Find the slope of the line defined by the parametric equations [tex]$x=-6$[/tex] and [tex]$y=5+8t$[/tex] without eliminating the parameter. If the slope is undefined, enter [tex]$\varnothing$[/tex].
Answer (2)
Find d t d x : Since x = − 6 , d t d x = 0 .
Find d t d y : Since y = 5 + 8 t , d t d y = 8 .
Calculate the slope d x d y = d t d x d t d y = 0 8 , which is undefined.
The slope is undefined, so the answer is ∅ .
Explanation
Problem Setup We are given the parametric equations x = − 6 and y = 5 + 8 t . We want to find the slope of the line defined by these equations without eliminating the parameter.
Applying the Chain Rule To find the slope, we need to compute d x d y . Using the chain rule, we have d x d y = d t d x d t d y .
Finding dx/dt First, let's find d t d x . Since x = − 6 , which is a constant, its derivative with respect to t is 0. So, d t d x = 0 .
Finding dy/dt Next, let's find d t d y . Since y = 5 + 8 t , its derivative with respect to t is 8. So, d t d y = 8 .
Calculating the Slope Now, we can find d x d y = d t d x d t d y = 0 8 . Since division by zero is undefined, the slope is undefined.
Final Answer Since the slope is undefined, we enter ∅ .
Examples
Parametric equations are used in physics to describe the trajectory of a projectile. For example, the position of a ball thrown in the air can be described by parametric equations where time is the parameter. Finding the slope of the trajectory at a given time helps determine the direction of the ball's motion.
The slope of the line defined by the given parametric equations is undefined due to the constant value of x resulting in a zero derivative. Therefore, the answer is ∅ .
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