(a) Suppose that f'(; y) = 0 for all x in a certain n-ball B(a) and...
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
B
is a connected ball of finite radius
2, Let f : U → Rm be Ci and let B be a compact connected subset of U Show that there exists a constant M such that for all a, y e B. (Hint: use the mean value theorem). Find an example which shows that the assumption that B was compact is essential
2, Let f : U → Rm be Ci and let B be a compact connected subset of...
THEOREM. Suppose that F(x, y) = (P(x, y), Q(x, y)) is a vector-valued function of two variables and that the domain of P(x,y) and Q(x,y) is all of R2. Then it is possible to find a function f(x,y) satisfying Vf = F if and only if Py = Q. Instructions: Use this Theorem to test whether or not each of the following vector-valued functions F(x,y) has a function f(x, y) that satisfies VS = F (that is, if there is...
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and g are smooth (Co) functions. Let 7(t) be the solution of the initial value problem with γ(0) = (1,2). Prove or give a counterexample to the statement that the w-limit set of γ can contain more than one critical point.
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and...
all a,b,c,d
1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on C with gradient Vg, we define the normal derivative Dng by and we define the Laplacian, V2g, of g by For this problem, assume D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f...
Suppose hat the joint probability distribution of the continuous random variables X and Y is constant on the rectangle 0 < x < a and 0 < y < b for a, b E R+. Show mathematically that X and Y are independent. Hint: (a) Recall JDx "lly f(r, y) dy dx-1 (b) Recall X, Y are independent if ffy fry
Suppose hat the joint probability distribution of the continuous random variables X and Y is constant on the rectangle...
a) A vector field F is called incompressible if div F = 0. Show
that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is
incompressible.
b) Suppose that S is a closed surface (a boundary of a solid in
three dimensional space) and that F is an incompressible vector
field. Show that the flux of F through S is 0.
c)Show that if f and g are defined on R3 and C is a closed curve
in R3 then...
If we start with o and form F from it, we are definitely creating a co Let's start there. 4. Suppose that Q(x, y?). Let F(x,y) = Vo(x,y). a. Find Vé(x,y). F.Tds if C is the quarter unit circle from (1,0) to (0,1). b. Let F(x,y)=VQ(x,y). Find otomo 19 Il Fundamental Theorem for Line Integrals Let F be a continuous vector field on an open region R in R. There exists a potential function o with F= Vo (which means...