(Hint: think of tan as a fraction!) lim (1 - .) tan 7. Suppose a function...
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
Suppose /(x) = Va.x+b, where a > 0 and a, b are constants in R. What is the inverse function / {x)? Find the parameters m, n in terms of a, b. The inverse function is given by x) = m(x + n)- where, Suppose f(x) = V3.+6. What is the domain off-tx)? Select one: a. (-0,6] b. Other c. [0,00) d. R e. [6,00) Consider the function 3-X 3x-4 if x <1 if 1sx52 if x > 2 Find...
If a quantity y satisfies the differential equation dy = kx(10-y), k>0 dx. when X = 2 and y = -7, the graph of yir increasing decreasing constant cannot be determined
Let fi and f2 be functions such that lim e s f1 (2) = + and such that the limit L2 = lim a s f2 (x) exists. Which one of the following is NOT correct? O limas (f1f2)(x) = 0 if L2 = 0. limas (fi + f2)(x) = too if L2 = -0. Olim as (f1f2) (x) = too if 0 <L2 5+co. lim a s (f1f2)(x) = - it L2 = -. Which one of the following...
Suppose that the piecewise function J is defined by f(2)= {**** -1<<3 - 3x2 + 2x + 23, 2> 3 Determine which of the following statements are true. Select the correct answer below: O f() is not continuous at I = 3 because it is not defined at I = 3. Of() is not continuous at 2 = 3 because lim f(x) does not exist. f() is not continuous at I = 3 because lim f() f(3). ->3 f(x) is...
please explain each step, give all the reasoning, don’t just
give the graph, I have already gotten the graph
1. Sketch the graph of the function that satisfies all the given conditions. (a) f"()>0 on (-0, -4) and (4,oo); f"(x) <0 on (-4,0) and (0,4); lim f()2, lim f(r) -2 ェ→00 (b) f(x) c0 on (-o,-3) and (0, 0) ()>0 on-3,0) f"(z) < 0 on (-00 ,-), f"(z) > 0 on (- 0) and (0,00) f,() = 0, f(-2)--21, f(0)...
7. The vector field F =< 3x2z In y + ze+2 +20, - 3y?, x° In y + ce2 +423 > is conservative. Find a potential function f(x, y, z) such that F=Vf. Y
Vx+1-1 Evaluate: lim x>0 х Please solve it in detail and show all your steps./
Suppose f'(x) = -1/3(x+3). On what open interval(s) is f(x) decreasing 0-3 < < 0 0-3 <x<0 0 - < I< -3 0 - < ?<-3 and 0 < x < 0