Consider the two parametrized paths: r(t) = (t² + 7, t + 1, 25t-1), s(t) = (8t, 2t – 2, t² – 8) What is true about these two parametrized curves?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Parametrized Curves Intersection Analysis
Consider the two parametrized paths:
\[
r(t) = (t^2 + 7, t + 1, 25t^{-1})
\]
\[
s(t) = (8t, 2t - 2, t^2 - 8)
\]
#### Question:
What is true about these two parametrized curves?
#### Options:
- ❍ They intersect, but they don't collide
- ❍ They collide, but they don't intersect
- ❍ They don't intersect
The provided parametrized paths \( r(t) \) and \( s(t) \) represent curves in 3-dimensional space defined over a parameter \( t \). To determine whether these curves intersect or collide, one would typically solve for conditions under which their corresponding equations yield equivalent points in space for any parameter value \( t \).
#### Detailed Explanation:
1. **Intersection** means there is at least one point in space (x, y, z) that both curves pass through, but potentially at different parameter values.
2. **Collision** means that there is at least one parameter value \( t \) where \( r(t) = s(t) \).
Evaluate the path equations:
- For \( r(t) \):
- \( x_1 = t^2 + 7 \)
- \( y_1 = t + 1 \)
- \( z_1 = 25t^{-1} \)
- For \( s(t) \):
- \( x_2 = 8t \)
- \( y_2 = 2t - 2 \)
- \( z_2 = t^2 - 8 \)
By setting the equations equal to solve for common \( x, y, z \) coordinates, you can determine intersection or collision points. This analysis can involve solving a system of equations to identify potential \( t \)-values that satisfy both paths simultaneously.
If the equations confirm both criteria at the same \( t \) value or different \( t \) values yielding the same (x, y, z) point, select the appropriate option above based on your findings.
### Analysis of Results
- **Intersects, but doesn't collide:** There exists at least one spatial point common to both curves but at different parameter values.
- **Collides, but doesn't intersect:** Unlikely in typical scenarios as intersecting at a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64775454-4845-4a2e-95ce-edb2bc3db137%2F100f29ef-62de-4616-a75c-15eb1b16aff6%2F1swqc4q.png&w=3840&q=75)
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