2. It can be shown that the inequalities x2 x sin x < 1 1 2 – 2 cos x hold for all values of x close to zero. What, if anything, does this tell you about x sin x lim x0 2 - 2 cos x

Q: 2. Suppose, f'(x) =2 cosx+sec x, -7<x<: and it is known that f()= =4. Find the function f(x)

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Q: (a) Apply the Comparison Theorem (Theorem 5 in Section 5.2) to the inequality sin x 0) to prove…

A: Given: Use the inequality  sinx≤x   (valid for all x≥0)To prove:a) 1-x22≤cosx≤1b) x-x36≤sinx≤xc)…

Q: - 1 X If lim ([sin ¹x] + [2²sin ¹' (2x)] + [3³ sin-¹ (30)] x →0 X X + [x²³sin-¹ (x2x)]) + ... +…

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Q: 2. Prove the following identity: sin (x+y)sin (x-y)=cos² y-cos²x X =COS - COS¯ x X.

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Q: In cos 2x 4. Find lim x→0 In cos 3x

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Q: 1- r + In (x) lim 41 1+ cos (3a x)

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Q: Prove the following inequalities (ii) -10|< sin 0 < |0| and -|0| < 1– cos 0 < |0|

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Q: If f(x) = = sin (3 – x), then f'(0) O 6 sin 3 cos 3 O 6 cos 3 O -2 sin 3 cos 3 O -2 cos 3 -2 cos O 2…

A: f(x)=sin2(3-x)

Q: The largest value of the non-negative integer a for which lim x →1 - ax + sin(x - 1) + a x + sin(x −…

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Q: 1 o. = 0. Show that lim x² sin -| *

A: Use the property that -1≤siny≤1. Take y=1x, then -1≤sin1x≤1

Q: Prove that the inequality|cos(x)|≥1−sin2(x) holds true for allx∈R

A: Given inequality is

Q: 1 9. In the proof that (sin-1x) = that sin(sin-1 x) = x. It is also true that sin-(sinx) = x; could…

A: Differentiation of function gives the change of function with respect any variable. The inverse…

Q: 2. The limit of the following function equals: a. 0 b. 1 c. -1 d. None of the above answers x2/3 +…

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Q: It can be shown that the inequalities 1- 6. x sin x <1 2 – 2 cos x hold for all values of r close to…

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Q: 2 Prove that lim x*cos 0.

A: Explanation: Given that,  limx→0x4cos2x=0     Taking LHS(Left Hand Side) limx→0x4cos2x

Q: 2 cos(5x) – 5 x² – 2 sin(3x) – x2 – 3x - lim - - (Write your answer in the given box)

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Q: H.W 1: Prove cos 4x = 1-8 sin² x . cos' x H.W 2: Prove that: sin 2x cos 2x sin x cos x sec x sin ( a…

A: 1) Prove that cos 4x=1-8sin2 x ·cos2 x. Consider the left hand side. cos 4x=1-2sin22x…

Q: Prove the following inequalities -10|< sin 0 < |0| and -10| < 1– cos 0 < |0|

A: Given the inequalities         -θ≤sin θ≤θ and -θ≤1-cos θ≤θ. Prove it.

Q: (d) Suppose that the inequalities 1 1- cos a 1 2 24 x2 2 hold for values of æ close to zero. What…

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Q: n+1 Evaluate lim cos“(x²)dx n→∞ (and show all work)

A: The required results are as follows: cos2 x=12+12·cos(2x) and cos 2x=1-(2x)22!+(2x)44!-(2x)66!+...…

Q: 15 and x terminates in guadrant III. 17 Find sin2x, cos 2x, and tan 2x if sin x= - sin 2x cos 2x %3D…

A: The solution is given in the next step.

Q: 4. Show that f(x) = sin(x² + 3x) is equal to g(x) = -x cos(x) at least once on Hint: Think about the…

A: Intermediate Value Theorem (IVT) : It states that if a continuous function f(x) on the interval…

Q: 23/ Solve the following. Choose two only. 1. lim x lnx³ X-0 2. lim x sin(1/x) 848

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Q: 8 and x terminates in guadrant III. 15 Find sin 2x, cos 2x, and tan 2x if tanx= sin 2x %3D cos 2x…

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Q: In evaluating 4 2 2 dx x√√x²-4' is same as evaluating lim cos-¹(f(x)). Determine the value of f(x)…

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Q: In evaluating 2 2 dx XV is same as evaluating lim_ cos-¹(f(x)). Determine the value of f(x) if x=2.9…

A: As per the question we are given a definite integral and we have to find the primitive of the…

Q: Solve 2cos (x)- sin(x)=2 exactly for all x in the interval [0,27).

A: 2cos2x-sinx=2⇒21-sin2x-sinx=2…

Q: 4 35₂² In evaluating 2 dx x√√x²-4 is same as evaluating lim cos-¹(f(x)). Determine the value of f(x)…

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Q: 15. Suppose that F'(x) = 3x + cos(r - 1) and F(1) = 5. Then find what F(x) is precisely.

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Q: 4. The graph of f (x) = x sin x defined on 0<x<n has an inflection point whenever 2 2 (b) tan x (a)…

A: Given: fx=xsinx  , 0&lt;x&lt;π     Definition: Inflection point: The point at which the second…

Q: Suppose, f'(x)= 2 cos x+sec x, -7<x<, and it is known that and it is known that f()=4. Find the…

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Q: 2. The limit of the following function equals: a. 0 b. 1 c. -1 d. None of the above answers x2/3 +…

A: See this solution in.

Q: 8 and x terminates in guadrant IV. 15 Find sin 2x, cos 2x, and tan 2x if tanx %3D

A: Given  tanx=-815,  x in quadrant IV

Q: The limit lim (18 x; - 28 sin (x;)) Ax i=1 on the interval 0, is equal Tt/4 | (9 x² + 28 cos (x)) dx…

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Q: x=cos¬! 1 prove that sec" X= cos

A: Given sec-1x=cos-11x

Q: Given f’(x)= 3 cos x - 9 sin x and f(0)= -6 find f(x)

A: We can solve this as follows:

Q: Can the Intermediate Value Theorem be used to prove that the equation 9 2(r + 9) sin*(x) – 4 cos"(x)…

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Q: Now, let f"(x) =0 and solve for x. Remember that f"(x) = - sin is a periodic function, and thus…

A: use the equation above for solving the equation for x.

Q: 5. Prove the following identities: (a) cos (x +) + sin (x = 0 6. sin(x – y) cos(x) cos(y) (c)…

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Q: Q3:A| Prove that:- cos x 1) 1+sin x 1+sin x = 2 sec x cos x tan? x 2) tan? x+1 = sin x

A: topic - trigonometric identities

Q: In evaluating 4 2 dx X√ X²-4 is same as evaluating lim cos ¹(f(x)). Determine the value of f(x) if…

A: We have to find the value of f(x).

Q: Find the value of lim ((n + 1) cos (n + 1) cos¯¹ (1) − n) n

A: Given: limn→∞2πn+1cos-11n-n To find: The value of given limit.

Q: Show that, for real values of x, 3 sin x f(x) 2 + cos x cannot have a value greater than 3 or a…

A: To show that the minimum and maximum value of the function is -√3 and √3

Q: 1. Evaluate 2. Evaluate sin(20) 30 x-2x cos x sin x limo lim, 0

A: Given is  limθ→0sin(2θ)3θ. L'Hospital rule: limx→af(x)g(x)=limx→af'(x)g'(x) NOTE: According to the…

Q: (2) elnx = x (1) sin-1x = V1-x2 (1) ax A- Show that : cos x (2) lim x- cosx (2) (1) limx-0 В- Find :

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Q: 3. Given f(x) = cot(x) sin(x) and g(x) = cos³(-x)(sec² (x)-1) sin² (x) a. Prove that f(x) = g(x).…

A: Given that,  fx=cotxsinx and gx=cos3-xsec2x-1sin2x

Q: -1 Find y' where y = 22 sin-(cos (9 æ)) – tan (6 æ) Answer: -198 sin(9 x) 1 || V1-cos2 (9 æ) 1+36 x2…

A: To find y'

Q: (a) Show that lim | x| cos = 0.

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: If g(x) f'(x) and if g(x) (cos x - x sin x)e*cosx, then f g(x) tan(f(x))dx is x cosx In(e*cosx) +€…

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