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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 |
- (3points) 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.)
- Let f◦n := f ◦ f ◦ ··· ◦ f, 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 |
- Sketch the graph of fi, i = 0,…,4 (by hand or software).
- 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.
- Sketch the graph of the Heaviside function.
- 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.
- Sketch the graph of the ramp function y = tH(t). 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.
- 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 Write a formula for V (t) in terms of H(t) for t ≤ 32.
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