1. Sketch the function
f(x) = |x| + |2 x - 1| .
Hint: Use the definition of the absolute value function for each term, and express the function f on separate intervals.
2. Calculate the following limits, or show that they do not exist. You are not allowed to use l'Hˆopital's rule (even if you already know it).
(a) lim
x→1-
|x2 - 1|
x + 1
(b) lim
x→0
sin 3x
sin 2x
(c) lim
θ-→0
1 - cos 2θ
θ cos θ
1
3. Let f be a function which
(a) is continuous everywhere, except at x = 1,
(b) is differentiable everywhere except at x = 0 or 3,
(c) satisfies
f′(x)
??
?
>0 if x < -2 or 0 < x <1 or x > 3,
= 0 if x = -2 or 1 < x < 2,
<0 if - 2 < x <0 or 2< x < 3,
(d) satisfies f(2) = 1, f(-2) = 2,
(e) has a graph which crosses the x-axis only at x = 5/2 and the y-axis only at y = 1/4. Draw a single possible sketch of the graph of f, which satisfies all of the properties listed above.
4. Two particles are spiralling outwards from the origin. At time t sec, t ≥ 0, the first particle is at the point (x, y) where
x = t cos πt , y = t sin πt ,
while the second particle is at the point
x = t cos πt -
1
π
sin πt , y = t sin πt +
1
π
cos πt -
1
π
.
The paths taken by the two particles during the first 8 seconds are illustrated.
-8 -6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
outward spiral paths taken by two particles
particle 1
particle 2
The speed of a particle is defined to be v(t) =
$%dx
dt
&2
+
%dy
dt
&2
.
Which particle has travelled the greater distance at any instant? (In other words, which path is the longer one?) Explain carefully.
Show that the difference between the two speeds tend to zero as t→∞. (Use a limit.)
5. In a beehive, each cell is a regular hexagonal prism, open at one end, with a trihedral angle, i.e., intersecting three planes, at the other end - see https://en.wikipedia.org/wiki/Honeycomb. It is believed that bees form their cells in such a way as to minimise the surface area for
a given volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle θ is amazingly consistent, with differences rarely more than 2o.
You can assume that the surface area A is given by
A(θ) = 6sh -
3
2
s2 cot θ +
3√3
2
s2cosec θ ,
where s (the length of the sides of the hexagon) and h (the height) are constants.
(a) What angle should the bees prefer? Using the methods of calculus, give your answer as an exact expression involving cos, as well as an approximate answer to the nearest degree. You are not required to show that this is indeed a minimum.
(b) Find the exact minimum area of the cell, Amin, in terms of s and h.
6. Consider the functions
y = fn(x) = √n + 1 cos x sinn x , 0 ≤ x ≤ π/2 ,
for positive integers n. The following illustrates the graphs for n = 1, 2, 3, 5, 8, 12, 20.
0 0.5 1 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y=sqrt(n+1)*cos(x)*(sin(x))n , 0<=x<=pi/2 , various n
2 3 5 8 12 20
n=1
As n increases, the maximum turning point is clearly moving to the right and appears to be levelling out.
(a) Solve f′n (x) = 0 and hence derive that the maximum value is
yn =
% n
n + 1
&n/2
.
(b) For what value of n does the maximum occur at x = π/3?
(c) Use the limit command in Matlab to determine lim
n→∞
yn . You must include a
printed copy of your Matlab commands and output.