Exercise 3: Let f and g be twice continuously differentiable, and f(a) = f(b) = g(a) = g(b) = 0. Show that: f(2)g" (x) dæ = | f" (x)g(2) dx.

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: z = eY/x + 2 log(2x²y)

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Q: That the value of the integration g(1) = 00 , g(2) = 0 1-e t) -dt 9(t) where g is a function that is…

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Q: f(x.y) =e^sin(y) + In(xy) ; әз of Әхд-у

A: Given: fx,y=exsiny+lnxy To find the partial derivative of ∂3f∂x∂2y Partial differentiation: Partial…

Q: dw Find the value of- dt at the point (1, In 2 ,0) on the curve x = cos t ,y= sin t, 0 <t< 2n where…

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Q: Which of (a) or (b) is equal to d/dx (x sin t)?

A: We have to evaluate ddxxsint

Q: Find dy/dx in Exercises 1–6. 1. y = x/4 3. y = 7Vx + 6 2. y = V2r %3D 4. y = (1 – 6x)/3 - 5. y =…

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Q: What is the maximum value of f(t) = sin t + cos t in the interval 0<t< =?

A: Given; f(t) = Sin t + Cos t.

Q: In Exercises 55 and 56, find both dy/dx (treating y as a differentiable function of x) and dx/dy…

A: Since you have submitted two questions, we'll answer the first question. For the second question…

Q: 2) Find a value of c for which f (x) is a pdf on the f(x) = cx + x², [0. 1]

A: Introduction ∫-∞∞fxdx=1 The total area under the curve fx of pdf is 1.

Q: If f(x) = xe , x >0 and (Y = ln(X)), then the PDF of Y is %3D (In(1)² (In(y))e ye

A: Answer: ⊙ ey-12e2y

Q: Find d2x/dt? as a function of x if d 6x sin x. dt? ||

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Q: Determine whether the equilibrium at x = 0 is stable, unstable, or semi-stable. dx = 3x7 dt The…

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Q: | f(t) dt = x*e* e + a

A: This question is based on integration.

Q: psin(rx) (b) Let g(æ) 1 and f(x) g(t) dt. Find f'(2). x4 + 1

A: We have to use derivative under the functional limit formulae first then put x=2.

Q: 2x,x X, +X, Find the total differential dy, given

A: Introduction: On an open set S, let z=f(x,y) be continuous. Changes in x and y are represented by dx…

Q: Find the average of the function g (x) = 2 cos 2x on the interval [0, r] -T

A: The function f(x)=2 cos 2x on the interval 0,π

Q: f(2) = | : t2 – 1 dt 1+ cos² (t) At what value of ¤| does the local max of f(x) occur?

A: Given that a function f(x) and we find value of x does the local maxima of f(x) occurred. We know…

Q: 8. Let y -e=y-x²y. Find dy dxlx=1 be a function of x such that e*-y

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Q: g(x) dt + 1 %3D

A: To evaluate∫1x1t3+1dt

Q: (a) Consider the function f(1) = 1 on 0 <x < a. r+1 Find the value of a for which f(r) is a valid…

A: Here AS PER POLICY I HAVE CALCULATED FIRST MAIN QUESTION ONLY  PLZ REPOST FOR REMAINING PARTS WITH…

Q: F- e'さてi -xyでk is non. Conserv ative DTrue O Fulse

A: We have given a vector field ,  F(x , y , z) = x2yz i + 0j- xyz2 k We need to determine whether…

Q: Exercise 4: Integrate the following. dx (a) /= 16 х dr (b) / 5 + 2x4 et dx (c) / 1+ 2e2x cot x (d)…

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Q: Use the fundamental theorem of calculus and the chain rule to find (A(x)) where dx e2x A(x) = In(t?)…

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Q: b) f(x) = In(1 – 3x) %3D c) f(x) = cos(t³) dt

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Q: Let f(x) = sin? (x). Define F(x) | f(t) dt. What is F" -) ?

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Q: ) Find the linearization of f(x) = sinx at x = 0.(b) Compute dy if y = x2 +x and x changes from x =…

A: Since you have submitted two questions, we'll answer the first question. For the second question…

Q: Recall the chain rule from differential calculus: f (g (æ)) = f' (9 (æ)) · g' (æ). d dx Find de tan…

A: We have to find

Q: In integrating the function fx* in Sx dx using integration by parts, what would you choose for your…

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Q: Let y = tan x. (a) Find the differential dy. (b) Evaluate dy and Δy if x = π/3 and dx = -0.2…

A: Let,  y=fanxa To find differential dy, differentiate y       with respect to x,        dy=sec2x dx

Q: Suppose X and Y have the joint pdf f (x, y) = c(1.57 – x – y), x+y 0, y > 0 - a) Find the value of…

A: Introduction: The joint density function of two random variables X and Y is given below: (a). Find…

Q: Ejercicio. Evalúe [1?+1 dt dx Ejercicio. Derive F(x) = cos t dt 2

A: The objective is to solve the above questions.

Q: (a) f(x) = e" centered at r = 0.

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Q: Show that f(t) dt ) dx (x - a)f(x) dx.

A: To prove: ∫ab∫xbftdtdx=∫abx-afxdx

Q: d Use the chain rule to calculate f(x) if f(x) In (1+ tº) dt. dx, 0. f(x) =

A: let's use chain rule and find derivative of integral.

Q: Determine whether the equilibrium at x = 0 is stable, unstable, or semi-stable. dx 3 dt The…

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Q: Find the differential dy of each function: a) f(x) = 2x^1/3 - 3x^1/2 b) f(x) = ( x^2 +1) / (x+1)

A: we will find the derivative of given function using basic differentiation rules.

Q: Suppose that f(a) = f(b) = 0 and the second derivatives of f exist on the closed interval [a, b].…

A: Prove that ∫abx-ax-bf"xdx=2∫abfxdx.

Q: 3. Let f(x)= :/ arcsin(t) dt and g(x)= / arccos(t) dt. 0. 0. I f'(x) g' (x) dx =? 4 - T (a) 2 - T…

A: Given Applying derivative with respect to x: Similarly:

Q: Find the linearization of x = -1 f(x) = 3+ S sec(t – 1)dt at

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Q: Exercise 11. For what values of p is the function f(x) = { {JzP sin (!) x #0 x = 0 0 differentiable?

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Q: Using linear approximation, estimate e^0.1 cos(0.1)

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Q: If F(x)=[ cos(t² +1)dt , calculate using the fundamental theorem of calculus derived version.

A: Given : F(x)=∫-xxcos(t2+1)dt Aim: to calculate using the fundamental theorem of calculus ,the…

Q: F(x) = [,**dt COS x 1 %3D 19

A: Answer is given below

Q: 3 a) Find the exact integer value of a for which tsin (at)dt == ft sin (xt) dt = -². where a1. T

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Q: Does the fundamental theorem of calculus imply that e' dt = e*?

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Q: Let f(x)= | arcsin(t) dt and g(x)= | arccos(t) dt. | "'(x) g'(x) dæ =? 4 - T (a) 2 - T (b) 2 4 - T…

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