Prove Theorem (A + B )T = AT + B T

Q: In Exercises 61-67,firstfind f' and then find f.

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Q: Use the First Fundamental Theorem of Calculus part one to find F'(x). F(x)=|, (t² +3t +2)dt J-3

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Q: Prove that flx)= (i+x)? + X and diverge Litx)? both

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Q: Determine whether the functions f1(r) = e2*, f2(x) = e-2" , and f3(x) = sinh 2x are linearly…

A: To determine whether the functions f1x=e2x, f2x=e-2x, and f3x=sinh 2x are linearly independent on…

Q: Fundamental Theorem: 2z Let f(x) = | Vcos t + t° dt . Find the value of f'(x). %3D 1

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Q: 21 (important) Find the fixed point for F(x) = ax + s. When is it attracting?

A: Given Fx=ax+s

Q: Solve for g'(x)?

A: Given:

Q: F(2) = (2t – 1)° dt using the Fundamental Theorem of Calculus.

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Q: Verify Young's Theorem y? x2 z = In(2x + y) + +e(x+2y) y

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Q: Let F (x) = (1 + t2)/4 dt - Then %3D F'(x) =

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Q: 2. Consider the vector-valued function R(t) = { Show that is continuous at t = 1. (el-¹, 1-², In (2-…

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Q: If g(x) J.J S. 1 -dt , find g'(2).

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Q: 41. Let r be a differentiable vector function of t. Show that if r•(dr/dt) = 0 for all t, then |r|…

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

A: This question is based on integration.

Q: d rsinx Catnx 2t3-t dt dx J4

A: Given Data The equation is d/dx(ꭍ4sinx (2t3-t))dt.   Solve the equation d/dx(ꭍ4sinx (2t3-t))dt by…

Q: The function f(x, y) = x2 – y² at the point (0,0) is saddle point. صواب İbi

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Q: Calculate the derivative using Part 2 of the Fundamental Theorem of Calculus. 29 de ²4 (2t5 - t) ²9…

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Q: 4. para t> 0, 0 <x < 2 u(0,t) = u(2,t) = 0 u(x,0) = x (2 – x), ди (x,0) = 0

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Q: Q2: Let f(z) = z² is f(z) satisfy Cauchy-Riemann equation ?

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: x - y -2x Prove by definition that for f(r, y) fy(x, y) = %3D x+ y' (x+ y)2

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Q: Differentiate. f(t)=  t2 in t

A: Given a function, f(t)=  t2 ln (t)

Q: Let H(x) integral ,for -2 ≤ x ≤2.Use geometry to evaluate H(2).

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: - (a x r) = a x r' for any constant vector a. dt Show that

A: To prove: ddt(a×r)=a×r'. Consider L.H.S. Differentiate with respect to t using product rule.…

Q: Show that the function u(x,t) = t/2e-*/4 (t > 0) satisfies u, = uxx.

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Q: Explain this calculus formula for acceleration a (t)=d/dt [v (t)] a (t)=d^2/dt^2  [x (t)]

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Q: 1 Find F'(0) given that F(x) = dt. In(t + 1) 2

A: F(x)=∫2x11ln(t+1)dt

Q: If f(x) I (t° + 2t + 6) dt 0. then %3D = (x), ,f

A: The function needs to be integrated first and then to the subsequent function we will be double…

Q: Let g(x) V1+ 8t dt. Use the Fundamental Theorem of Calculus to find g'(x). g'(z) =

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Q: Find f(t). s(s + 1).

A: L-1[F(s−a)]=eatf(t) From first shifting property, L{e^(at)*F(t)} = f(s-a)…

Q: Use Green's Theorem to evaluate F = (xy + y²)i+ (x – y)j | (1, 1) x = y² y = x² (0, 0)

A: we have to evaluate the following by using greens theorem

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: Find a value of x such that

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Q: (a) Prove that f (x) satisfies the properties of the Pdf 8. (i) fx (x) 2 0 for all x and (ii) Jfx…

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Q: 5. Let A(r) = In(t) dt. Using the Fundamental Theorem of Calculus Part 2, find A'(x). Show all %3D…

A: This question is related to calculus.solve in step 2.

Q: Assume that z = ƒ(x, y), x = g(t), y = h(t), ƒx(2, -1) = 3, and ƒy(2, -1) = -2. If g(0) = 2, h(0) =…

A: We use the partial derivatives to find dz/dt  Then plug x=g(t) and y=h(t)

Q: Verify Stoke's Theorem for F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection between z=1…

A: Given: F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection between z=1 and z² = x² + y².

Q: Find the linearization L(x) of f(x) = cot x at x = 4 %3D -

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Q: Prove that J3 (x) dx + J2 (x)+J, (x) = 0

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Q: 6. let h be the function defines by h(x) = 2e***dt, Find h'(2) %3D

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: ✓(3) Prove Green's theorem for F = Y Z + x²y³j y (1,1) i (1,0)

A: Solution of the given problem is below...

Q: 9. Explain why dt r1/I dt 1+t2 1+t? is valid for any x > 0.

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Q: Apply the first part of the fundamental theorem to determine the value of F'(x) if: e) F (x) = S…

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Q: Show that f(z) = x³+i(1−y)³ is differentiable only at x = 0, y = 1 and f'(z) = 0.

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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: 4. Let f € L'(R). Prove that the function F(x) = S f(t)dt is continuous.

A: Let, f∈L1ℝ. Since,  L1=f:ℝ→ℂ∫ℝftdt&lt;∞∫ℝftdt≤∫ℝftdt&lt;∞Therefore,∫ℝftdt&lt;∞ ∫ℝftdt is a finite…

Q: b) Let f(x) be the linear function so that the straight line y=f(x) passes through the points (0,1)…

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Q: F(z) = (2t – 1)° dt %3D sing the Fundamental Theorem of Calculus. (z) =

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Q: Let g(x) 8 In(t) dt. Use the Fundamental Theorem of Calculus tc g'(x) =

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Q: Find the linearization of f(x,y) at the indicated point (Xo.Yo). f(x,y) = In (x +y): (4, 5) L(x,y) =

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Q: 3. Use the Fundamental Theorem of Calculus I to find h'(x) if h(x) = S,(t7 – t/2) dt

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.