22. If f(x) 10 2x*1 and x is a nonnegative integer, show that f(x+ 2)- f(x) is divisible by 99.

Q: Let h(x)=(x2−3x+2)/(−x2−4x+4,) what is h′(−5)? Do not include "h′(−5)=" in your answer. For…

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Q: Let {(=) = In ="(z +2)*(z² + 9)] f'(x) =

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Q: 8. Construct a natural cubic spline that interpolates the function f(x) = 1/(1+ x²) at x = -2, –…

A: Let the cubic spline polynomial of f(x) be H(x). Then Hx=fa+fa,ax-a+fa,a,bx-a2+fa,a,bx-a2x-b…

Q: How can I show that f(x) is O(x2), where f(x) = 6x2 + x + 1

A: Given: f(x) = 6x2 + x + 1

Q: 2. Let h(x) = |9 – x²].

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Q: 1. Differentiate. Show your work to get credit. a) g(x) = x* In(2.x² +7) by using the product rule.

A: SInce multiple subparts are asked, I am ansering only first 3

Q: 9. For |x|<1, prove that 8x 16x 24x 1-x1+x 1-x 2 8x 8x = 1+ (1 – x)? (1+x2 (1-x)? ...

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Q: Prove that for x≠0 the inequality ex&gt;1+x holds.

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Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

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Q: Prove that 2 1 1 5 2 (4-2x)² for all relevant x-values

A: We have to prove that 1215-x&gt;24-2x2, for all relevant x-values.

Q: 7. There is aremovable discontinuity at x = 1 andf(1) = 4

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

Q: Verify that 2. 1 Hint: Consider x 0, x 0 separately

A: To verify

Q: In(x-1) 33 Differentiate with respect to x. 2x + 3

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Q: 5. Check the continuity of r+5| 2x+10 x + -5 f(x) = x = -5 at x = -5.

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Q: 7. Suppose f(x) = x³ – 27x + 1. Prove that f has exactly one root (zero) in the interval [0, 2].

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Q: 12 f(x) = X ¹²³ + x² + x² + x² Show that whether f(x) is irreducible or not

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Q: 1 Expand f(2) = -in the domain 4 < |2 < 12 (z-4)(z–12)

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Q: Suppose that .3 f(3) = }, and f(x) > 0, and juw <0. < 0. Is it possible that f(2) = . %3D False O…

A: You have asked two different questions. As per the company guidelines we are supposed to solve first…

Q: Prove f:N - N. F(X) = 5x is injoctive.

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Q: 2.59 Find || f |sup if a) f(x) = x on (-1, ). So if x E Q, b) f(x) = – x² .2 if x ¢ Q. -

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Q: 9. Use the given graph of f (x) x2 to find a number 8 such that %3D if |x – 1| < 8 then |x2 – 1| <;…

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Q: if ニ if 1-X fCx +2) = f(x) Hx f on [-3,3] and we Suppose draw the funch'on

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Q: Find the domain of the function. h(x) = In x + In(2 - x)

A: Given h(x)=ln x+ln (2-x) Find the domain by finding where the expression is defined.

Q: @ f(x)= x²-3x-4 2 @ f(x) = x² - 4x +3 Show that whether f(x) reduceable Poly or not

A: The given problem is solved below in detail.

Q: Find positive integers that satisfy x + y + z = 6 x + 5y + 8z = 35 x = i ! y = i

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Q: Show that f (x) = 3x² - x - 1 is O(x²) with suitable C and k.

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Q: 5. Show that Ax) = Inx² - 1- In/3, if f'(x)= and (2) = 0. x- 1 Justify your answer.

A: To show that f(x)=ln|x²-1|-ln√3

Q: (х — т 11- тх if x < 3 if x 2 3 5) Find m so that g(x) = Is continuous for all x?

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Q: Give a proof by cases that [4x] = [x] + [x + ¼ ]+ [x + ½ ] + [x + ¾ ] %3D

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Q: 3. Show that f(2) = (z+ 1)³ – 3z is nowhere analytic.

A: The given function is, fz=z+13-3z The function z is given by, z=x-iy Substitute values in above…

Q: y3-3ax2+x3=0,then

A: Given:  y^3 - 3ax^2 + x^3 = 0

Q: Compute f' (3). f (x) = –3.3x² + 3.9 O f' (3) = –19.8 %3D O f' (3) = –3.3 f' (3) = -15.9 O f' (3) =…

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Q: A³ f2 = V³ fs 3

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Q: Indicate the least integer n such that f(x) = (3æ³ + 2x²)(5æ5 + 4æ*) € O(x").

A: Finding f(x) explicitly using multiplication of polynomial. The highest power of x or degree of f…

Q: 3. Determine the value of a so that f(x) is continuous on the entire real number line when: f(x) =…

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Q: (It v5)" (1 Vsy (b) Prove that f. for all integers n> 1.

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Q: 12. Let f(x) = x + 2 and g(x) = x² - 1. Find (gᵒf)(4)

A: Given two functions 1.f(x)=x+2 g(x)=x^2-1 lets find g(f(x)) at 4

Q: 10. Find all of the discontinuities that exist for g(x) = 2*+3x-4 Show any work required to %3D x+1…

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Q: 10. Determine the upper bound on the error in the approximation = x - 16x5 + 256x° if |x| < 0.2…

A: We need to find upper bound on the error.

Q: Prove that y = e-3* (Acos 5x - Bsin 5x) + Cx; A, B and C are arbitrary constants will equalto y"(17x…

A: Given,            y =e-3xAcos 5x -Bsin 5x + Cx

Q: Find any restrictions on x. ITS 1/2x - 5= 10

A: The given function is 12x-5=10 Solve 12x-5=10 as follows: 12x-5+5=10+512x=15x=30

Q: 1 Find "(x) if Ax) -. %3D X+1. 1 (x+1) (x+ 1) -2 (x+ 1)3 12

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Q: Show that f (r) = x³ – 3x2 + 2 is monotonically increasing on every interval.

A: Given that: f(x)=x3-3x2+2

Q: 15. Show that (x+ 5) is a factor of f(x) =x3 -x² -21x + 45 then factor f (x) completely.

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Q: Use intermediate theorem to explain P [x] = x* – x' – 10 has a zero in [2, 3]

A: According to intermediate value theorem of a function f(x) is continuous on [a, b] and L is any…

Q: "8 does not divide x^2– 1 if x is even". prove it directly

A: We need to prove  8 does not divide x2 - 1 , if x is even. OR if x is even, the 8 does not divide x2…

Q: Find 2 for all of: 1- y = xvInx 2- y = 3 x(x-2) x²+1

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Q: 29. For |x|<1, prove that 16x 24x 8x 1 + 1-x1+x 1-x ..... .. 8x 8x = 1+ (1- x)? (1+x)²' (1-x)?

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.