1. Show by defintion that f: R R, f(x) = cos(a), is continuous on R. Hint: You may try to mimick the proof of conitinuity of f(r) = sin(æ) on R. You may need to use the formula a + b COS a - cos b = -2 sin sin

Q: -) Prove that there is a one-variable Cl function y = g(x), defined near r = 0, such that %3D Sr2 In…

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Q: Find General Solution on this Diffential E Qution by using (uC set) =e +Sin X

A: Given differential equation is y''+y'-2y=ex+sinxfind the general solution

Q: 1. If y= tan x, prove that y"= 2(y² +1)(3y² +1).

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Q: Find a value of c that will make the following function continuous at x = 0. sin² 4x x ≤ 0 x² f(x) =…

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Q: 1. If y= tan x, prove that y``= 2(y² +1)(3y² +1).

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Q: 3. Let z = x + iy. Prove that the function f (z) = e-* sin y – 10 + ie-* cos y is analytic in C AND…

A: analytic  function

Q: 3. Let G = {z = x+iy : x < 0 or y # e" sin x}. Show that G is simply connected.

A: Given: G=z=x+iy; x&lt;0 or y is not equal to ex sin x

Q: /hich of the following does not require the use of the dy nain rule to find ? dx A y = cos1 (10x –…

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Q: What is the limit of sin when x approaches 0? Select one: O a. 1 O b. 0 O c. Does not exist

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Q: is stricly increasing on (0, T /2). sin x Prove that the function f(x)=

A: Given that: f(x)=xsinx

Q: let Po)- tar'c) + tant (L). show that Plx)=C,where cis a constant and then Aind C

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Q: For f(x): cos(x) + 2, choose all values of c in the interval TT 37 guaranteed by Rolle's Theorem. 2'…

A: Given, fx=cosx+2 f'x=-sinx By Rolle's theorem, fx=cosx+2 is defined in the interval -π2,3π2, such…

Q: The Rolle's theorem is applicable for f(x) = cos x on [0, T]. True False A Moving to another…

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Q: E Q4) Evaluate cach of the following: 1- Prove lim sin3xcot5x X-0 xcot4x 2- Find dx Zx+3/ 3- Find…

A: Questiion 1,Given,            limx→0sin3xcot5xxcot4x=125To find : To prove the limit.

Q: The following DE sin(x) +y cos(x) = 1 can be written as: dx Cos(x) sin(x) None of These 1-cos(x).…

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Q: 4. Use the inequality | sin r|< |x| for x E R to show that the sine function is continuous at r = 0.…

A: Sine function and it's continuity at 0

Q: Let f(x) = In(cos(x)). For what value of c is f'l =CX + TC 1? 4. C =

A: We can solve it for the value of c as below

Q: Find a and b which make the following function differentiable at x = 0; ax+b, x < 0 sin x, f(x) = =…

A: We know a function is differentiable if and only if Right hand derivative and left hand derivative…

Q: Given that: z = y sin(2x) + (1+x²)e* + cos x tan¬1(y* + 1) In(y6 + 137)

A: z=ysin2x+1+x2ex+cosxtan-1y4+1lny6+137 Differentiating wrt x, we have…

Q: (a) Restrict f(x) = sin x to [TT/2,3TT/2] (b) Restrict f(x) = cos x to [-T, T] %3D (c) Restrict f(x)…

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Q: Gren thint sin (A+i6) = xtiy, 28 Prove that, 9父+ =1 snh'B 1) x essec?A -y' sec'A =1

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Q: of the tangent line to the graph of f(x) = 7 sin(x) + 6 cos(x) at x = in the form y = mx + b.

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Q: If it was f(x, y) = ln(xy) + tan(xy) so Fxy fxy = sec(xy) (2y tan(xy) + 1) - A fxy = sec (xy) (2x…

A: Given fx,y=lnxy+tanxy

Q: DIfferentiate the following fuh ction with respect to K In X %3D Sin a

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Q: Consider the function w = f(x,y), where %3D x = r cos 0 and Y = r sin 0 Verify each of the…

A: To find the function in terms of x and y, use the trigonometric identity, cos2x+sin2x=1 and…

Q: 24 8. (10) Let g(x) = sin (x) if a +0 %3D 0. if r=0 Is g' is continuous at 0?

A: Topic:- Continuity  g'(x) is continuous at 0.

Q: The Wronskian, W, of the functions y1 = sin(2x) and y2 cos(2x) is %3D

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Q: 2 If s Sinants COs ant 3 an= ০ । A. limitdoes not exist 2.

A: Series and sequences

Q: on f(z) = e cos x+ ie-Y sin x is entire.

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Q: 1. Determine the absolute extrema of f(x) = (sin x – 1) cos x on 2' 2 if they exist, and identify…

A: Given:-f(x)=sinx-1cosx,        x∈-π2, π2f'(x)=sinx-1ddxcosx+cosxddxsinx-1…

Q: Verify that the given function is Soluiion of the corresponding inūegral equasion. p(x) = xe; 9) =…

A: Solution of an integral equation: If a function ϕx satisfies the integral equation, then ϕx is the…

Q: If ƒ is continuous everywhere and 1| f(x) dx = 6, find | sec? 2x · f(tan 2r) dr.

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Q: 2. Find a closet rank-1 approximation in the f norm to A = cos 8 - sin e sin e cos 0

A: According to the given information it is required to find closet rank-1 approximation in the l2 norm…

Q: Given f(x,y) = e X cos y, Evaluate fx(0,2) cos 2 -sin 2

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Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ () <0. f(x) has a root in [0]. If we use…

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Q: 22: Find dyldx for the following functions * (40 marks) · y = In (3 – 4 cos x) + (4x – 5)5 2. y = x²…

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Q: on [ 0, 2î] ,determine all velafive the Finction. .If f(x) = sin x t cos X extrema for

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Q: If f(x,y,z)=In(x²y+sin?(x+y))+111x112y2z113, then at (1,1,1) is equal to əx²əyəz

A: We have to find derivative

Q: (b) Approximate the definite integral f"/² cos(x)dx by employing the Composed Midpoint rule with n =…

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Q: Given that the function g(r) is defined on R by g(x): A a · sin(cos(1/a)) , if r +0 , if r = 0 What…

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Q: Given +2 – 25 dt 1+ cos2(t) f(x) : At what value of x does the local max of f(x) occur? Preview

A: Consider the given function.

Q: -ツ() The function f(x) is invertible. Find given that f(x) = 5x +4 cos(x).

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Q: Find F'(x) given that F(x) 3+sin (5t) dt. |3D In (t8) Provide your answer below: F'(x) =

A: The objective is to find F'(x).

Q: So Evaluate F. dr where C F = (– 2z, – y, – 4x), and C is given by 7(t) = (sin(t), cos(t), t), 0 < t…

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Q: 1. Consider the function f defined on [0, ∞), 1 f(x) = sin- x + 0 x = 0

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Q: 1. Prove: a. csch x In +c %3D b. sech x = 2 Arctan(e*) + c %3D

A: 1. (a). To prove: ∫cschx=lnex-1ex+1+C Proof: We know that, sinhx=ex-e-x2 The left hand side can be…

Q: 5. Suppose x = r(0) cos(0) and y = r(0) sin(0), where r(0) is sufficiently smooth, and suppose x = a…

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Q: d Prove, through the limit process, that (cos x) = - sin x. dx

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Q: Is there a value of c that will make sin² 3x f(x) = C, x = 0 continuous at x = 0? Give reasons for…

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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.