2 Using the inequality tanz く, show, without using your calculator, that z for 0 tan zdx 2 0.12. 0.1 Clearly state the theorem that you are using. 2 Using the inequality tanz く, show, with...
2. Evaluate without using a calculator. Draw a picture. Show and label where your answer lies. What is the reference angle? Write your answer as an exact value. You must show your work. Do not skip steps. V2 2 -1 a. sin I b. tan (43)
Show how you would plug into the calculator to get answer se the calculator calculate the z and p values. What do the p values and z values mean? State clearly what test means in non-technical terms. 5) 35 points se the calculator calculate the z and p values. What do the p values and z values mean? State clearly what test means in non-technical terms. 5) 35 points
Seard 2. Evaluate without using a calculator. Draw a picture. Show and label where your answer lies. What is the reference angle? Write your answer as an exact value. You must show your work. Do not skip steps. v2 2 a. sin b. tan (13) 3. Find the length of the arc s intercepted by a central angle 0 = 330 in a circle of radius r =6 in. Round your answer to the nearest tenth. 1 4. If cos...
Here you are asked to prove the Fundamental Theorem of Algebra a different way by using Rouché's Theorem. Where n E N, consider the polynomial n-1 Pn (z)z" k-0 Using the circular contour C-[z : zR with R appropriately chosen, (a) prove that pn(2) has (counting multiplicity) precisely n zeros in the open disc D(0, R); (b) also show that Pn(z) has no zeros in C \ D(0, R) Here you are asked to prove the Fundamental Theorem of Algebra...
Show that the equation z 3 -1- 2e‘ = 0 has exactly one solution. State clearly every result you use and show that the conditions are satisfied. (Hint: Start by showing that there is at least one solution and then exclude the possibility of more solutions.)
7. (a) State Chebyshev's inequality and prove it using Markov's inequality. 151 (b) Let (2, P) be a probability space representing a random experiment that can be repeated many times under the same conditions, and let A S2 be a random event. Suppose the experiment is repeated n times. (i) Write down an expression for the relative frequency of event A 131 ) Show that the relative frequence of A converges in probability to P(A) as the number of repetitions...
. Show all your steps to evaluate without using a calculator. 8log, 49 log216 Blogg1 The following is a graph of a rational function. Vertical and horizontal asymptotes are graphed as dotted lines. <- and y- intercepts are shown. Find the rational function. X-1 1 101 (0,8) S y-2 -5 (-2,0) (2,0) 5 X *---0, f(x) Using the graph, as *+-1,5(x)→
Q 9. In this question, you must clearly set out your working, describing your process with full English sentences. (a) Using Euclid's Algorithm 14.13, show that ged(25, 33) = 1, and hence find integers x, y E Z such that x · 25+ y: 33 = 1. Clearly state the values of x and y. (b) By the Multiplicative Inverse Theorem 14.22, it follows from (a) that 25 has a multiplicative inverse, 25(-1), in Z33. Calculate 25(-1) using your answer...
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using the Lyapunov function V(x, y, z) = ρ「2 + ơy2 + ơz?, show that the origin is globally asymptotically stable. (Hint. You may need to use the Invariance Principle as well.) στ 3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using...