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