Vv156 Homework 1 -Solved

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
 
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? 

Exercise 1.3 [Ste10, p. 57] Given f : R → R, f(x) = 5x, show that for h = 06                     ,

Exercise 1.4 Given functions e,τ,τ0,τ00,σ,σ0 : {1,2,3} → {1,2,3} as follows,

x
e(x)
τ(x)
τ0(x)
τ00(x)
σ(x)
σ0(x)
1
1
2
1
3
2
3
2
2
1
3
2
3
1
3
3
3
2
1
1
2
(i) Complete the following composition table of functions using elements from the set {e,τ,τ0,τ00,σ,σ0}.

(For example, σ0 ◦ τ00 should be replaced with τ.)

(ii)  Let f◦n := f ◦ f ◦ ··· ◦ f, n ∈ N. For each f ∈ {e,τ,τ0,τ00,σ,σ0}, find the smallest number n ∈ N

|      {z            } n times

such that f◦n = e.

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)  ketch the graph of fi, i = 0,...,4 (by hand or software).

(ii)   Calculate |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.

Exercise 1.6 [Ste10, p. 44] The Heaviside function H is defined by

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)  Sketch the graph of the Heaviside function.

(b)  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).

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)   Sketch the graph of the ramp function y = tH(t).

(b)   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)    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.
image from internet.