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Calculus 3  Exam 1 ( Semester A ) 
This is the first exam for third semester calculus. 
Exam Details 

Tools 
Time  2 hours  
Calculators  no 
Questions  5  
Formula Sheet(s) 
1 page, 8.5x11 or A4 
Total Points  75  
Other Tools  none 
Instructions:
 Show all your work.
 For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
 Correct notation counts (i.e. points will be taken off for incorrect notation).
 Give exact, simplified answers.
[ Click on the question to reveal/hide the solution. ]
Question 1 
(20 points) For \( \vec{r}(t) = 2\cos(t^2) \hat{i} + 2\sin(t^2)\hat{j} \), \(t\ge0\), compute
a. \(\vec{T}(t)\); b. \(\vec{N}(t)\); c. curvature; d. \(a_T\) and \(a_N\);
e. \(\vec{r}(s)\) 

a. \(\displaystyle{ \vec{T} = \frac{\vec{v}(t)}{\\vec{v}(t)\ } }\)
\(\vec{r}'(t)=\vec{v}(t)=2\sin(t^2)(2t)\hat{i}+2\cos(t^2)(2t)\hat{j}\)
\( \\vec{v}(t)\ = \sqrt{ [4t\sin(t^2)]^2+[4t\cos(t^2)]^2 } = \)
\( \sqrt{ 16t^2\sin^2(t^2) + 16t^2\cos^2(t^2) } = \)
\( \sqrt{ 16t^2[\sin^2(t^2) + \cos^2(t^2)] } = \sqrt{16t^2} = 4t\)
\(\displaystyle{ \vec{T} = \frac{\vec{v}(t)}{\\vec{v}(t)\ } = \frac{4t\sin(t^2)\hat{i}+4t\cos(t^2)\hat{j}}{4t} }\)
\( \boxed{ \vec{T} = \sin(t^2)\hat{i}+\cos(t^2)\hat{j} } \)
b. \(\displaystyle{ \vec{N}(t) = \frac{ d\vec{T}/dt }{ \d\vec{T}/dt\ } }\)
\(\displaystyle{ \frac{d\vec{T}}{dt} = \cos(t^2)(2t)\hat{i} + (sin(t^2))(2t)\hat{j} }\)
\( \ d\vec{T}/dt \ = \sqrt{ 4t^2\cos^2(t^2) + 4t^2\sin^2(t^2) } = \) \(\sqrt{4t^2} = 2t\)
\(\displaystyle{ \vec{N}(t) = \frac{2t\cos(t^2)\hat{i}2t\sin(t^2)\hat{j}}{2t} }\)
\( \boxed{ \vec{N}(t) = \cos(t^2)\hat{i} \sin(t^2)\hat{j} }\)
c. \(\displaystyle{ K=\frac{1}{\\vec{v}\} \left\\frac{d\vec{T}}{dt}\right\ = }\)
\(\displaystyle{ \frac{1}{4t}(2t) = \frac{1}{2} }\) \(\boxed{ K=1/2 }\)
d. \(a_N = k \\vec{v}\^2 = (1/2)(4t)^2 = 8t^2\) \(\boxed{a_N=8t^2}\)
You may be tempted to use the equation \(\displaystyle{ a_T = \frac{\vec{v} \cdot \vec{a}}{\\vec{v}\} }\), which will work but it is the hard way. However, if you look ahead at the next part of this problem, you are asked to find \(\vec{r}(s)\), the arc length parameterization of \(\vec{r}(t)\). During the course of solving that, you will find that \(s=t^2\). So, to find \(a_T\), we can use \(\displaystyle{ a_T=\frac{d^2s}{dt^2} }\).
\(\displaystyle{ a_T=\frac{d^2s}{dt^2} = \frac{d}{dt}[4t] = 4 }\) \(\boxed{a_T=4}\)
e. \(\displaystyle{ s(t) = \int_{a}^{t}{\\vec{v}(u)\~du} }\)
Since \(\\vec{v}(u)\ = 4u\), we have \(\displaystyle{ s(t)=\int_{0}^{t}{4u~du} = \left. 2u^2\right_0^t = 2t^2 }\)
So \(s=2t^2\). Solving for t, we have \(s=2t^2 \to s/2=t^2 \to t=\pm\sqrt{s/2}\). Since \(t\ge0\), we choose the positive square root, giving us \(t=\sqrt{s/2}\). Substituting for t in the original position function, we get \(\vec{r}\) in terms of s.
\(\boxed{ \vec{r}(s) = 2\cos(s/2)\hat{i} + 2\sin(s/2)\hat{j} ~~~ s \ge 0}\)
Question 2 
(5 points) Compute the indefinite integral of \(\vec{r}(t)= (5t^{4}t^2)\hat{i} + (t^64t^3)\hat{j} + (2/t)\hat{k}\) 


\(\displaystyle{ \int{ \vec{r}(t)~dt } = \left( \frac{5}{3}t^{3}\frac{t^3}{3} \right)\hat{i} + \left( \frac{t^7}{7}t^4 \right) \hat{j} + \left( 2\ln \abs{t} \right)\hat{k} + \vec{C} }\) 
Evaluating each integral separately, gives us
xcomponent: \(\displaystyle{ \int{ 5t^{4}t^2~dt } = \frac{5t^{3}}{3}  \frac{t^3}{3} + c_1 }\)
ycomponent: \(\displaystyle{ \int{ t^64t^3 ~dt} = \frac{t^7}{7}\frac{4t^4}{4} + c_2 }\)
zcomponent: \(\displaystyle{ \int{ \frac{2}{t}~dt} = 2\ln \abs{t} + c_3 }\)
Let \(\vec{C}= c_1\hat{i} + c_2 \hat{j} + c_3 \hat{k} \).
Question 2 Final Answer 
\(\displaystyle{ \int{ \vec{r}(t)~dt } = \left( \frac{5}{3}t^{3}\frac{t^3}{3} \right)\hat{i} + \left( \frac{t^7}{7}t^4 \right) \hat{j} + \left( 2\ln \abs{t} \right)\hat{k} + \vec{C} }\) 
Question 3 
(10 points) For the trajectory \(\vec{r}(t)=\langle e^t\sin(t), e^t\cos(t), e^t\rangle\), find
a. the speed associated with the trajectory;
b. the length of the trajectory on the interval \(0\leq t \leq \ln(2)\) 

a. speed \(\\vec{v}(t)\\)
\( \vec{v}(t) = \vec{r}'(t) = \langle e^t\cos(t)+e^t\sin(t), e^t\sin(t)+e^t\cos(t), e^t \rangle \)
\( \\vec{v}(t)\ = \sqrt{ (e^t\cos t + e^t \sin t)^2 + (e^t\sin t+e^t\cos t)^2 + e^{2t} } = \)
\(e^t\sqrt{ \cos^2 t + 2\sin t \cos t + \sin^2 t + \sin^2 t  2\sin t \cos t + \cos^2 t + 1 } = \)
\(e^t \sqrt{1+1+1} \) \(\boxed{ \\vec{v}(t) \ = e^t\sqrt{3} }\)
b.
\(\displaystyle{ L = \int_{a}^{b}{ \\vec{v}(t)\ } = \int_{0}^{\ln 2}{ e^t\sqrt{3}~dt } = }\)
\(\displaystyle{ \left. e^t\sqrt{3}\right_{0}^{\ln 2} = }\)
\(\sqrt{3}e^{\ln 2}  \sqrt{3}e^0 = 2\sqrt{3}\sqrt{3} = \sqrt{3}\) \(\boxed{L=\sqrt{3}}\)
Question 4 
(15 points) For the vectors \(\vec{u}=\langle 1,3,2 \rangle \) and \(\vec{v}=\langle 4,2,1 \rangle \), find
a. the cosine of the angle \(\theta\) between the vectors
b. the projection of \(\vec{v}\) onto \(\vec{u}\);
c. \(\vec{u} \times \vec{v}\) 

a. To find the cosine of the angle, use \(\vec{u}\cdot\vec{v}= \\vec{u}\~\\vec{v}\ \cos \theta \)
\(\vec{u}\cdot\vec{v}=1(4)+3(2)+(2)(1)=4+62 = 8\)
\( \\vec{u}\ = \sqrt{1^2+3^2+(2)^2} = \sqrt{14} \)
\( \\vec{v}\ = \sqrt{4^2+2^2+1^2} = \sqrt{21} \)
\(\displaystyle{ \cos \theta = \frac{8}{\sqrt{14}\sqrt{21}} }\)
\( \boxed{ \displaystyle{ \cos \theta = \frac{8}{7\sqrt{6}} } }\)
b. \(\displaystyle{ proj_{\vec{u}}\vec{v} = \\vec{v}\ \cos\theta\left( \frac{\vec{u}}{\\vec{u}\} \right) = }\)
\(\displaystyle{\sqrt{21}\frac{8}{7\sqrt{6}} \frac{\langle1,3,2\rangle}{\sqrt{14}} = }\)
\(\displaystyle{ \frac{4}{7}\langle 1,3,2 \rangle }\)
c.
\(\displaystyle{ \vec{u}\times\vec{v} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & 3 & 2 \\
4 & 2 & 1
\end{vmatrix}
= }\)
\(\displaystyle{ \hat{i}(3+4)  \hat{j}(1+8) +\hat{k}(212) }\)
\(\boxed{ \vec{u}\times\vec{v}=\langle 7,9,10\rangle }\)
Question 5 
(25 points) A baseball is hit 3 ft above home plate with an initial velocity of \(\langle 65,85,85\rangle\) ft/sec. The spin on the baseball produces a horizontal acceleration of the ball of 12 ft/sec^{2} in the eastward direction. Assume the xaxis points east, the yaxis points north and the zaxis is vertical with g=32 ft/sec^{2}.
a. Find the velocity vector for \(t\ge0\).
b. Find the position vector for \(t\ge0\).
c. Clearly explain how you would determine the time of flight and range of the ball (do not calculate).
d. Clearly explain how you would determine the maximum height of the ball (do not calculate).


a. \(\vec{a}=\langle 12,0,32\rangle\) ft/sec^{2}
\(\vec{v}=\int{\vec{a}~dt}=\langle 12t+c_1, c_2, 32t+c_3 \rangle\)
\(\vec{v}(0)=\langle c_1,c_2,c_3 \rangle = \langle 65,85,85 \rangle\)
\(\boxed{ \vec{v}(t)=\langle 12t+65, 85, 32t+85 \rangle ~ ft/sec }\)
b. \( \vec{r}(t) = \int{\vec{v}~dt} = \)
\( \langle 6t^2+65t+k_1, 85t+k_2, 16t^2+85t+k_3 \rangle \)
\(\vec{r}(0) = \langle k_1, k_2, k_3 \rangle = \langle 0,0,3 \rangle\)
\(\boxed{\vec{r}(t)=\langle 6t^2+65t, 85t, 16t^2+85t+3 \rangle ~ ft }\)
c. To calculate the time of flight, set the zcomponent of the position vector to zero an solve for t. You will get two values since the equation is a quadratic. The small value is negative, so it has no significance. The large value will be the time when the ball hits the ground, which is the time of flight.
To find the range, plug the value of t found for the time of flight into the position and find the magnitude of that vector to get the range.
d. To calculate the maximum height, we take the zcomponent of the velocity vector and set it equal to zero. This is equivalent to taking the derivative of the position of vector and set it equal to zero that you are familiar with from calculus 1 to find the critical value. Solving for t gives us the time when the ball is at it's maximum value. We take that value of t and plug it into the zcomponent of the position vector to get the maximum height.