Mathematical Methods for Social Scientists Math 195 (Sec 45), Autumn 2005 Revision Sheet for Mid-term 1

This mid-term will cover those sections in the text book which we have studied in class. From Chapter 11 we studied parametric equations, sketching a parametric curve and calculus with parametric curves. We studied all the sections in chapter 13, and vector functions, and their derivatives and integrals in chapter 14. The following questions are of the style you can expect in the exam.

1. A cycloid is the curve γ in the xy-plane swept out by a point on the edge of a wheel as the wheel rolls along the x-axis.

(a) Draw a diagram of the wheel after it has turned an angle θ to help explain how the cycloid is obtained. Label the point on the edge of the wheel P, the centre of the wheel C and the point where P was when the wheel began to turn O.

(b) Use the diagram to derive a parametric equation for the cyloid γ, parametrised by the angle θ.

(c) Calculate dy/dx for this curve using your parametrisation. Where is dy/dx not defined?

2. (a) Write down the expression that gives the arc length of a curve γ in R^{2} given by the parametric equations

x = f(t) and y = g(t),

for t ∈ [t_{1}, t_{2}].

(b) Suppose f(t) = t/(1 + t), g(t) = ln(1 + t), t_{1} = 0 and t_{2} = 2. Calculate the arc length of the curve.

3. (a) Write down the equation for a sphere of radius r and centre r_{0} = (x, y, z).

(b) Consider the points P (x, y, z) such that the distance from P to A(−1, 5, 3) is twice the distance from P to B(6, 2, −2). Show that the set of all such points is a sphere, and find its centre and radius.

(c) Find an equation for the set of points equidistant from the points A(−1, 5, 3) and B(6, 2, −2). Describe the set.

4. (a) Define the length |a| of a vector a ∈ V_{3} and the dot product a · b of two vectors a, b ∈ V_{3}

(b) Prove that a · a = |a|^{2}

(c) Write down an expression relating the dot product of a and b and the angle θ between the two vectors. When are two vectors orthogonal?

(d) Find a c ∈ R such that < 3, 4, 8 > and < −3, 1, c > are orthogonal.

5. (a) Define the cross product a × b of two vectors a, b ∈ V_{3}.

(b) Describe a geometric meaning of |a × b| for two vectors a and b.

(c) The equation 2x + 3y + 4z = 12 is the equation of a plane. Find a normal vector to the plane.

(d) Find an equation for the plane which passes through the point (3, 6, −1) with

normal vector < 3, 2, −1 >.

6. (a) For a vector-valued function r: R → V_{3} with component function r =< f_{1}, f_{2}, f_{3} >

define what it means to say

lim_{t→t_{0}} r(t) = a.

Define the derivative r′(t), for t ∈ R.

(b) Give a formula for

\frac{d}{dt}(f(t)r(t)).

where r is a vector-valued function and f is a real-valued function.

(c) If r(t) =< cos t, sin t, t > and s(t) =< t, t^{2}, 1 >, compute

\frac{d}{dt}(r(t) · s(t)).

(d) For r as in the previous part of the question compute

∫0^{\frac{π}{2}} r(t) dt.