WEEK #5: Trig Functions, Optimization

Transcription

1 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 How does the circle definition lead to the trigonometric identity sin 2 (θ) + cos 2 (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.

3 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.

4 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. Question The graph of y = cos(x) is shown in which of the following diagrams? A B C D Show the amplitude and the average on the correct graph.

5 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.)

6 Consider the graph of the function y = cos(π(x 1)). properties of the function: What are the following amplitude period average phase Sketch the graph on the axes below. Include at least one full period of the function.

7 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. Sketch the graph of y = x, and the graph of y = x cos(x) on the axes below. Use only x 0

8 Use your intuition to sketch the graph of y = e x cos(x) on the axes below.

9 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 web 1. However, it is in some ways more useful to derive the formula in a graphical manner. 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/ π /2 π π/2 0 π/2 π 3 π/2 1 From this sketch, we have evidence (though not a proof) that Theorem d dx sin x = 1 For example,

10 Most students will also be familiar with the other derivative rules for trig functions: d cos(x) = sin(x) dx d dx tan(x) = sec2 (x) d sec(x) = sec(x)tan(x) dx d csc(x) = csc(x)cot(x) dx d dx cot(x) = csc2 (x) Prove the secant derivative rule, using the definition sec(x) = derivative rules. 1 cos(x) and the other Question: Find the derivative of cos(πx 2 + 1) A. 4 6 sin(πx 2 + 1) (2πx) B. 6 cos(πx 2 + 1) (2πx) C. 6 sin(πx 2 + 1) (2πx) D. 6 sin(πx 2 + 1) (πx 2 + 1) E. 6 sin(2πx)

11 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). What is the domain of arcsin(x)? What is the range of arcsin(x)?

12 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 should end up with an equation involving d arcsin x. dx Solve for d dx arcsin x, and simplify the resulting expression by means of the formula which is valid if θ [ π 2, π 2 ]. cosθ = 1 sin 2 θ, d dx arcsin x =

13 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 d 2 dt2p = p This is our first differential equation. Now we ll try to see how this type of equation can arise out of an application. 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? Use 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?

14 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, dt2p = p Are there other possible solutions? Think about the spring system. d 2 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?

15 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. Example: Graph the function f(x) = x, and say whether x = 0 is a critical point of f(x).

16 Example: Consider the function f(x) = 1 : is x = 0 a critical point? x Sketch the graph of f(x) = 1 x for reference. Example: Identify the critical points of the graph shown below.

17 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. 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.

18 While graphing is one way to identify the type of critical point, it is useful to have algebraic tests as well. 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: local minimum at c local maximum at c sign of f to left of c sign of f to right of c Example: Use the first derivative test to identify the nature of the critical points of f(x) = x 3 (1 x 2 ) 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.

19 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. 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. 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.

20 Example: On the graph below, assuming that the graph is only defined on the domain shown, identify the local extrema global extrema

21 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.