[(r* - 1) (4 – x* )ax

Q: Prove that 1/r + 1/r2 + 1/r3 + . . . = 1/r − 1, for |r|  > 1.

A: we have to prove that: 1r+1r2+1r3+. . . . . .=1r-1  ,  for r>1

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

A:

Q: Prove or disprove the statement: For all x E R, |x + 1| + |x – 1| > 2.

A:

Q: Find k such that 2x = ekx for all x.

A: To find k 2x=ekx

Q: Let f(2) = In[="(= + 7)*(z² + 7)°] f'(x) =

A: Given : fx = lnx8x+72x2+76

Q: 5 Prove for Vx > 0 0<1+x - 1 x + - 81 1/9 1/3

A: Prove for ∀ x&gt;0  0&lt;1+x3-1-x3+19x2&lt;581x3

Q: Show that for all x>0, e* >1+x+÷ 2.

A:

Q: 2ア+ 2z+ 11 (2ェ+3) (z +4)

A:

Q: Prove that 1/ r + 1/ r2 + 1/ r3 + . . . = 1/ r−1 , for ∣r∣ &gt; 1.

A: Consider, It is a geometric series with,

Q: : 2 < x < 4} and B = {x € R : –4 < x < 1}. Prove

A:

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

A:

Q: Q\ Determine a condition on |x - 1| that will assure that * 1 |x² – 1| <

A:

Q: Q\ Determine a condition on |x - 1| that will assure that * 1 |x² – 1| < |3

A: Given that |x2-1|&lt;13 That is -13&lt;(x2-1)&lt;13

Q: 1. • If r + 1| < o, prove that |r³ + 1| < 0.331.

A: Here we use the fact  | x |&lt; a implies  -a&lt; x &lt; a .

Q: Which of the following is a critical number of f(x)= 4. 2x %3D

A: The given functIon is: f(x)=14x2-2x

Q: 1. Expand (r+ 1)(x – 2)(x + 3)

A: We will explain this using commutative law and associative law of multiplication.

Q: Given that 32a × 11º = 274x × 332× express a in terms of x?

A: Find your answer below

Q: 11. Evaluate the integral. +3 x² + 4

A:

Q: 2x-3 for *く-S メ+4 %3D ーメ+3 for マ-S メ+2 for x) -5 メ+ヒ find *ビ" し 1n for order f(x) to メ- -5

A:

Q: What is the expansion of 1 f (x) = %3D Ο Σε' Σ 2n 2n Σ Next

A:

Q: f(x) = (x² − 9) -

A: Given :f(x)=x2-923

Q: Evaluate (x + 5)(x − 1)(x − 2) (x - 1)(³) — 2(x − 1)(²) O A (x − 1)(³) + 2(x − 1)(²) (x − 1)(²) —…

A:

Q: -a = Y express L and K as expressions of

A:

Q: Derive a formula for fx* In x dxwhere a.) n+ -1 b.) n = -1

A:

Q: f(x) x* + 3x - 3x + 2 %3D as x -00 f (x) → 0 as x -of (x) → -∞ as x +∞f(x) → -∞ as x → +of(x) → +o

A: Given polynomial f(x) =-x4+3x2-3x+2

Q: ind the Wronskian of: 18x - 6 18x + 6 6x - 18 6x + 18 f₁ = x² - 6x f₂ = x² + 6x f3=x²-3

A:

Q: B: Evaluate the integral: TUTO (x + 1)-/2 (x – 1)5/2

A:

Q: 1 What is the expansion of f (x) = 1-r OE(-1)"x" Ο Σ 2n

A:

Q: 6. (a) Express f (x), where x3 + 1 f(x) = %3D (x – 1)2 (x² + 1) Hence, or, otherwise, find HE AND…

A: f(x)=x3+1(x-1)2(x2+1)

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

A:

Q: Excrcise 8. Evaluate -In 2 xe d.x 1

A:

Q: Which of the following is S(x) = x2 - 7x + 10 %3D 11-5x S(x) = 3x-8 S(x) = 6- x f(x) = /2x – 5

A: We can find which option is not one to one as below.

Q: 3. Let f(x) = { ,x<2 ax + b, x22 Find a and b so that f (2) exists.

A: To find the value of a and b

Q: ( Σα - xi-1) = x = x for n > 2. i=2 11 IDDLE: following "proof" th

A: Given that c, x1, x2, ..., xn, y1, y2, ..., yn are 2n+1 given numbers.

Q: - Give the domain for k(a) in interval notation. x2 + 3x +2

A:

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

A:

Q: HW KLX) = 2x+5 7 -1. Find ke-²¹ (x) 2. Prove by kok orklok

A: Let's find inverse function.

Q: The expansion 1 + x + x² ... approximates (1 - x)1 if O x is not equal to 1 本 O absolute value of x…

A: The expansion 1+x+x2+.........approximtes 1-x-1

Q: Prove: for n > 1, F; – Fn+1 Fn-1 = (-1)".

A: F_n are the Fibonacci numbers which are defined by F_0=1,F_1=1 and F_n = F_{n-1}+F_{n-2} for…

Q: Verify that Al2 = I½A = A for an arbitrary A E M2x2(C).

A:

Q: Consider f(x) = 2x +3. Is f : N → N one-to-one? Is it onto? Prove, or give counterexample.

A: As per our guidelines we can answer only one question. please repost other seperately

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: ) Show that |2x - 2|-|x+1| +2≥0 for every x € R.

A:

Q: any n > N(\epsilon) | fn(x) - f(x) | < \epsilon

A: The Weierstrass approximation theorem states that every continuous function defined on a closed…

Q: ? Suppose that * f(3) = }, and f(x) > 0, and f(x) < 0. Is it possible that Í(2) = . 3 O False True

A:

Q: Discuss the continuity of : 1 for x 2

A: Continuity

Q: B.Discuss the continuity of 1 x +- for x <0 y = -x3 for 0 < x < 1 -1 for 1 < x

A: The objective is to discuss the continuity of the given function.

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)?

A:

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.