r(t)= <t^3, ln(3-t), sqrt(t) >
find domain of r and answer in interval notation
The domain of r is all values of t that ensure that the components of r are real numbers:
- t^3: defined for all t (i.e. domain is (-infinity, infinity))
- ln(3-t): defined only for t < 3, because ln(x) is only defined for x > 0
- sqrt(t): defined only for t ≥ 0, because the square root of a negative number is not real.
Therefore, the domain of r is:
(0, 3)
Note that we exclude 0 because the square root of 0 is defined but not differentiable, which violates the assumptions of the problems we will consider. The domain in interval notation is:
(0, 3)
the domain of r is [0,3)
we do not care whether it is differentiable or not at t=0
If the domain of r is [0,3), then the components of r must be defined and real for all values of t in that interval. Let's check:
- t^3: defined for all t in [0,3)
- ln(3-t): defined only for t < 3, so it is defined for all t in [0,3)
- sqrt(t): defined only for t ≥ 0, so it is defined for all t in [0,3)
Therefore, the domain of r is indeed [0,3). Note that the function r is not differentiable at t=0 because the square root function is not differentiable at t=0. However, this may or may not be relevant depending on the context of the problem.
To find the domain of the vector function r(t) = <t^3, ln(3-t), sqrt(t)>, we need to determine the values of t for which the function is defined.
Let's analyze each component of the vector function individually to find their domains:
1. For the x-component, t^3, there are no restrictions on the domain of a polynomial function. Therefore, the x-component is defined for all real numbers.
2. For the y-component, ln(3-t), we need to consider the domain of the natural logarithm function. The natural logarithm is defined only for positive real numbers. So, 3-t > 0. Solving this inequality for t, we get t < 3. Therefore, the y-component is defined for all t values less than 3.
3. For the z-component, sqrt(t), the square root is defined only for non-negative real numbers. Therefore, t ≥ 0. The z-component is defined for all non-negative t values.
Now, to find the domain of r(t), we need to consider the intersection of the domains of each component. So, the domain of r(t) is defined where all three components are defined.
Considering the analysis above, we have:
- The x-component is defined for all real numbers.
- The y-component is defined for t < 3.
- The z-component is defined for t ≥ 0.
To find the common interval where all three components are defined, we take the intersection of the intervals: (-∞, 3) ∩ [0, ∞). Simplifying this interval, we have (0, 3).
Therefore, the domain of r(t) is (0, 3) in interval notation.