Calculus: Analytic Geometry and Calculus, with Vectors

118 Functions, limits, derivatives

(o)Ax + Ox)-AX) g(x +

Ox)

• g(x) = 2x + Ax
  6x Ax

(p)

4(x + Ax) - ¢(x) -2x- Ax

AX [1 + x^2 ][1 + (x + AX)^2 ]

2 The signum function having values sgn x (read signum x, almost like sine x)

1

is defined by the formula

sgn x = 1 (x > 0)

sgn x =^0 (x = 0)

sgn x = -1 (x < 0).

x Show that Figure 3.191 displays the graph of sgn x and

then draw the graph of sgn (x - 2). Show that IxI = x

• -1 sgn x. Hint: Consider separately the cases in which

Figure 3.191 x>0,x = 0, andx <0.

3 The Heaviside (1850-1925) unit function having

ly values defined by

1

Figure 3.192

H(x) = 1 (x > 0)

H(x) ffl (x = 0)

H(x) = 0 (x < 0)

is named for the mighty electrical engineer who popu-

x larized its use. Show that Figure 3.192 displays the

graph of H(x) and then draw the graph of H(x - 2).

Show that

H(x) =1 -+ -2gn x, sgn x = 2H(x) - 1.

i

4 We need more evidence that not all functions

have simple graphs that are easily drawn. Let D be

the dizzy dancer function, defined over the interval

x 0 <= x 5 1, for which

Figure 3.193 D(x) = 0 (x irrational)

D(x) = 1 (x rational).

Think about this matter and acquire the ability to make a figure more or less

like Figure 3.193 to "represent" the graph of D.

5 A function g is defined by the formulas

g(x) = x2 (0 <_ x 5 1)

g(x) = x (otherwise).

Plot its graph.

6 A function f is said to be an even function if f(-x) = f(x) whenever x

belongs to the domain off and is said to be an odd function if f(-x) _ -f(x)

whenever x belongs to the domain of f. Prove that the polynomial having values

2 - 3x2 + 5x' (with only even exponents appearing) is even. Prove that the

polynomial having values x - 7x5 + 2x7 (with only odd exponents appearing)

is odd. Prove that the polynomial having values I - 2x + 3x2 is neither even

nor odd.