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How do you find the slope of the tangent line of a parametric curve at a given value of t?
1. Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \). 2. Calculate \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \). 3. Substitute the given value of t into \( \frac{dy}{dx} \).
How do you determine where a parametric curve has a horizontal tangent?
1. Find \( \frac{dy}{dt} \). 2. Set \( \frac{dy}{dt} = 0 \) and solve for t. 3. Verify that \( \frac{dx}{dt} \neq 0 \) for those values of t.
How do you determine where a parametric curve has a vertical tangent?
1. Find \( \frac{dx}{dt} \). 2. Set \( \frac{dx}{dt} = 0 \) and solve for t. 3. Verify that \( \frac{dy}{dt} \neq 0 \) for those values of t.
Given x(t) and y(t), how do you eliminate the parameter t to find the Cartesian equation?
1. Solve one of the parametric equations for t. 2. Substitute that expression for t into the other parametric equation.
How do you find the equation of the tangent line to a parametric curve at a specific point (x₀, y₀)?
1. Find the value of t corresponding to the point (x₀, y₀). 2. Find \( \frac{dy}{dx} \) at that value of t. 3. Use the point-slope form of a line: \( y - y₀ = \frac{dy}{dx}(x - x₀) \).
How do you find the second derivative \( \frac{d^2y}{dx^2} \) for parametric equations?
1. Find \( \frac{dy}{dx} \). 2. Find \( \frac{d}{dt}(\frac{dy}{dx}) \). 3. Divide the result by \( \frac{dx}{dt} \): \( \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}} \).
How to solve for t given x(t) = a and y(t) = b?
1. Solve x(t) = a for t. 2. Plug the value(s) of t into y(t). 3. If y(t) = b, then that t value is the solution.
How do you find the points of intersection between two parametric curves?
1. Set the x-equations equal to each other and the y-equations equal to each other. 2. Solve the system of equations for the parameter values. 3. Substitute the parameter values back into the original equations to find the points of intersection.
How do you find the arc length of a parametric curve given x(t) and y(t) from t=a to t=b?
1. Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \). 2. Calculate \( \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \). 3. Integrate the result from a to b: \( \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt \).
How do you find the area under a parametric curve?
1. Express y as a function of t, y(t). 2. Find \( \frac{dx}{dt} \). 3. Integrate \( y(t) \cdot \frac{dx}{dt} \) with respect to t over the appropriate interval.
Explain the concept of differentiating parametric equations.
Finding the rate of change of y with respect to x, where both x and y are defined in terms of a third variable t.
Explain why \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \).
It's an application of the chain rule. It relates the rate of change of y with respect to t to the rate of change of x with respect to t to find the rate of change of y with respect to x.
What does \( \frac{dy}{dx} \) represent graphically?
The slope of the tangent line to the parametric curve at a specific point.
Why is it important to check that \( \frac{dx}{dt} \neq 0 \) when finding \( \frac{dy}{dx} \)?
Division by zero is undefined. If \( \frac{dx}{dt} = 0 \), the tangent line is vertical, and the slope is undefined.
Explain how parametric equations provide more freedom in manipulating horizontal motion compared to Cartesian equations.
Parametric equations allow the x-coordinate to change direction and speed independently of the y-coordinate, unlike Cartesian equations where x typically increases at a constant rate.
Describe the relationship between parametric equations and real-world phenomena like projectile motion.
Parametric equations can model projectile motion by separately defining the horizontal and vertical positions (x and y) as functions of time (t), allowing for the incorporation of factors like gravity and initial velocity.
Explain the concept of a tangent line to a parametric curve.
The tangent line is a straight line that touches the curve at a single point and has the same slope as the curve at that point, representing the instantaneous direction of the curve.
How does the parameter 't' affect the graph of a parametric equation?
The parameter 't' dictates the order in which points are plotted, thus defining the direction and speed at which the curve is traced.
What is the geometric interpretation of \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)?
\( \frac{dx}{dt} \) represents the horizontal velocity, and \( \frac{dy}{dt} \) represents the vertical velocity of a point moving along the curve.
Explain how parametric equations can be used to describe circular motion.
By using trigonometric functions (sine and cosine) for x(t) and y(t), parametric equations can describe circular motion with the parameter t representing the angle or time.
What is the formula for finding \( \frac{dy}{dx} \) given parametric equations x(t) and y(t)?
\( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \)
How to find the slope of a tangent line to a parametric curve?
\( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \)
What is the condition for \( \frac{dy}{dx} \) to be valid in parametric equations?
\( \frac{dx}{dt} \neq 0 \)
What is the formula for finding the second derivative \( \frac{d^2y}{dx^2} \) of parametric equations?
\( \frac{d^2y}{dx^2} = \frac{d/dt(\frac{dy}{dx})}{\frac{dx}{dt}} \)
How do you find the equation of the tangent line to a parametric curve at a specific value of t?
Find \( \frac{dy}{dx} \) at the given t, then use the point-slope form: \( y - y(t) = \frac{dy}{dx}(x - x(t)) \)
What formula do you use to find the points where the tangent line is horizontal?
Set \( \frac{dy}{dt} = 0 \) and \( \frac{dx}{dt} \neq 0 \) and solve for t.
What formula do you use to find the points where the tangent line is vertical?
Set \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} \neq 0 \) and solve for t.
How do you find the arc length of a parametric curve?
\( \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt \)
What is the formula for the area under a parametric curve?
\( \int_{a}^{b} y(t) \cdot x'(t) dt \)
How do you find the speed of a particle moving along a parametric curve?
Speed = \( \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \)