Find the domain of the vector-valued function r(t)=F(t)+G(t), where F(t)=cos⁡ti−sin⁡tj+tk, G(t)=cos⁡ti+sin⁡tj

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Find the domain of the vector-valued function r(t)=F(t)+G(t), where  F(t)=cos⁡ti−sin⁡tj+tk, G(t)=cos⁡ti+sin⁡tj

Sharolyn Larson · Accepted Answer

Step1 Given F(t)=cos⁡(t)i−sin⁡(t)j+tk G(t)=cos⁡(t)i+sin⁡(t)j Now, r(t)=F(t)+G(t) =(cos⁡(t)e−sin⁡(t)j+tk)+(cos⁡(t)i+sin⁡(t)j) =2cos⁡(t)i+tk Step2 Since, The domain of the vector valued function in the form ∣r(t)=f(t)i+g(t)j+h(t)k is intersection of all the components (f (t),g(t) and h(t)). So, By comparing: f(t)=2cos⁡(t) h(t)=t Now, The domain of f(t): f(t)=2cos⁡(t) The function has no undefined points nor domain constraints. Therefore, the domain is: −∞&amp;lt;x&amp;lt;∞ And The domain of h(t): h(t)=t The function is defined only when f≥0. Therefore, the domain is:0≤x&amp;lt;∞ Step3 Now, The intersection of domains of f(t) and h(t) is the required domain of a vector-valued function r(t). Therefore, Domain of r(t):[Solution:t≥0Interval Notation[0,∞)]

Find the domain of the vector-valued function \r(t) = F(t)

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