Vector calculus: midterm

Grading

These are exercises for Vector calculus: course.

1. True or false?
   1. The intersection of two linear subspaces is a linear subspace.
   2. The empty set is a linear subspace.
   3. The product of two linear functions is a linear function.
   4. The acceleration is always perpendicular to the velocity.
   5. There is only one natural parametrization of a straight line.
2. Give an example of:
   1. a curve with curvature equal to $0$,
   2. a parametric curve that traces a circle $3$ times clockwise,
   3. a linear subspace in ${\bf R}^3$ with no vectors perpendicular to it,
   4. a system of $2$ linear equations with no solutions,
   5. a system of $2$ linear equations with exactly $2$ solutions.
3. (a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\bf R}^3$.
4. Find the projection of $v = (1,1,1)$ onto the plane spanned by $(1,0,1)$ and $(0,1,1)$.
5. Find all vectors perpendicular to the plane spanned by $(-1,1,0)$ and $(0,1,1)$.
6. For the linear map $L(x_1,x_2) = (3x_1 + x_2, -3x_1 - x_2)$ find the basis of the image space.
7. Find the function $f: {\bf R} \rightarrow {\bf R}^2$ such that $f' '(t)=(\cos t, \sin 3t)$ and $f(0)=(1,0), f'(0)=(0,0)$.
8. Prove the formula: $<x,y>=||x||||y|| \cos \theta$ in $R^2$.
9. Find the best affine approximation of the function $f(t)=(e^{t}sin t,t^2 + 1, \cos t)$ at the point $(0,1,1)$.
10. Let $f(t)=(t^3-3t,t^2)$. (a) Find the derivative $f'$ of $f$. (b) Use $f'$ to plot the parametric curve $f$.
11. Use the arc-length formula to compute the length of the circle of radius $3$ centered at $(1,1)$.
12. (a) Find the natural parametrization of the helix $f(t)=(\cos t, \sin t,t)$. (b) Find its curvature. (c) Find the center of the osculating circle at $(1,0,0)$.
13. The curve $(t^3-t,t^2-1)$ has a self-intersection at $0$. Compute the angle.

1. Calculate $X \cdot Y$ for $X=3(1,0,1)$ and $Y=(2,1,-2)$.
2. Give the number $t$ that makes $X=(1,1+t,2+t)$ and $Y=(2,1,1)$ perpendicular.
3. At time $t$ with $0 \leq t$, an object is at the position $$P[t]=( \cos(t+1),t+1,te^{t}).$$ Calculate its velocity, $v[t]$, and its acceleration, $a[t]$, as functions of $t$.
4. Give a parametric formula for the 3D-line through the point $(2,3,4)$ and parallel to the line through the points $(1,1,1)$ and $(-1,-2,-3)$.
5. Find a vector perpendicular to both $X=(0,1,1)$ and $Y=(-1,2,0)$.
6. Here are parametric equations for two lines: $(x,y,z)=(1+2t,3t,2+5t)$ and $(x,y,z)=(-1,-1,-1)+t(1,0,1)$. Say how you can tell that these lines are NOT parallel.
7. A plane has an $xyz$-equation $2(x-1)+3(y-2)+4(z-3)=0$. Give a vector PARALLEL to the plane.
8. In an effort to find the line in which the planes $$x-y+2z=0$$ $$-2(x-2)+2(y+2)-4(z+2)=0$$ intersect, a student multiplied the first one by $(-2)$ and then added the result to the second. He got $0=0$. Explain the result.
9. Calculate the gradient $grad\, f[x,y,z]$ of $f[x,y,z]=xyz+ \frac{x}{y}$.
10. Does $f[x,y]=-x^2e^{x^2+y^2}$ have a maximizer or a minimizer? How do you know?
11. Calculate $\frac{\partial f}{\partial z}$ for $f[x,y,z]=e^{xyz}+x$ at the point $(1,2,3)$.
12. You are at the point $(0,0)$. In the direction of what vector should you step off $(0,0)$ in order to get the greatest initial increase in the function $f[x,y]=x^2+\frac{1}{4}y^2$. Explain.
13. Let $f[x,y]=\sin (x-y)$. Give a formula for the function $h[x]$ defined by $$h[x]=\displaystyle\int_0^{x}f[s,y]ds. $$
14. Measure by hand calculation the volume of the solid lying above the rectangle $R$ in the $xy$-plane consisting of all points $(x,y,0)$ with $-1 \leq x \leq 1,0 \leq y \leq 1$ and bounded from above by the surface given by $z=x^2y$.
15. There is a 2D region $R$ with bottom boundary curve (a line) $low[x]=1$ and with top boundary curve (a line) $high[x]=2-x^4$. Calculate

$$\displaystyle\int\int_{R}2 dxdy.$$