Download WEEK #5: Trig Functions, Optimization Goals: • Trigonometric

WEEK #5:
Trig Functions, Optimization
Goals:
• Trigonometric functions and their derivatives
• Optimization
Textbook reading for Week #5: Read Sections 1.8, 2.10, 3.3
2
Trigonometric Functions
From Section 1.8
There are two fundamental interpretations of the sine and cosine functions, from
which all the other trigonometric functions are defined:
• the point on the unit circle versus an angle
• the traditional oscillating graphs
θ
Week 5 – Trig Functions, Optimization
3
How does the circle definition lead to the trigonometric identity sin2(θ) +
cos2(θ) = 1?
The trigonometric functions can also be defined on triangles (recall the mneumonic
device, “SOHCAHTOA”).
Use the 45/45 and 60/30 triangles to compute the sine and cosine of these
common angles.
4
Show how the circle and triangle definitions define the same values in the first
quadrant of the unit circle.
It is useful to understand both definitions of trig functions (circle and triangle) as
sometimes one is more helpful than the other for a particular task.
Week 5 – Trig Functions, Optimization
5
Sine and Cosine as Oscillating Functions
Despite the geometric source of the trigonometric functions, they are used more
commonly in biology and many other sciences as functions because they show periodicity and oscillations. For many cyclic behaviours in nature, using trigonometric functions is a natural first choice for modeling.
6
Question The graph of y = 10 + 4 cos(x) is shown in which of the following
diagrams?
14
14
12
12
10
10
8
6
8
4
6
2
4
−2
2
−4
−6
A
B
12
12
10
10
8
8
6
6
4
4
2
2
C
D
Week 5 – Trig Functions, Optimization
Show the amplitude and the average on the correct graph.
7
8
Period and Phase
How can you find the period of the function cos(Ax)?
How can you reliably determine where the function cos(Ax + B) ‘starts’ on
the graph? (For a cosine graph, where the ‘start’ represents a maximum, the
starting time or x value is sometimes called the “phase” of the function.)
Week 5 – Trig Functions, Optimization
Consider the graph of the function y = 5 + 8 cos(π(x − 1)).
following properties of the function:
• amplitude
• period
• average
• phase
9
What are the
10
Sketch the graph on the axes below. Include at least one full period of the
function.
Week 5 – Trig Functions, Optimization
11
More complicated amplitudes
In the form y = A + B cos(Cx + D), the B factor sets the amplitude. In many
interesting cases, however, that amplitude need not be constant.
Sketch the graph of |y| = 5, and the graph of y = 5 cos(x) on the axes below.
12
Sketch the graph of |y| = x, and the graph of y = x cos(x) on the axes below.
Use only x ≥ 0
Week 5 – Trig Functions, Optimization
Use your intuition to sketch the graph of y = ex cos(x) on the axes below.
13
14
Derivatives of Trigonometric Functions
From Section 2.10
Having covered the graphs and properties of trigonometric functions, we can now
review the derivative formulae for those same functions.
The derivation of the formulas for the derivatives of sin and cos are an interesting
study in both limits and trigonometric identities. For those who are interested,
many such derivations can be found on the web1. However, it is in some ways more
useful to derive the formula in a graphical manner.
1
For example, http://www.math.com/tables/derivatives/more/trig.htm#sin
Week 5 – Trig Functions, Optimization
15
Below is a graph of sin(x). Use the graph to sketch the graph of its derivative.
1
−3 π /2
−π
−π/2
0
π/2
π
3 π/2
π/2
π
3 π/2
−1
1
−3 π /2
−π
−π/2
0
−1
16
From this sketch, we have evidence (though not a proof ) that
Theorem
d
sin x =
dx
Week 5 – Trig Functions, Optimization
17
Most students will also be familiar with the other derivative rules for trig functions:
d
cos(x) = − sin(x)
dx
d
tan(x) = sec2(x)
dx
d
sec(x) = sec(x) tan(x)
dx
d
csc(x) = − csc(x) cot(x)
dx
d
cot(x) = − csc2(x)
dx
18
1
and the
Prove the secant derivative rule, using the definition sec(x) =
cos(x)
other derivative rules.
Week 5 – Trig Functions, Optimization
Question: Find the derivative of 4 + 6 cos(πx2 + 1)
A. 4 − 6 sin(πx2 + 1) · (2πx)
B. −6 cos(πx2 + 1) · (2πx)
C. −6 sin(πx2 + 1) · (2πx)
D. −6 sin(πx2 + 1) · (πx2 + 1)
E. 6 sin(2πx)
19
20
Inverse Trig Functions
In addition to the 6 trig functions just seen, there are 6 inverse functions as well,
though the inverses of sine, cosine, and tangent are the most commonly used.
Sketch the graph of sin(x) on the axes below
On the same axes, sektch the graph of arcsin(x), or sin−1 x, or the inverse of
sin(x).
Week 5 – Trig Functions, Optimization
What is the domain of arcsin(x)?
What is the range of arcsin(x)?
21
22
In the next few questions you will obtain the formula for the derivative of arcsin x.
Simplify sin(arcsin x)
Differentiate both sides of this equation, using the chain rule on the left. You
d
arcsin x.
should end up with an equation involving
dx
Week 5 – Trig Functions, Optimization
23
d
Solve for
arcsin x, and simplify the resulting expression by means of the
dx
formula
p
cos θ = 1 − sin2 θ,
π π
which is valid if θ ∈ [− , ].
2 2
d
arcsin x =
dx
24
Application of Trig Functions: Simple Harmonic Motion
Take two derivatives of the function p(t) = cos(t). What do you notice?
From this observation, we note that if p(t) = cos(t), we can say that
d2
p = −p
dt2
This is our first differential equation. Now we’ll try to see how this type of
equation can arise out of an application.
Week 5 – Trig Functions, Optimization
25
Consider the spring/mass system shown below.
Hooke’s Law: The force exerted by a spring is proportional to the amount of
spring stretch or compression.
Using p(t) as the position of the mass away from equilibrium, what is the
magnitude of the force exerted by the spring, if the spring proportionality
constant is k?
26
Use
P
F = ma to write a differential equation involving p(t)
From your intuition about such a system, what kind of function do you expect
p(t) to be?
Week 5 – Trig Functions, Optimization
27
Set the values of k = m = 1 in the F = ma equation. What do you notice?
From this, we infer that p(t) = cos(t) is what we’ll call a solution to the
differential equation,
d2
p = −p
dt2
Are there other possible solutions? Think about the spring system.
28
Show that this new solution satisfies the differential equation by plugging it
in, and checking that LHS = RHS.
Using your intuition, how would using different masses, or different springs,
affect the solution?
Week 5 – Trig Functions, Optimization
29
Optimization
From Section 3.3
In almost every discipline, the points where a function reaches its maximum or
minimum values are of interest. We will start our study by reviewing the concept
of critical points, how they appear in a graph, and how they help to locate maxima
and minima of a function.
If f (x) is defined on the interval (a, b), then we call a point c in the interval a
critical point if:
• f ′(c) = 0, or
• f ′(c) does not exist.
We will also refer to the point (c, f (c)) on the graph of f (x) as a critical point.
We call the function value f (c) at a critical point c a critical value.
Note that f (c) must be defined for c to be a critical point.
30
Example: Graph the function f (x) = |x|, and say whether x = 0 is a critical
point of f (x).
Week 5 – Trig Functions, Optimization
31
1
Example: Consider the function f (x) = : is x = 0 a critical point?
x
Sketch the graph of f (x) =
1
x
for reference.
32
Example: Identify the critical points of the graph shown below.
Week 5 – Trig Functions, Optimization
33
Local Extrema
Defining Local Minima and Maxima
After identifying the critical points, we realize that not all of them are necessarily
local minima or maxima. We say that f has a local maximum at p if f (p) is a
maximum as compared to nearby points on either side of p.
34
Example: Identify the local maxima and minima of the graph shown below.
Assume that the graph continues on in the same fashion to the right and left
of the part shown.
Week 5 – Trig Functions, Optimization
35
While graphing is one way to identify the type of critical point, it is useful to have
algebraic tests as well.
36
First Derivative Test
One way to decide whether at a critical point there is a local maximum or minimum
is to examine the sign of the derivative on opposite sides of the critical point. This
method is called the first derivative test. Complete this table:
sign of f ′ to left sign of f ′ to right
of c
of c
local
minimum
at
c
local maximum
at
c
Week 5 – Trig Functions, Optimization
37
Example: Use the first derivative test to identify the nature of the critical
points of
f (x) = x3(1 − x2)
38
The Second Derivative Test is another way to test the type of critical point,
and it is described in the textbook (page 268). You may use either approach in
this class.
Week 5 – Trig Functions, Optimization
39
Global Extrema
The First and Second Derivative Tests only indicate whether a point is higher or
lower than other points nearby. To determine if the point is higher than any
other point can be trickier.
40
Determining the Global Maximum and Minimum
To determine the global maximum and minimum (as opposed to local max and
min) for a continuous function on an interval, you will need to evaluate the function
at
• all critical points (if any), and
• the end points of the interval, if the interval is closed (includes its end points)
If the function is discontinuous, you also need to check for asymptotes.
• The global maximum will be the critical point or end point with the largest
y value, and
• the global minimum will be be the critical point or end point with the smallest y value.
Week 5 – Trig Functions, Optimization
41
For the purposes of our class, we will not refer to the end points of an interval
as critical points (or local maxima or minima), though they can still be global
maxima or minima.
42
Example: On the graph below, assuming that the graph is only defined on
the domain shown, identify the
• local extrema
• global extrema
Week 5 – Trig Functions, Optimization
43
Example: Now repeat the analysis, but under the assumption that the graph
continues on as shown on the left, and continues to have shrinking oscillations
on the right.