In applications it is sometimes useful to use an approximation for factorials. The factorial of a positive integer n is defined to be the number n! = 1 · 2 · 3.. (n – 1) · n, e.g., 4! = 1 · 2 · 3·4 = 24. Noticing that F(x) = x ln x – x is an antiderivative of f(x) =In x, use the definition of the definite integral (Riemann sums) to show that In(n!) = n In n – n. is a good approximation of ln(n!) for large values of n.

Q: Which of the following is correct for F (۲) ce (1) 0 s+) cety (1) - (۱)

A: We have to find the correct choice.

Q: For what values of a is Jaʼx – 2a, x2 2 f(x) ( 12, x< 2 continuous at every x?

A: Given : The function is f(x) =a2x - 2a        , x ≥212,                     x&lt;2 .

Q: 2. Suppose that f(x) = ((.x² + x·5).5 +.r.5).5 when x ≥ 0. Find df(x)/dx. •5 (@ï¥2³ )² +x²)+…

A: The answer is given in step 2

Q: The number of units in Z31[x]

A: See the solution below.

Q: 8. Using the Mean Value Theorem and Rolle's Theorem, show that3 + x – 1 =0 has exactly one real…

A:

Q: (c) Show that f(x)>0 for all values of x, x # -2.

A:

Q: Prove that x² +¸µ111² +21112 where prime denotes differentiation w.r.t.s. = 1 o² + 1+p¹² P

A:

Q: Which of the following does not satisfy the conditions of the Mean Value Theorem? Select one: O a.…

A: We recall the theorem and then proceed to sovle the problem

Q: Let x2-2x-3 f(x) = キ3 if For what Values of x is f Contin uo4s? Find the absolate min values of the…

A: Given function is

Q: Prove that coth-la = In x +1 for |a| > 1 x - 1

A:

Q: f(b)-f(a) 4. Use the Mean Value Theorem to find c e [a, b] such that f'(c) = [a, b] = [-2, 2]. given…

A:

Q: (d) Prore that the function f(x) =X is Continous but not unifomly Continous on the intenal

A: Use the  absolun delta definition for uniformly continuous

Q: If f(x) = x² - ²x² 5 4 5 4 has a relative maximum? 3 -x³+60, then what is the set of x values for…

A:

Q: Are the following functions monotone in x? Explain your answer. (a) ƒ(x) = x²

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: 6. Prove that f(x)=√x is continuous for all x ≥ 0.

A:

Q: Show that x + is increasing on [1, 0).

A:

Q: Choose a prime number, n, between 10 and 99, and using n, prove why the function ƒ(e (2πix)/n )=…

A: G is the group of nth roots of unity under complex multiplication.Zn is the group of integers mod n…

Q: If we apply Rolle's Theorem to the function f(x) = 2x² - 12x - 4 on the interval [1, 5], how many…

A:

Q: Evaluate Hint: Use differentiation to show that n=1 (1 – x)-2 = nx"-! (for |x| < 1)

A: We have to show that 1-x-2=∑n=1∞nxn-1

Q: Wich of te following is not a factor of f(x)=x^3+2x+3 in Z5[x] a. x+1 b. x+2 c. x+3

A: Which of the following is not a factor of f(x)=x^3+2x+3 in Z5[x]

Q: Show that the following function is decreasing on [1, +∞)

A: Let, f(n)=2-nn Now, to find f'(n) using the quotient rule for derivative…