Homework # 8 - Math 250
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
1. Find the unit tangent vector to the curve \({\bf r}(t) = \langle 3te^t, \frac{4}{\ln 2} 2^t \rangle\) at \(t = 0\)
a. \({\bf T}(0) = \langle 0, 1 \rangle \)
b. \({\bf T}(0) = \langle 3, 4\rangle \)
c. \({\bf T}(0) = \langle 0, 1\rangle \)
d. \({\bf T}(0) = \langle \frac{3}{5}, \frac{4}{5} \rangle \)
e. \({\bf T}(0) = \langle \frac{1}{2}, \frac{\sqrt{3}}{2}\rangle \))
2. Find the principal unit normal vector to the curve \({\bf r}(t) = \cos(t^2), \sin(t^2) \rangle\) where \(t > 0\)?
a. \({\bf N}(t) = \langle -\cos(t^2), -\sin(t^2)\rangle \)
b. \({\bf N}(t) = \langle t^2\cos(t^2), t^2\sin(t^2)\rangle \)
c. \({\bf N}(t) = \langle -t^2\sin(t^2), t^2\cos(t^2)\rangle \)
d. \({\bf N}(t) = \langle \sin(t^2), \cos(t^2)\rangle \)
e. \({\bf N}(t) = \langle -t^2\sin(t^2), -t^2\cos(t^2)\rangle \)
3. Evaluate the indefinite integral. \(\int \langle t e^t, e^{-t} \rangle dt\)
a. \(\langle \frac{1}{2}t^2 e^t + c_1, -e^{-t} + c_2 \rangle \)
b. \(\langle \frac{1}{2}t^2 e^t + c_1, e^{-t} + c_2 \rangle \)
c. \(\langle te^t - e^t + c_1, -e^{t} + c_2 \rangle \)
d. \(\langle te^t + e^t + c_1, e^{-t} + c_2 \rangle \)
e. \(\langle te^t - e^t + c_1, -e^{-t} + c_2 \rangle \)
4. What does it mean if an object has a constant acceleration of \(\langle 0, 0\rangle \)?
a. The object is stopped.
b. The object is moving in a circle at constant speed.
c. The object is moving in a straight line at constant speed or is stopped.
d. The object is moving in a circle at constant speed or is stopped.
e. The object is moving in a circle at constant speed or a straight line at constant speed.
For questions 5 and 6, consider the function \( f(x, y) = \frac{x}{y} + e^{xy^2}\)
5. \(f_x(x,y) = \)
a. \(\frac{1}{y} + y^2 e^{xy^2}\)
b. \(\frac{x}{y^2} + 2xy e^{xy^2}\)
c. \(\frac{1}{y} + y^2 e^{y^2}\)
d. \(\frac{x}{y^2} + 2xy e^{2xy}\)
e. \(\frac{1}{y} + 2y e^{2xy}\)
6. \(f_y(x,y) = \)
For questions 7 through 11, consider the limit $$ \lim_{(x,y)\rightarrow(0,0)}\frac{x^2y}{x^4+y^2}$$
7. Evaluate the limit along the x-axis
a. \(0\)
b. \(\frac{1}{4}\)
c. \(\frac{1}{2}\)
d. \(1\)
e. DNE
8. Evaluate the limit along the y-axis
9. Evaluate the limit along the line \(y = x\)
10. Evaluate the limit along the parabola \(y = x^2\)
11. What is the value of the original limit?
12. Describe the level set for the function \(z = f(x, y) = xy\) at height \(z = 0\).
a. A hyperbola opening in quadrants 1 and 3.
b. A hyperbola opening in quadrants 2 and 4.
c. The x and the y axes.
d. A circle of radius 1 centered at the origin.
e. A parabola opening on the positive y-axis.
13. Describe the level set for the function \(z = f(x, y) = xy\) at height \(z = 1\).