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Problem: 15 Consider the function f(x)-x" on 0sxs 1, where α>0. Suppose we want to approximate...
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where...
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the...
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a)...
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a) For independent observations In show that the log-likelihood is given by, (b) Hence derive an expression for the maximum likelihood estimate for α. (c) Suppose we observe data such that n 6 and Σ61 log(xi) 12. Show that the associated maximum likelihood estimate for α is given by α = 1.5.
Suppose that f is twice differentiable function where f(0)=f(1)=0. Prove that strategy Suppose that f is a twice differ...
Suppose that f is twice differentiable function where
f(0)=f(1)=0.
Prove that
strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b.
strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
2. (a) We want to find the root x of the function f(x); that is, we need f(r) = 0 . This can be done using Newton...
2. (a) We want to find the root x of the function f(x); that is, we need f(r) = 0 . This can be done using Newton's method, making use of the iterative formula f(xn) Show that the sequence ofiterates (%) converges quadratically if f'(x) 0 in some appropriate interval of x-values near the root χ 9 point b) We can get Newton's method to find the k-th root of some number a by making it solve the non-linear cquation...
3. Suppose we want to use the ri-term trapezoid rule to approximate Sinde (a) (3 points)...
3. Suppose we want to use the ri-term trapezoid rule to approximate Sinde (a) (3 points) Make a graph of y= between = 2 and 3 = 4. Draw on your graph the trapezoids used to apply the Trapezoidal Rule with n = 3. (So, your graph should have 3 trapezoids.) (b) (2 points) Does the Trapezoidal Rule overestimate or underestimate the value of justify your answer. 1 dx? No need to (c) (5 points) For the Trapezoidal Rule, the...
b) Suppose we wish to minimize the function, f(X)-0.5XTCX +bTX+ 1, where b and C 1...
b) Suppose we wish to minimize the function, f(X)-0.5XTCX +bTX+ 1, where b and C 1 ,using Steepest Descent optimization method starting from X Please carry out the first iteration by hand and check for convergence. If the above search direction is called S1 and the one to be used for the second iteration is called S2, what is the relationship between Si and S2, or more specifically, what is ST S2?
b) Suppose we wish to minimize the function,...
00 1 for convergence. It turns out that a good 7. Suppose that we want to...
00 1 for convergence. It turns out that a good 7. Suppose that we want to test n(In n) test to use is the Integral Test. So, let f(x) n=2 1 Therefore, we want to consider X(In x)2" Z 1 1 dx = lim t00 dx. The next best step is which of the following? x(In x)2 x(In x) (a) Use l'Hospital's Rule. (b) Find the derivative of x(In x)2. (c) Let u = ln x. (d) Let u =...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider th...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider the improper integrals separately. (The choice of π above is arbitrary.) By considering f (x) lim an show that 11 converges iff p< 1. Next, by considering lim J(z) an -p- dx show that /2 converges iff p +q>1. Finally, combine these results to show that I converges iff p < 1 and p+q1.
(d) The function f(x)1 is locally integrable on (0, oo)....
We define a function by: and we suppose that f (x + 2) = f (x)...
We define a function by:
and we suppose that f (x + 2) = f (x) for all x ∈ R.
(a) Draw the graph of the function f (x) over the interval [−3,
3].
(b) Find the Fourier series for the function f (x).
f(x) = { x +1 si -1 < x < 0; si 0 < x <1, 1
Suppose you have follow ing utility function : UX1,X2)-Xx2 where 0<0<1; 0<B< 1; 0<B+0<1 The price...
Suppose you have follow ing utility function : UX1,X2)-Xx2 where 0<0<1; 0<B< 1; 0<B+0<1 The price of commodity X, is P, >0; the price of good X, is P, >0 a) Set up the expenditure minimizati on problem. b) Obtain compensated demand curves c) Calculate cross price elasticiti es of demand curves d) Calculate marginal cost of utility e) Obtain the optimal expenditure function Is the expenditure function increasing with respect to prices and utility?
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the...
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a) For independent observations In show that the log-likelihood is given by, (b) Hence derive an expression for the maximum likelihood estimate for α. (c) Suppose we observe data such that n 6 and Σ61 log(xi) 12. Show that the associated maximum likelihood estimate for α is given by α = 1.5.
Suppose that f is twice differentiable function where
f(0)=f(1)=0.
Prove that
strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b.
strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
2. (a) We want to find the root x of the function f(x); that is, we need f(r) = 0 . This can be done using Newton's method, making use of the iterative formula f(xn) Show that the sequence ofiterates (%) converges quadratically if f'(x) 0 in some appropriate interval of x-values near the root χ 9 point b) We can get Newton's method to find the k-th root of some number a by making it solve the non-linear cquation...
3. Suppose we want to use the ri-term trapezoid rule to approximate Sinde (a) (3 points) Make a graph of y= between = 2 and 3 = 4. Draw on your graph the trapezoids used to apply the Trapezoidal Rule with n = 3. (So, your graph should have 3 trapezoids.) (b) (2 points) Does the Trapezoidal Rule overestimate or underestimate the value of justify your answer. 1 dx? No need to (c) (5 points) For the Trapezoidal Rule, the...
b) Suppose we wish to minimize the function, f(X)-0.5XTCX +bTX+ 1, where b and C 1 ,using Steepest Descent optimization method starting from X Please carry out the first iteration by hand and check for convergence. If the above search direction is called S1 and the one to be used for the second iteration is called S2, what is the relationship between Si and S2, or more specifically, what is ST S2?
b) Suppose we wish to minimize the function,...
00 1 for convergence. It turns out that a good 7. Suppose that we want to test n(In n) test to use is the Integral Test. So, let f(x) n=2 1 Therefore, we want to consider X(In x)2" Z 1 1 dx = lim t00 dx. The next best step is which of the following? x(In x)2 x(In x) (a) Use l'Hospital's Rule. (b) Find the derivative of x(In x)2. (c) Let u = ln x. (d) Let u =...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider the improper integrals separately. (The choice of π above is arbitrary.) By considering f (x) lim an show that 11 converges iff p< 1. Next, by considering lim J(z) an -p- dx show that /2 converges iff p +q>1. Finally, combine these results to show that I converges iff p < 1 and p+q1.
(d) The function f(x)1 is locally integrable on (0, oo)....
We define a function by:
and we suppose that f (x + 2) = f (x) for all x ∈ R.
(a) Draw the graph of the function f (x) over the interval [−3,
3].
(b) Find the Fourier series for the function f (x).
f(x) = { x +1 si -1 < x < 0; si 0 < x <1, 1
Suppose you have follow ing utility function : UX1,X2)-Xx2 where 0<0<1; 0<B< 1; 0<B+0<1 The price of commodity X, is P, >0; the price of good X, is P, >0 a) Set up the expenditure minimizati on problem. b) Obtain compensated demand curves c) Calculate cross price elasticiti es of demand curves d) Calculate marginal cost of utility e) Obtain the optimal expenditure function Is the expenditure function increasing with respect to prices and utility?
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