Find the domain of the following vector-valued functions. 𝐫(t)=cos2 t 𝐢+e^√(t) 𝐣+(12)/(t) 𝐤

Name: Problem 40, Chapter 14: Calculus: Early Transcendentals
Uploaded: 2020-06-18T16:32:44-08:00
Duration: 2 min 39 s
Channel: Howard Francis
Description: Find the domain of the following vector-valued functions. 𝐫(t)=cos2 t 𝐢+e^√(t) 𝐣+(12)/(t) 𝐤

step 1 Let's find the domain of the following vector valued function. R of t equals cosine 2 t plus e t

Find the domain of the following vector-valued functions....

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Key Concepts

Functions Involving Radicals and Rational Expressions

Some functions involve operations like square roots and divisions, which impose specific restrictions. Square root operations require the radicand to be non-negative, while rational expressions necessitate that denominators are non-zero. Identifying and applying these restrictions is crucial in determining the domain of the function.

Domain of a Function

The domain refers to the set of all input values for which a function is defined. In the context of real-valued functions, this means including only those values that do not lead to undefined operations like division by zero or taking square roots of negative numbers.

Vector-Valued Functions

A vector-valued function returns a vector for each input value, meaning that each component of the vector (often corresponding to different coordinate functions) must be defined for the same input. When determining the overall domain, one must consider the domains of all individual components.

Component-wise Domain Analysis

To find the domain of a vector-valued function, one analyzes each component function independently to determine any restrictions. The overall domain is then the intersection of these individual domains, ensuring that every component yields a valid output for any input in the overall domain.

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