Determine the O-class of the function h(n) defined by if n = 1 h(n) h(l#)+h([])+log n if n >1°

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Q: Find the first six terms of the expansion of the function e^x log(1+ y)

A: We have to find the first six terms of the expansion of the function              exlog1 + y Here ,…

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A: Given

Q: Q2 (a) Explain the difference between even and odd Fourier series function.

A: First of, we know about Odd numbers are : 1,3,5,7,….. Even numbers are: 2,4,6,8,…..   =>…

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A: To prove,  loga1N=-logaN where, a  and  N are positive real numbers and a≠1. Now, take the left…

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Q: The value of c which satisfy the conclusion of Rolle's Theorem for f)= log(x²+1)-log(2) on [-1,1]is:…

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Q: 2. The function f(x) = log, x is the logarithm function %3D So f(9) = -- f()- with base %3D f(1) =…

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Q: Let f be a function defined by f(x) = ln(3 − 2x) With the first 5 terms of the Maclaurin expansion…

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Q: () N f(x) = , show that f(0) and its leading differences form a geometr 5. (i) Prove that A log x…

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Q: Consider the function g defined by 1 g(x, y) = cos (Tx)+ log3 (x - y) Do as indicated. 1. Determine

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Q: Let f be the exponential function defined by (2+2) f(2) = () and determine the folloving information…

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Q: ) a) Give the estimation of Big O of f(n) where f(n) = 3n log(n!) + (n² + 3) log n

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Q: f (x) = In(x + 4) + 3 %3D f(2) 3 e2-6- 4 %3D f (x) = log(a- 2) - 8 %3D f (x) = 108+2 ----- f (x) =…

A: Some functions are given and the inverse of the functions are also given. We have to check that…

Q: Let f R*R, defined by f(x) = 3*, then f(x) = %3D O log,3 3* o log3x log3

A: See solution below.

Q: Show that log,(n) € O(log,(n)) for any b>0 with b 1· Generalize this result.

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Q: Find a formula for f n  x if fx = ln (x-1)

A: Given, f(x)=ln(x-1) differentiate with respect to x, f'(x)=1x-1 Again differentiate with respect to…

Q: f(x)=3x+3-18 and f-1(x)=log3(x+18)-3 Show that (f•f-1)(x)=x

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Q: [E] Prove with usual notations that (i) hD = log(1 +A) = – log(1 – V) = sinh '(u8)

A: We will prove the following result.

Q: The value of c which satisfy the conclusion of Rolle's Theorem for f(x)=log(x²+1)-log(2) on [-1,1]…

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Q: Write down the Fourier series for an odd function f(x) such that f(x + 2π) = f(x).

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Q: O Give a big-O estimate for f(n) = 3n log(n!) + (n²+ 3) logn.

A: Big O notation is a mathematical notation that describes the limiting behavior of a function when…

Q: The derivative of f(z) = Log(2iz – 4 + 6i) at z = -3 + i is O Undefined. O Log(-6). O None of them O…

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Q: For which value(s) of k, if any, is the function 1 (²) = x [kx² - log₂ (x) 2xk²x + 5 continuous on…

A: Use continuity definition of function

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Q: 4. Prove the formula below using mathematical induction log 2k u + zu log 2 k=1

A: Given: Formula: ∑k=1n log2k=n2+n2log2

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Q: Find the smallest positive integer n so that f(x) = x8-log(x³ ) + x6-log(x5) is big-O of xr.

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Q: When comparing the f(x) = x² + 2x and g(x) = log(2x + 1), on which interval are both functions…

A: Let's find interval when both functions are negative.

Q: Determine the O-class of the function h(n) defined by if n = 1 h(n) |h(L)+h([1) + log n if n > 1

A: We will first do a clever substitution to convert the given piece-wise function into a recurrence…

Q: If n is a power of 2, prove that M(n) = (n log n) / 2 , using Master of Theorem method. Show all…

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Q: Find the big O estimate for functions f(x)

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Q: 5. Prove that if {a,}f° converges to 1, then {an}° converges to 1. Hint: You may use a logarithm…

A: If an1∞ converges to 1, then an1/21∞ converges to 1.

Q: Solve the following d. In 3- Inx - In (x + 5) = 0 Log4 (x + 2) – log4 (x - 1) = 1 e. f. Log2 (x-1) –…

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Q: 3. Compute the value of log(2875) from the following values of log(x) with the help of Newton…

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Q: log x" -[x] [x] Evaluate : lim where, n e N and [] denotes the greatest integer function.

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Q: The domain of f(x)= log4(1 – I2) is Select one: O a. ( -1,1) O b[ =1,1] O c. (0, 00) O d. [1, 00)

A: In the question it is asked to calculate the domain of f(x)=log4(1-x2).

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Q: Use l’Hôpital’s rule to find the limits

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Q: Find the least integer n such that f(x) is O(x") for the function 3x +x log ax

A: This is a problem of finite math, and complexity.

Q: Find of, af , for f(x, y) = log₂ (3г² - +y)

A: Solution of given question is written in below steps.

Q: Expand the function shown as a Fourier Series: f(x) 27

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Q: Represent the function below as Fourier Series: y 1 -1

A: The graph of the function is given below At θ=0, function value is 0 and when θ is π2, function…

Q: (a) Prove that log₁ x= -log₁ x where a > 1. a

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