2. Prove that a) the function f(x) = tan x+x can be replaced by g(x) =x if x→0. b) the function f(x)= /1+x –1 can be replaced by g(x)=x/n if x→0 (n is an integer

Q: Suppose f"(x) = - (sin(x)), ƒ'(0) = –2, and f(0) = -3. %3D Find f(1/4).

A: Given,          Suppose f''x=-sinx , f'0=-2 and f0=-3.

Q: 2. (a) Use algebra to verify that (1+ (x – 1) (b) Differentiate arctan(x+1) – arctan(x- vital thing…

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Q: Does the function f(x) = 5x − cos(x) + 4 have a root on [−π, 0]? Justify your answer.

A: Explained below

Q: Letf (x) be defined by f (x) x sin(-) for x # 0 and f (0) = 0. Find f' (0). O 1 Does not exist.

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Q: 4 35₂ In evaluating 2 dx X√√x²-4 is same as evaluating lim cos-¹(f(x)). Determine the value of f(x)…

A: We have to find the value of f(x) at x= 2.9.

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ (²) < 0. ƒ (x) has a root in [0]. If we…

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Q: -x² +1 Let f(x) be defined as f(x)={* - | 1–sin² x if x20 if x<0

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Q: Show that the function is continuous. (x) = 3x cos 3x

A: Show that the function is continuous. (x) = 3x cos 3x

Q: 2. Verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval then…

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Q: If f (x) = A (В B C D Stams. 0 sec²x tanx 1+1² Ite¹ 1+e tanx sec ³x 1+e -dt, then f'(x) equals tanx…

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Q: Determine whether f '(0) exists. {: x2 sin 4 if x + 0 f(x) = if x = 0 f '(0) does exist. O f '(0)…

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Q: prove that f(x) = sin*x + cos*x is a periodic function with periods

A: We use the Given Equations Prove that f(x) is a periodic function with period pie/2.

Q: Prove that for all x € [1, 00), – sinh (- |cosh¯'(x)|) = Vx² – 1.

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Q: 1. Sketch the graph of f(x) = cosx on 0x< and use your sketch to find the absolute and local minimum…

A: Lets make a table for y=cos xWe pick known values of x and find y for them. Then convert them into…

Q: Evaluate tan" (x) sec4x dx. 1 ¹__tan (m + 1)(x) + ___¹__tan (™− 1)(x) + C A. m+1 m-1 1 1 -tan (m…

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Q: Prove that sin x can be approximated by x − (x^3/6) within 0.01 on the interval [-1,1].

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

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Q: Evaluate tan" (x) sec¹x dx² A −tan (m+2)(x) + — tan” (x) + C m+2 1 −tan (m+3)(x) +_¹__tan (m+1)(x) +…

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Q: Let f(x) = 2x? cos(4r). %3D f'(x) = - 8x sin(4x) + 4x cos(4x) COS f'(2) = %3D f"(x) =|| 0 f"(2) =| |

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Q: Give an example of a (discontinuous) function that does not satisfy the conclusion of the IVT on…

A: Given: f(x)=sin1x    x≠00            x=0

Q: Evaluate using the result of Exercise 54

A: Given: limh→0cos3h-1h2 we know that 1-cos2θ=2sin2θ…

Q: Evaluate tan" (x) secx dx. 1 A n(m + 1)(x). m+1 1 (В -tan (m + 2)(x) + m+2 1 -tan (m+1)(x). m+1 1…

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Q: Suppose f(x) = π − 2x − cos x. (a) Show that f is strictly decreasing on R. (b) Hence, justify that…

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Q: Evaluate tan" (x) sec¹x dx. 1 1 (A) -tan (m+2)(x) +· ¹tan" (x) + C - m+2 m 1 1 (в -tan (m+3)(x) +…

A: On solving this we will get Option (B) is correct

Q: 19. Find the points of inflection on the graph of f(x) = x+2 cosx on the interval [0, 27].

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Q: Consider the function f(x)=2.5x−cos(x)+1 on the interval 0≤x≤1 The Intermediate Value Theorem…

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Q: Let f(x) = cos(x). (a) On [0,2n) solve f(x) = 0. 4.

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Q: 9. Differentiate y = cos(x3/2 In x), do not simplify. x2-4 Using Calculus, show on which intervals…

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Q: Find g'(x) given that g(x) = x° cos(Tx) os (πα)

A: Given,  g(x)=x3cos(πx)

Q: Evaluate tan tan" (x) sec¹x dx O 1 −tan (™+2)(x) + — tan” (x) + C m+2 m O 1 1 -tan (m + 1)(x) — — ¹…

A: To evaluate the integral

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: If f(x) = (cos(√) - In (x²))³, then f'(x) = 3

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Q: 1-Show that the function :defined by the base f(x, y) = x³ 3xy² - harmonic function

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Q: Evaluate tan" (x) sec4x dx. 1 1 (A) =tan(+3)(x) + _____tan ( + 1)(x) + C m+3 m+1 1 B) (m+3 -tan…

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Q: 3. (2') Let A+ cos x x 2. Given that q(x) is a continuous function, determine the value of A and B.

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Q: Find the points of discontinuity of cos () for |x| 1 Determine the type of discontinuity and…

A: Given: gx=cosπx2, for x&lt;1x-1, for x≥1 To Determine: Type of discontinuity and whether g is left…

Q: Given f(x) = |x| for x € [-n,1]| Show that cos(2n – 1)x |x| 2 \xl = -- - (2n – 1)2 n=1 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: For the function f(x)=sin(x)/x defined on [-1,1], there is some -1<c<1 such that f′(c)=0.f′(c)=0.…

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Q: 4. Find f'(x) if f(x)= In (2x) + cos (6x)

A: Given- f(x)=ln(2x)+cos(6x) To find - f'(x) Here we will use the following formulas-…

Q: Determine the extreme values of the function f(x) = 2 cos(x) –1 on the interval [0, 27].

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Q: If f(x) = A B C D tanx S 0 1+tan²x 1+1² Ite' -dt, then f'(x) equals 1+ e tanx In (secx)√√1+tan²x…

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

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

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

A: We have f(x)=cos(x)-3x+1,  x0=0.5, x1=0.6 then secant method formula:…

Q: 5. Find Oz дz дх and if x - z = arctan(yz) defines z implicitly as a function of x and y. ду

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Q: If f is the function defined by f(x) = x + cos 2x, then f"(x) = %3D -8sin 2x O -sin 2x sin 2x 8cos…

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Q: If f(x)=sin-'x. Show that (1-x2) fx(x)-xf (x)=0. HENCE ,prove that f^(n+2)(0)=n²f^n(0)

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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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