Vv156 Honors Calculus II
Assignment 1
This assignment has a total of (31 points).
Exercise 1.1 [Ste10, p. 23] Given f, g : R → R, determine the parity of f + g and f · g based on the parities of f
and g. Fill in the following table.
f g f + g f · g
even even
even odd
odd even
odd odd
(4 points)
Exercise 1.2 [Ste10, p. 44] Given linear functions f, g : R → R with f(x) = m1x + b1 and g(x) = m2x + b2. Is f ◦ g
also a linear function? If so, what is the slope of its graph?
(2 points)
Exercise 1.3 [Ste10, p. 57] Given f : R → R, f(x) = 5x
, show that for h 6= 0,
f(x + h) − f(x)
h
= 5x
5
h − 1
h
(2 points)
Exercise 1.4 Given functions e, τ, τ 0
, τ 00, σ, σ0
: {1, 2, 3} → {1, 2, 3} as follows,
x e(x) τ (x) τ
0
(x) τ
00(x) σ(x) σ
0
(x)
1 1 2 1 3 2 3
2 2 1 3 2 3 1
3 3 3 2 1 1 2
(i) (3 points) Complete the following composition table of functions using elements from the set {e, τ, τ 0
, τ 00, σ, σ0}.
◦ e τ τ
0
τ
00 σ σ
0
e
τ
τ
0
τ
00
σ
0 σ
0 ◦ τ
00
σ
(For example, σ
0 ◦ τ
00 should be replaced with τ .)
(ii) (6 points) Let f
◦n := f ◦ f ◦ · · · ◦ f
| {z }
n times
, n ∈ N. For each f ∈ {e, τ, τ 0
, τ 00, σ, σ0}, find the smallest number n ∈ N
such that f
◦n = e.
(9 points)
1
Exercise 1.5 Given f : R → R, f0(x) = 3 + x/2, and fi
, i = 1, . . . , 4 as follows,
x f1(x) x f2(x) x f3(x) x f4(x)
10.0 8.04 10.0 9.14 10.0 7.46 8.0 6.58
8.0 6.95 8.0 8.14 8.0 6.77 8.0 5.76
13.0 7.58 13.0 8.74 13.0 12.74 8.0 7.71
9.0 8.81 9.0 8.77 9.0 7.11 8.0 8.84
11.0 8.33 11.0 9.26 11.0 7.81 8.0 8.47
14.0 9.96 14.0 8.10 14.0 8.84 8.0 7.04
6.0 7.24 6.0 6.13 6.0 6.08 8.0 5.25
4.0 4.26 4.0 3.10 4.0 5.39 19.0 12.50
12.0 10.84 12.0 9.13 12.0 8.15 8.0 5.56
7.0 4.82 7.0 7.26 7.0 6.42 8.0 7.91
5.0 5.68 5.0 4.74 5.0 5.73 8.0 6.89
(i) (5 points) Sketch the graph of fi
, i = 0, . . . , 4 (by hand or software).
(ii) (4 points) Calculate P11
k=1|f0(xk) − fi(xk)|
2
for each i = 1, . . . , 4, where x1, . . . , x11 are taken from the x-column
of the above table for different fi
’s respectively.
(9 points)
Exercise 1.6 [Ste10, p. 44] The Heaviside function H is defined by
H(t) = (
0, t < 0
1, t ≥ 0
It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch
is instantaneously turned on.
(a) (1 point) Sketch the graph of the Heaviside function.
(b) (1 point) Sketch the graph of the voltage V (t) in a circuit if the switch is turned on at time t = 0 and 120 volts
are applied instantaneously to the circuit. Write a formula for V (t) in terms of H(t).
(2 points)
Exercise 1.7 [Ste10, p. 44] The Heaviside function defined in Exercise 1.6 can also be used to define the ramp
function y = ctH(t), which represents a gradual increase in voltage or current in a circuit.
(a) (1 point) Sketch the graph of the ramp function y = tH(t).
(b) (1 point) Sketch the graph of the voltage V (t) in a circuit if the switch is turned on at time t = 0 and the
voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for V (t) in terms of
H(t) for t ≤ 60.
(c) (1 point) Sketch the graph of the voltage V (t) in a circuit if the switch is turned on at time t = 7 seconds and
the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V (t) in terms of
H(t) for t ≤ 32.
(3 points)
image from internet.
References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).
2
