Find the domain of the vector-valued function. 𝐫(t)=𝐅(t)-𝐆(t), where 𝐅(t)=lnt 𝐢+5 t 𝐣-3

step 1 We're going to find the domain of a vector valued function. Notice we have a F of T minus G of T

Find the domain of the vector-valued function. ...

Find the domain of the vector-valued function.
$$\begin{aligned}&\mathbf{r}(t)=\mathbf{F}(t)-\mathbf{G}(t), \text { where }\\&\mathbf{F}(t)=\ln t \mathbf{i}+5 t \mathbf{j}-3 t^{2} \mathbf{k}, \quad \mathbf{G}(t)=\mathbf{i}+4 t \mathbf{j}-3 t^{2}\mathbf{k}\end{aligned}$$

Key Concepts

Component-wise Operations

Operations on vector-valued functions, such as addition or subtraction, are performed on a component-by-component basis. Consequently, the domain of the resulting vector-valued function is determined by the common domain for which each of the individual component operations is well-defined.

Logarithmic Function Properties

The natural logarithm function, ln(t), is defined only for positive values of t, meaning that its domain is (0, ?). This fundamental property is crucial when the logarithmic function appears as a component, as it restricts the domain of the overall function based on this condition.

Vector-Valued Functions

A vector-valued function is a function that outputs a vector rather than a scalar. Such functions are typically defined in terms of their component functions, with each component defined over a subset of the real numbers. Understanding the vector-valued function requires analyzing each component, particularly when operations like addition or subtraction are applied to other vector-valued functions.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When determining the domain of a vector-valued function, one must consider the domains of all the individual component functions and take their intersection, ensuring that every component yields a valid output for all values in the resulting domain.

Transcript

Please give Ace some feedback