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.
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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.
Practice
Unless otherwise instructed, determine the values of \(t\) where the vector function is smooth.
\( \vec{r}(t) = t^3\vhat{i}  t^5\vhat{j} \)
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 

\( \vec{r}(t) \) is smooth everywhere except for \( t = 0 \)
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 \( t = 0 \)
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\( \vec{r}(t) = (t^2e^{t})\vhat{i}  2(t1)^2\vhat{j} \)
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 

\( \vec{r}(t) \) is smooth for all \(t\)
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 \(t\)
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You CAN Ace Calculus
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1  basic identities  

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) 
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) 
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) 
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) 
Set 2  squared identities  

\( \sin^2t + \cos^2t = 1\) 
\( 1 + \tan^2t = \sec^2t\) 
\( 1 + \cot^2t = \csc^2t\) 
Set 3  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle formulas  

\(\displaystyle{ \sin^2(t) = \frac{1\cos(2t)}{2} }\) 
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) 
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\) 
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
Trig Integrals
\(\int{\sin(x)~dx} = \cos(x)+C\) 
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\) 
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) 
\(\int{\csc(x)~dx} = \) \( \ln\abs{\csc(x)+\cot(x)}+C\) 
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Practice Instructions
Unless otherwise instructed, determine the values of \(t\) where the vector function is smooth.