / dx/dt we saw earlier! It’s important to note that the above equation is only valid if dx/dt is not equal to zero at the point of interest, as a vertical tangent line would not have a well-defined slope.

Now, this explanation still seems abstract when only talking about the theory. Why don’t we work this out with some examples? 😁


📝 Practice Differentiating Parametric Equations

Here are some concrete examples!

🥇 Differentiating Parametric Equations: Example 1

Find the slope of the tangent line of the parametrically defined equation at t = 3.

x(t)=t22t,y(t)=t2+1x(t)=t^2-2t, y(t)=t^2+1

We first need to find dy/dt (based on y(t)) and dx/dt (based on x(t)):

dxdt=2t2\frac{dx}{dt}=2t-2
dydt=2t\frac{dy}{dt}=2t

To get our final ratio dy/dx:

dydx=dydtdxdt=2t2t2=tt1\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{2t}{2t-2}=\frac{t}{t-1}

Normally, we’d stop here but we have a given parameter value (aka t = 3). Plug this into t:

dydxt=3=331=32\frac{dy}{dx}|_{t=3}=\frac{3}{3-1}=\frac{3}{2}

Therefore, the slope of the tangent line at t = 1 is 32\frac32!

🥈 Differentiating Parametric Equations: Example 2

Find the slope of the tangent line of the parametrically defined equation at t = -1.

x(t)=ln(t),y(t)=3t4+2t5+3t8x(t)=ln(t), y(t)=3t^4+2t^5+3t-8

You know the drill: find dy/dt and dx/dt by deriving x(t) and y(t), respectively:

dxdt=1t\frac{dx}{dt}=\frac{1}{t}
dydt=12t3+10t4+3\frac{dy}{dt}=12t^3+10t^4+3

To get our final ratio dy/dx:

dydx=dydtdxdt=12t3+10t4+31t=12t4+10t5+3t\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{12t^3+10t^4+3}{\frac{1}{t}}=12t^4+10t^5+3t

Plug t = -1 into our result:

dydxt=1=12(1)4+10(1)5+3(1)=1\frac{dy}{dx}|_{t=-1}=12(-1)^4+10(-1)^5+3(-1)=-1

Therefore, the slope of the tangent line at t = -1 is equal to -1!


💫 Closing

Feeling better? Like most of AP Calc, this unit becomes more and more navigable with practice and repeated exposure! It’s highly encouraged that you brush up on your derivatives and tangent line calculations from the differentiation-focused sections in earlier units. 🗒️

If you have limited time, the key takeaway for this section is the following:

As usual, good luck! 💯

Untitled

Image Courtesy of CalcWorkshop.

Key Terms to Review (16)

Cartesian Graph

: A Cartesian graph, also known as an xy-plane or coordinate plane, is a two-dimensional grid system used to plot points and graph functions. It consists of two perpendicular number lines called the x-axis (horizontal) and y-axis (vertical), which intersect at the origin (0, 0).

Derivative

: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.

df/dx

: df/dx represents the derivative of a function f with respect to x. It measures the rate at which the function is changing at any given point and provides information about its slope or steepness.

Dummy Variable

: A dummy variable is a placeholder variable used in mathematical equations or statistical analysis to represent different categories or conditions. It does not have any real meaning or value, but it helps organize and analyze data.

dy/dt

: The notation dy/dt represents the derivative of a function y with respect to the independent variable t. It measures the rate at which y is changing with respect to t.

Horizontal Motion

: Horizontal motion refers to the movement of an object in a straight line parallel to the ground, without any vertical displacement.

Limit Definition

: The limit definition provides a precise mathematical way to describe what happens as x approaches some value. It determines the behavior and value that a function approaches as its input gets arbitrarily close to a certain number.

Parameters

: Parameters are variables or constants that are used to define or describe mathematical objects or relationships. They allow us to customize or adjust certain aspects of these objects or relationships.

Parametric Curve

: A parametric curve is a set of equations that describe the coordinates of points on a curve in terms of one or more parameters. These equations allow us to represent curves that cannot be easily expressed as functions.

Parametric Equations

: Parametric equations are a set of equations that express the coordinates of points on a curve or surface in terms of one or more parameters. They allow us to represent complex shapes and motions by breaking them down into simpler components.

Parametric Function

: A parametric function describes a set of equations that define coordinates as functions of an independent parameter (usually time). Instead of expressing y directly in terms of x, both x and y are expressed separately as functions with respect to another variable.

Power Rule

: The power rule is a calculus rule used to find the derivative of a function that is raised to a constant power. It states that if f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is equal to n*x^(n-1).

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Quotient Rule

: The quotient rule is a formula used to find the derivative of a quotient or division between two functions. It states that the derivative of a fraction is equal to (the denominator times the derivative of numerator) minus (the numerator times the derivative of denominator), all divided by (the square root of denominator squared).

Real-valued functions

: Real-valued functions are mathematical functions that take real numbers as inputs and produce real numbers as outputs.

Tangent Line

: A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.