Find 'a' and 'b' such that f is differentiable everywhere: f(x) =  (cos x, x < 0           ax + b, x ≥ 0)

Q: The set of functions {sin nx}, n = 1, 2, 3, ... , is %3D orthogonal on the interval [ -n, n]. Show…

A: Given- The set of functions sin nx, n = 1, 2, 3, ⋯, is orthogonal on the interval -π, π To Show-…

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Q: Find the absolute extrema of the function on the closed interval. y = 5ex sin x, [0, π]

A: Solution:-Given functiony=5exsin x in [0,π]∴f(x)=5exsin xf'(x)=ddx5exsin x∴f'(x)=5ex cos x+5ex sin…

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Q: Differentiate. f(x) = ex sin(x) + cos(x)

A: The answer is found in step-2

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A: Given Data The first equation is x=cos θ+ θ sin θ. The second equation is y=sin θ- θ cos θ.…

Q: Differentiate : a. y= x^2 sinx +sin^2x + cos^2x + tan 2x/sin2x/cos2x b. y= In √x^3

A: a) y=x2sinx+sin2x+cos2x+tan2xsin2xcos2xb) y=lnx3

Q: x sin if x # 0 Show that the function f(x)= is differentiable at x = 3D0 and f '(0)= 0. if x = 0

A: A  function is  differentiable at  point c if L f'c=R f'cwhere L f'c=limx→c-fx-fcx-c   =left hand…

Q: Q2/ Find f (x) : 1- f(x) = 4 cos x + 2 sin x 2- f(x) = cos x - x csc x 3- f(x) %3D %3D cot x 1+cscx…

A: As per guidelines we can solve only one question at a time. So the first question can be solved as…

Q: Use Linear Approximation to estimate Aƒ = f(3.02) – f(3) for f(x) = sin(). (Use decimal notation.…

A: Given: fx=sinπx16 We need to estimate ∆f=f3.02-f3 using Linear Approximation.

Q: Limit as x approaches 0 1-cos(x)/sin(x)

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Q: In the interval [0, pi], find the absolute extrema of: g(x) = ex sin x - sin x + cos x

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Q: sin (2x) cos (2x) csc (x) 2. Determine the limit of · xVx4 as x approaches 0

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Q: Explain why the function f(x)=cos.x-xmust have at least one zero on the interval [0,5]

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Q: Suppose f is a differentiable function with f(1) = 1 and f'() = -1. Let y = f(x). tanx + ex. cosx.…

A: According to question,Given Function,y = fx.tan x + ex.cos xfπ = 1andf'π =- 1

Q: find f'(x). 5x(x+2 tan(x)) sec (x) Find f'(3). f(x)= 5x² tan z sec z

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Q: What is special about the functions f(x) = sin x - 1 x + 1' x 2 0, and g(x) = 2 tan Vĩ? -1 Explain.

A: Given that functions are f(x)=sin-1x-1x+1, x≥0 and g(x)=2tan-1(x). The graph of the function…

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Q: Integrate to find F as a function of x, and demonstrate the Second Fundamental Theorem of Calculus…

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Q: Differentiate. f(x) = 4 sin(x) − 7 cos(x)

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Q: Find f" (x), given f(x) = sec x. O f" (x) = sec³ x tan x Of" (x) = sec a tanz Of" (x) = sec³ x + sec…

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Q: Let f : R→ R be a differentiable function. Then for all x € R, the derivative of f ( sin (4x)) with…

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Q: Find the absolute maximum value and absolute minimum value of the function f(x) = x- 2 cos(x) on the…

A: fx=x-2cosx   ,-π2,π2 We need to find the absolute maximum and absolute minimum value of the function…

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A: Given function is g(x)=cosx and h(x)=x/2

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A: Given :f(x)=x3-sin2x+1=22.688

Q: Show that the function is continuous on its domain. State the domain. Vx + x cosx

A: fx=x4+x3 cosx The domain is the set of all positive real numbers including zero. That is, the domain…

Q: Find all critical point of the function f(x)= sin^26x+cos6x , 0<x< pi/3

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Q: Sketch flx) = 2 cos x-Sin 2x , osx<2o

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Q: Find any points of inflection (as ordered pairs) on the graph of f (x) = x+ 2 cosx in the interval…

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Q: Find the absolute maximum and minimum values for f (x, y) = 4 sin(x) + 9 cos(y) on the rectangle R…

A: Given that: fx,y=4sinx+9cosy Limit: 0≤x≤2π, 0≤y≤2π We know that if a,b is a critical point and…

Q: Differentiate the function. f (x) = sin x · In(5x) O f' (x) = cos · In(5x) sin z – 257 sin a O f'…

A: Proceed as shown

Q: Find where w= cos z - cos x cos 2y + sin x sin 2y and z x+2y. Me.

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Q: The derivative of sec² 2x (cos² r - sin² x) with respect to a is O2 sec 2x tan 2x O4 sec² 2x cot 2x…

A: Find the derivative

Q: Differentiate the function. f(x) = sin(x) In(3x) sin (x) f '(x) + cos (x) In (3x)

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Q: At which points does f(x) = 2 sin x - cos 2x have an absolute maximum value and what is the maximum…

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Q: Use Linear Approximation to estimate Af = f(3.02) – f(3) for f(x) = sin(). (Use decimal notation.…

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Q: Of of Let f(x, y) = x°y² tan y Then x +y. equals

A: The derivative of a function of two or more variables with respect to one variable, while the others…

Q: Find the absolute maximum of the function. fAx) = sin(6x) + cos(6x) on the interval 0. 18 %3D

A: Given: f(x)=sin(6x)+cos(6x) Differentiae the function.…

Q: Find the absolute extrema of the function on the closed interval. y = 9 cos x, [о, 2л] (x, y)…

A: Given,

Q: (b) Verify that f(x) = 2 cos(x) sin(x). f(x) = sin(7x) + sin(8x) 7x + 7x = 2 sin 2 = 2 sin |co(-)

A: Let the function is  f(x)=sin(7x)+sin(8x)   Formula used: Sin(a)+sin(b)=2sina+b2cosa-b2

Q: Find a value of c that makes Continuous extension the function 9x - 3 sin 3x 5x f(x) = c, continuous…

A: The given function is

Q: find the intergral of f(x) = sin(x) + cos(2x) on the interval [0,pi]

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Q: sinx +cos, x 0 Q4) Show that the function f(x)= is continuous at x 0 2, x 0 Q5) Determine the value…

A: See the attachment