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.

fgf+gf·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 ̸= 0, f(x+h)−f(x) = 5x􏰉5h −1􏰊

hh Exercise 1.4 Given functions e,τ,τ′,τ′′,σ,σ′ : {1,2,3} → {1,2,3} as follows,

x e(x) τ(x) τ′(x) τ′′(x) σ(x) σ′(x) 1121323 2213231 3332112

(i) (3points) Complete the following composition table of functions using elements from the set {e,τ,τ′,τ′′,σ,σ′}.

◦ e τ τ′ τ′′ σ σ′ e
τ

τ′ τ′′

σ′ σ′ ◦τ′′ σ

(For example, σ′ ◦ τ ′′ should be replaced with τ .)
(ii) (6points) Let f◦n := f ◦f ◦···◦f, n ∈ N. For each f ∈ {e,τ,τ′,τ′′,σ,σ′}, find the smallest number n ∈ N

(

such that f◦n = e. (9 points)

􏰑 􏰐􏰏 􏰒

n times

1

Exercise 1.5 Given f : R → R, f0(x) = 3 + x/2, and fi, i = 1, . . . , 4 as follows,

(3 points)

k=1
of the above table for different fi’s respectively.

x f1(x) 10.0 8.04 8.0 6.95 13.0 7.58 9.0 8.81 11.0 8.33 14.0 9.96 6.0 7.24 4.0 4.26

12.0 10.84 7.0 4.82 5.0 5.68

x f2(x) 10.0 9.14 8.0 8.14 13.0 8.74 9.0 8.77 11.0 9.26 14.0 8.10 6.0 6.13 4.0 3.10 12.0 9.13 7.0 7.26 5.0 4.74

x f3(x) 10.0 7.46 8.0 6.77

13.0 12.74 9.0 7.11 11.0 7.81 14.0 8.84 6.0 6.08 4.0 5.39

x f4(x) 8.0 6.58 8.0 5.76 8.0 7.71 8.0 8.84 8.0 8.47 8.0 7.04 8.0 5.25

1. (i)  (5 points) Sketch the graph of fi, i = 0, . . . , 4 (by hand or software).
2. (ii)  (4 points) Calculate 􏰃11 |f0(xk) − fi(xk)|2 for each i = 1, . . . , 4, where x1, . . . , x11 are taken from the x-column

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

􏰧0, t<0 H(t)= 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.

1. (a)  (1 point) Sketch the graph of the Heaviside function.
2. (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.

1. (a)  (1 point) Sketch the graph of the ramp function y = tH (t).
2. (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.
3. (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.

19.0 12.50 12.0 8.15 8.0 5.56 7.0 6.42 8.0 7.91 5.0 5.73 8.0 6.89

References

image from internet.

[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2). 2