Yet Another Calculus Text, 2007a

20 CHAPTER 1. DERIVATIVES
y
1
0.8
0.6
0.4
0.2
y = t
y =sin(t)
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−0.2
t
−0.4
−0.6
−0.8
−1
Figure 1.5.4: Comparison of y =sin(t) withy = t
Example 1.5.5. For a numerical comparison, note that for t =0.1, cos(t) =
0.9950042, compared to 1 − t2 2
=0.995, and sin(t) =0.0998334, compared to
t =0.1.
Exercise 1.5.3. Verify that the triangle with vertices at A, B, andD in Figure
1.5.2 is an isosceles triangle with base angles of t 2
at A and B.
Exercise 1.5.4.
Verify the half-angle formula,
cos(θ) = 1 (1 + cos(2θ)),
2
for any angle θ, using the identities cos(2θ) =cos 2 (θ) − sin 2 (θ) (a consequence
of the addition formula) and sin 2 (θ)+cos 2 (θ) =1.
1.5.3 Compositions
Given functions f and g, wecallthefunction
f ◦ g(x) =f(g(x)) (1.5.39)
the composition of f with g. Ifg is continuous at a real number c, f is continuous
at g(c), and ɛ is an infinitesimal, then
since g(c + ɛ) ≃ g(c).
f ◦ g(c + ɛ) =f(g(c + ɛ)) ≃ f(g(c)) (1.5.40)