Calculus: Analytic Geometry and Calculus, with Vectors

3.1 Functional notation 119

7 If h(x) = x + 1/x when x 76 0, show that h(11t) = h(t) when t 0 0 and

that [h(x)]2 = h(x2) + 2. Work out a formula for h(h(x)) and check the formula

by setting x = 2.

8 If f(x) = x2 + 3x + 1, show that f(-3) = 1, f(-1) = -1, f(O) = 1,

f() _, f(2) = 11, and

f(x+Ax) =x2+3x+1+(2x+3)Ax+Ax2

when Ax2 means (Ax)2. It is quite appropriate to use this formula as a basis

for a feeling that, when x has a particular fixed value such as 0 or -2 or a, the

value of f (x + Ax) is nearly the same as the value of f(x) whenever Ax is nearly 0.

9 If f(x) = mx + b, show that

f(x2) - f(xi)= m

whenever x2 0 xi. Sketch a figure and comment upon the result.

10 If f(x) = x2, show that

f(x + h) - f(x)= 2x + h

h

when, as always when we make calculations of this kind, h 0 0. Sketch a

graph of the function and use the above formula to find the slope of the line L

passing through the two points on the graph for which x = 1 and x = 1.001.

The answer is 2.001, and it is quite appropriate to have a feeling that this is

nearly the slope of the line tangent to the graph at the point (1,1).

11 If f(x) = x2, simplify

f(x+Ax) +f(x -Ax) - 2f(x).

Ax2

12 If f(x) = 1/x, and if x and x + Ax are both different from 0, show that

f(x+Ax)-f(x)_ -1

Ax X (X -+ Ax)

13 Make appropriate use of the trigonometric formulas

sin (a+0) = sin a cosf+cosa sinO

cos (a + e) = cos a cos # - sin a sin 0

to obtain the formulas

sin (x+h)-sinx_sinhcosx-I-Coshsinx

cos (x + h) - cos x sink I - cos h

h = - h sin x - h cos x.

14 Show that y will be a function of x for which

x2 + xy(x) + [y(x)I2 = 3