On this page we explain how to determine where a vector function is smooth. You need to understand what vector functions are and how to take the derivative of vector functions in order to understand this page.
For many of our calculations with vector functions, we will require that the vector function be smooth. A smooth vector function is one where the derivative is continuous and where the derivative is not equal to zero. This is comparable to what you already know from basic continuity where a graph is continuous and does not contain any sharp corners. Here is a good video clip explaining this in more detail.
video by MIP4U 

Okay, so you are ready to work some practice problems on your own.
Problem Statement 

Determine the values of t where the vector function \( \vec{r}(t) = t^3\vhat{i}  t^5\vhat{j} \) is smooth.
Final Answer 

Problem Statement 

Determine the values of t where the vector function \( \vec{r}(t) = t^3\vhat{i}  t^5\vhat{j} \) is smooth.
Solution 

video by PatrickJMT 

Final Answer 

\( \vec{r}(t) \) is smooth everywhere except for \( t = 0 \) 
close solution

Problem Statement 

Determine the values of t where the vector function \( \vec{r}(t) = (t^2e^{t})\vhat{i}  2(t1)^2\vhat{j} \) is smooth.
Final Answer 

Problem Statement 

Determine the values of t where the vector function \( \vec{r}(t) = (t^2e^{t})\vhat{i}  2(t1)^2\vhat{j} \) is smooth.
Solution 

video by PatrickJMT 

Final Answer 

\( \vec{r}(t) \) is smooth for all t 
close solution

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