(a) Use the Constant Difference Theorem (4.8.3) to show that if f ′ x = g ′ x for all x in the interval − ∞ , + ∞ , and if f and g have the same value at some point x 0 , then f x = g x for all x in − ∞ , + ∞ . (b) Use the result in part (a) to confirm the trigonometric identity sin 2 x + cos 2 x = 1 .
(a) Use the Constant Difference Theorem (4.8.3) to show that if f ′ x = g ′ x for all x in the interval − ∞ , + ∞ , and if f and g have the same value at some point x 0 , then f x = g x for all x in − ∞ , + ∞ . (b) Use the result in part (a) to confirm the trigonometric identity sin 2 x + cos 2 x = 1 .
(a) Use the Constant Difference Theorem (4.8.3) to show that if
f
′
x
=
g
′
x
for all
x
in the interval
−
∞
,
+
∞
,
and if
f
and
g
have the same value at some point
x
0
,
then
f
x
=
g
x
for all
x
in
−
∞
,
+
∞
.
(b) Use the result in part (a) to confirm the trigonometric identity
sin
2
x
+
cos
2
x
=
1
.
Equations that give the relation between different trigonometric functions and are true for any value of the variable for the domain. There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.
Find the smallest interval in x over which the functions sin(x/xo) and
cos(Tx/xo) are orthogonal. On the same interval, are the following pairs
of functions orthogonal?
a) sin(x/x₁) and sin(2πx/xo).
b) sin(x/xo) and sin(x/xo).
c) sin(x/xo) and sin(x/2x).
6. Let L be any positive number. Consider the functions sin(x) and cos(x) in
L² ([-L, L]) for which values of L are these functions orthogonal to each other
in this space?
Consider the function f(x) = cos(x2 – 2) + 1 on the interval (-3,3). Calculate the two x-intercepts of this function on the given interval. Express your answers in exact form, as a comma separated list. You may not use the zero/root functionality of a calculator
to do this; you must show all of your work and explain your reasoning, where necessary.
X =
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