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Questions and Answers
What is the first derivative $ rac{dy}{dx}$ for the parametric equations $x = t^2$ and $y = t^3 - 3t$?
What is the first derivative $ rac{dy}{dx}$ for the parametric equations $x = t^2$ and $y = t^3 - 3t$?
For the parametric equations $x = 4 + t^2$ and $y = t^2 + t^3$, what is the expression for $ rac{dy}{dx}$?
For the parametric equations $x = 4 + t^2$ and $y = t^2 + t^3$, what is the expression for $ rac{dy}{dx}$?
What is the second derivative $ rac{d^2y}{dx^2}$ for the parametric equations $x = 2 ext{sin} t$ and $y = 3 ext{cos} t$?
What is the second derivative $ rac{d^2y}{dx^2}$ for the parametric equations $x = 2 ext{sin} t$ and $y = 3 ext{cos} t$?
When given the parametric equations $x = t^2$ and $y = t^3 - 3t$, which statement about the derivatives is accurate?
When given the parametric equations $x = t^2$ and $y = t^3 - 3t$, which statement about the derivatives is accurate?
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If $y = 3 ext{sin} t$ and $x = 4 + t^2$, what is the correct expression for $ rac{dy}{dx}$?
If $y = 3 ext{sin} t$ and $x = 4 + t^2$, what is the correct expression for $ rac{dy}{dx}$?
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In the context of parametric equations, which formula correctly represents the second derivative of $y$ with respect to $x$?
In the context of parametric equations, which formula correctly represents the second derivative of $y$ with respect to $x$?
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For the parametric equations $x = t^2$ and $y = t^3 - 3t$, what is the value of the second derivative at $t = 1$?
For the parametric equations $x = t^2$ and $y = t^3 - 3t$, what is the value of the second derivative at $t = 1$?
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What is the formula used to find the slope of the tangent line to the parametric equations $x = t^4 + 1$, $y = t^3 + t$?
What is the formula used to find the slope of the tangent line to the parametric equations $x = t^4 + 1$, $y = t^3 + t$?
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Given the parametric equations $x = 1 + ext{ln}(t)$ and $y = t^2 + 2$, what is the value of $t$ at the point $(1, 3)$?
Given the parametric equations $x = 1 + ext{ln}(t)$ and $y = t^2 + 2$, what is the value of $t$ at the point $(1, 3)$?
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At which point does the curve defined by $x = 2t^3$ and $y = 1 + 4t - t^2$ have a tangent line with slope equal to 1?
At which point does the curve defined by $x = 2t^3$ and $y = 1 + 4t - t^2$ have a tangent line with slope equal to 1?
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What is the partial derivative $ rac{ ext{d}z}{ ext{d}y}$ if $z = 2x + 3y - 6$?
What is the partial derivative $ rac{ ext{d}z}{ ext{d}y}$ if $z = 2x + 3y - 6$?
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How do you calculate the partial derivative $f_x(1,1)$ for the function $f(x,y) = 4 - x^2 - 2y^2$?
How do you calculate the partial derivative $f_x(1,1)$ for the function $f(x,y) = 4 - x^2 - 2y^2$?
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If $z = x^3 + x^2y^3 - 2y^2$, what is the value of the partial derivative $ rac{ ext{d}z}{ ext{d}x}$?
If $z = x^3 + x^2y^3 - 2y^2$, what is the value of the partial derivative $ rac{ ext{d}z}{ ext{d}x}$?
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To find the partial derivative $f_y$, if $f(x, y) = y^5 - 3xy$, what is the calculated expression?
To find the partial derivative $f_y$, if $f(x, y) = y^5 - 3xy$, what is the calculated expression?
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What is the result of the partial derivative $ rac{ ext{d}z}{ ext{d}y}$ for the function $z = (2x + 3y)^{10}$?
What is the result of the partial derivative $ rac{ ext{d}z}{ ext{d}y}$ for the function $z = (2x + 3y)^{10}$?
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If $f(x, y) = 4 - x^2 - 2y^2$, which expression represents the partial derivative $f_y$ at $(1,1)$?
If $f(x, y) = 4 - x^2 - 2y^2$, which expression represents the partial derivative $f_y$ at $(1,1)$?
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For the function $z = f(x, y)$, what is the interpretation of $f_x(a, b)$?
For the function $z = f(x, y)$, what is the interpretation of $f_x(a, b)$?
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Study Notes
Parametric Equations
- Parametric equations represent x and y as functions of a third variable (parameter), t.
- x = f(t) and y = g(t) are examples of parametric equations.
- Derivatives of parametric equations can be found using:
- dy/dx = (dy/dt) / (dx/dt)
- d²y/dx² = [d(dy/dt)/dt] / (dx/dt)
Exercises 13.1
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Problem 1: Find dy/dx and d²y/dx² given x = t² and y = t³ – 3t.
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Solutions provided in the text:
- dy/dx = (3t² – 3) / (2t)
- d²y/dx² = 6t/(2t)³ = 3/(2t²)
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Problem 2: Find dy/dx given x = 4 + t² and y = t + t³.
- Solution: dy/dx= (2t + 3t²)/2t=(1+3t^2/2t)
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Problem 3: Find dy/dx and d²y/dx² for the parametric equations; x = 2 sin t and y = 3 cos t
- Solution:
- dy/dx = -3 tan t/2
- d²y/dx² = -3 sec³t/4
- Solution:
Applications of Parametric Equations (Exercises 13.2)
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Problem 1: Find the tangent equation to the curve defined by x = t⁴ + 1 and y = t³ + t at t=-1.
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Note: The solution to the problem involves finding y and x at t=-1.
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Problem 2: Find the tangent equation to the curve specified by the parametric equations x = 1 + ln t, y = t² + 2 at the given point (1,3).
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Note: The solution involves finding dy/dx and the tangent line at the given point.
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Problem 3: Find the points on the curve with parametric equations x = 2t³ and y = 1 + 4t – t² where the tangent lines have a slope of 1.
Partial Derivatives
- Partial derivatives are used when a function depends on multiple variables (e.g., z = f(x, y)).
- fx (partial derivative of f with respect to x) is found by treating y as a constant.
- fy (partial derivative of f with respect to y) is found by treating x as a constant.
- Examples:
- z = 2x + 3y – 6 : fx = 2 and fy = 3
- f(x, y) = x³ + x² y³ – 2y²: fx = 3x² + 2xy³ and fy = 3x²y² - 4y
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Description
This quiz focuses on parametric equations, their derivatives, and applications. It includes problems with solutions to find dy/dx and d²y/dx² for given parametric functions. Test your understanding of the concepts and calculations involved in working with parametric equations.
