Section Practice
¶These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.
Vectors
Let \(\oa{x} = (-1, 3)\) and \(\oa y = (4,1)\) and graph \(\oa x, \oa y, \oa {xy},\) and \(\oa{x-y}.\)
Find the norm of each vector in the previous problem.
Graph \(\oa{(1,-2,1)}\) and \(\oa{(-2,4,3)}\) and find the angle between them.
Lines
Sketch the vectors \(x=(1,3,5)\) and \(y = (2,4,-3)\) and the line through these points.
Find a parametric function, \(\oa{l},\) for the line in the previous problem so that \(\oa{l}(0)= x\) and \(\oa{l}(1) = y.\)
Plot \(\oa l(t) = (2, -1, 4) + (1,0,0)t.\) Assuming this represents the position of an object, compute the speed of the object.
Find an equation for the position of the object that has: the same speed as the object in the previous problem, the same position at time 0, and is travelling in the opposite direction.
Functions
Sketch a graph of \(f(x,y) = 2x^2 + 3y^2.\)
Sketch the intersection of \(f\) from the previous problem with the plane, \(z=4.\)
Sketch the function \(g(x,y) = x^3 + y^2.\) Be sure and label a few points.
Compute the composition \(g \circ \oa{l}\) where \(g\) is from the previous problem and \(\oa l(t) = (2, -1) + (1,0)t.\)
Compute \((g \circ l)'(3)\) and indicate its meaning.
Sketch \(l(t) = (0,1) + (1,0)t\) for \(t \ge 0.\)
Sketch \(c(t) = \big( 3\cos(t) , 3\sin(t) \big)\) for \(0 \le t \le 2\pi.\)
Sketch \(e(t) = \big( 4\cos(t) , 2\sin(t) \big)\) for \(0 \le t \le 2\pi.\)
Graph \(\oa r(t) = ( \cos(t), 2t, \sin(t) )\) for \(t \in [0, 6\pi];\) if \(\oa r(t)\) is the position of an object at time \(t\) then show that the speed of the object is constant.