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) = ln x, use the definition of the definite integral (Riemann sums) to show that ln(n!) ≈ n ln n − n. is a good approximation of ln(n!) for large values of n.

Q: Which of the functions g(x) are such that 4" is (g(x))? (Select all that apply.) Og(x)=2+3+4 Dg(x) =…

A: Solution 5 : the correct options are : A and E Solution 6 : the correct options are : A

Q: State in your own words the general power rule for differentiating the function h(x)=[f(x)]^n, where…

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Q: 3 Given the function: f(x) = (1+ x)³/2 . x - 1 Prove that it is increasing on (0, +00) analytically,…

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Q: Use logarithmic differentiation to find the derivative of y y = X X+7 OA. OB. O C. (3)√ 2 7 O D. X…

A: topic- differentiation properties of log used:log(ab)=blog(a)logab=log(a)-log(b)derivative formula…

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: Finally, use the Generalized Power Rule to determine[x -x], noting that x-x is the difference of two…

A: Here we have to find differentiation of f(x)

Q: Pls help ASAP.

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Q: The function f(n) = 1.5n+√√n is (n). O True O False

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Q: Find a formula for f (z) if f(z) = I+1 f '(z) =O ( )'(2) =D-

A: We will use concept of inverse function and differentiation of a function.

Q: 14. Simplify the following functions using the properties of logs. a) f(x) = ln (e5x) - 3x + 5 ln(1)…

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Q: Set f(x) = In(9x - 8). Question 1 For k > 1, f (k) (1) =

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Q: GIVEN: The function, f(x) = (2x+ 7a)(x – a)², where a>0 and a constant. Find f'(x) in FACTORED FORM…

A: This question is related to derivative.

Q: Calculate Rs for f(x) = 6 - x over ver [3,5]. R8 IE

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Q: Suppose f f (x) da = 8 and f g (x) da = -3. 3 Evaluate f (-4 f (x) + 2 g (x)) dz. 3

A: Using integral property,

Q: (a) Let f(x) = In x and g(x) = Va. Show that f(x) and g(x) are both increasing when x > 0. (b) Let…

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Q: d [In(x² + 1)] = d [In(x2 + 1)] x2 + 2.x Given the derivative find the derivative dx Simplify 4. dx…

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Q: If f(x) = 8x + e5x, use the formula (f¯¹)'(x) = (f¹) '(1) = [ 1 f'(F-¹(x)) - to find (f-¹) '(1).

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Q: f(x) = - [x + 1 for xs0 lx -1 for x > 0 a. Show that f(-1) - S(1) = 0 but that there is no number c…

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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…