65. The proof of the latter part of the Second Fundamental Theorem of Calculus in the reading covered only the case h → 0+. Rewrite this proof in your own words, and then write a proof of what happens as h → 0-.

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

Q: 9. Suppose g(u) = | V1+t³dt. Find the value of (g¬1)'(0) if it exists. If it does not exist, explain…

A:

Q: 1. Explain why Corollary 1 of Theorem 9-7 is true. That is, explain why L1 = L2.

A: Given that ∠1=∠2 is true statementsHint: Major arc; Minor arcand…

Q: (1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute…

A: First let xy be positive  Then either both x and y are positive or both are negative  If both are…

Q: 13. Prove by Math Induction ex > 1+x for all positive integers x > 1

A:

Q: a. Prove that ex>= 1 + x if x>= 0. b. Use the result in part (a) to show that ex>= 1 + x +…

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Q: X sec10 can 1. Let u = 7 In х2 - 1

A: Given: ∫x sec107ln2x2 - 13tan7ln2x2 - 13dxx2 - 1.

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

A:

Q: 4. (a) Use Gauss's lemma to evaluate (), showing all working.

A: The given term is, 1119

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

A: Given is a expression  1215-x&gt;24-2x2. It is required to prove the given expression for all…

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: 4. Prove Theorem 2.6.5.

A: Let p,q be any two distinct points on l. Let r be any point on l. Let m be the line containing p and…

Q: 1. Prove that ƒ (2) = sin z is analytic.

A: "Hey, Since there are multiple questions posted, we will answer first question. If you want any…

Q: 1 1. -to prove that x+y|# | x|+ | y| for all real numbers d) Use x = and y = %3D x and y.

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Q: Prove Theorem 11.3

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Q: (a) Prove or disprove: f: Z3 → Zs, [r]3 + [r*]5 is well defined.

A:

Q: To prove that dx In /1+ (In x)² + In x + C x/1+ (In x)2 we use the substitutions

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Q: Prove that 223 a E lo,1) く 22 + 1영 ( 1-계 3(1-x2)

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Q: 6) Show that the function f:Z → N given by 2n +1, n20 f(x) = -2n, n<0. is 1-1 and onto. I

A:

Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

Q: 34. Prove parts (a) and (c) of Theorem 3.2.3.

A:

Q: Prove the following inequality. Do not evaluate the integral. 1 ,1 - x² + x +1 | - 2 0, 1+ x4 4

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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: prove that 2,

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Q: Prove that (0.01)(100sin2π(5000)t)2 is equal to 0.5+0.5cos2π(10000)t

A:

Q: Prove thot if yef(x) is continyous and f(x) =0 when x is a ratsonal number, then f(x) =0 for all x €…

A: Given that y = f(x) is continuous and f(x) = 0, x is a rational number. To prove f(x) = 0 for all…

Q: 12 २ (४) = dt | २स Prove !- \) ২-০০) = २(-०० ) 3D \

A: Given that Qx=12π∫x∞e-t22dtTo prove:1. Q-∞=12. Q-x=1-QxSolution:we know that ∫-∞∞e-x2dx=π       ...…

Q: 5) Prove that d tanh ) |-x2

A: Prove that  d/dx[tanh-1x] =1/(1-x2)

Q: 6. Prove from first principles (with ac- 6 argument ) that लेग ½ lim 7--1

A: 1x2+1-12=2-x2+12x2+1                               =2-x2+12x2+12+x2+1…

Q: 5. The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard…

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Q: 21. Discuss the continuity of : 1 x + for x 2

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Q: r37/4 -Зп/4 sinod0 Jn/4

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 5. Use Cauchy's Residue theorem to evaluate dz; Có lz-il =1 (2+1)

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Q: -5 3x+1 dp: 12

A:

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: 1) Prove that 0|a - a=0

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Q: 2 (1 – :-) d.x

A: Given ∫141-2x2dx

Q: Prove that JI = J [1 0 0] I = |0 1 0 Lo 0 -1 2 -31 J = 4 5 -6 %3D –7 8 -9 1.

A: Considering the matrix multiplication of J and I

Q: 5. Evaluate the integral JI. dA, R = [-2, 4] × [1, 3]. R

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Q: Frngl. prove that x

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Q: Q1. Prove that KT = = -.

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Q: a) Prove or disprove: f: Z3 → Z5, (r]3+ [r*]5 is well defined.

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Q: By expanding both sides of the identity (1+ x)"+4 = (1+x)"(1+x)* prove that ("**) - (") + «(,",) +…

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Q: 2 e4

A: The given integral is: I=∫e4e72xdx

Q: (18) (a) Prove that R* is isomorphic to R>0 x Z2. (b) Is Q* is isomorphic to Qo x Z2? Prove your…

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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…