L ectur e 5| 1 Limits and Derivatives involving
Trigonometric Functions
(1) , Dom , Rng
(2) , Dom , Rng
L ectur e 5| 2
L ectur e 5| 3 (5) , Dom , odd integer, Rng
(6) , Dom , , Rng
L ectur e 5| 4
Limits and Continuity
(1) The trigonometric functions and are continuous on . So
and
Others trig, are continuous on their domains.
(2)
L ectur e 5| 5 Proof (1) is true by the graphs.
(2) Recall
for , so
Applying the squeeze theorem, we get
Similarly, one can show the left-hand limit, therefore
L ectur e 5| 6 EX Evaluate the limits
and
L ectur e 5| 7 EX Show that
using the conjugate of .
L ectur e 5| 8
Derivative formulas
Proof Using
we get
L ectur e 5| 9 EX (Differentiation formulas and rules) If , evaluate
L ectur e 5| 10 EX (Chain rule revised) Using the chain rule to show that
and
where is a differentiable function of . Apply these formulas to evaluate
L ectur e 5| 11
Other formulas
where is a differentiable function of .
L ectur e 5| 12 Proof We only show the formula
The extra term is from applying the chain rule.
By the quotient rule, we have
L ectur e 5| 13 EX (Differentiation formulas)
Differentiate
L ectur e 5| 14 EX (Chain rule) Find
if is given as follows.
1.
2.
3.
4.
L ectur e 5| 15 EX (Chain rule twice) Let
Find
.
