Find the following for the given equation. r(t) = ( + 6t)i + (t² – 6t)j %3D (a) r'(t) = (b) r"(t) = (c) Find r'(t) - r"(t).

Q: Write the expression as a single Integral 1. f f(x)dx + f(x)dx =, 2. S f(x)dx – , f(x)dx = 3. f…

A: Use the property of the definite integral; 1). ∫abfxdx=-∫bafxdx 2). ∫acfxdx=∫abfxdx+∫bcfxdx   1).…

Q: ||r(t) × r"(t)|| Use the formula k(t) to find k(t). 13 r(t) = t'i+ t3j K(t) =

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Q: Find all relative extrema of the function. Use the Second-Derivative Test when applicable. (If an…

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Q: Find the Inverse Laplace Transform of the following functions: 16 a. L4. s(s2-14) b. L4. (25-32)2

A: we have to find       L-1{16s(s2-14)}   we can re-write the function 16s(s2-14) =1614( -1s +ss2-14 )…

Q: 10 Find div F and curl F if F(x, y, z) = xz°i+ 5y"x²j+ 4z?yk. div F= curl F=

A: Solution:

Q: 1. Do the integration by substitution for the following integral and find the value of definite…

A: We have to find the given integral value.

Q: Find V · (V × F), if F(x, y, z) = e2²i+ 7xe'j – 2e²k. - V·(V × F) =

A: Solved below

Q: Let I be the flux of G = (2e", 4x°e*“, 0) through the upper hemisphere S of the unit sphere. (a)…

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Q: Find the curvature of the plane curve y = -t at the point t = 2. к(2) -

A: Given:y = -t4differentiating with respect to t, we get:            y' =     d(-t4)dt…

Q: For each of the following series determine if the series converges or diverges. 00 In(n) Σ n n=2 n3…

A: The given series,1. ∑n=2∞ln(n)n2. ∑n=0∞n3n4+1

Q: Determine the volume generated when the area bounded by the curves y = sin x and y = cos x from x =…

A: Given problem:- Determine the volume generated when the area bounded by the curves y = sin x and y =…

Q: 2. Prove that the p-series converges for p > 1.

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Q: cos (0 + 11) y = Use symbolic notation and fractions where needed.) dy -4 cos (0+11) sin(0+11)| do

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Q: e2x - e 2x 3. lim, 0 sinx

A: To evaluate the given limit.limx→0 e2x-e-2xsin x

Q: Find the particular antiderivative of the following derivative that satisfies the given condition.…

A: I have stated the used formula in the solution

Q: Area A = 1,427 Calculate the definite integral f(x) dx by referring to the Area B = 2,460 Area C =…

A: Here we will find the definite integral,

Q: Find div F and curl F if F(x, y, z) = 7y°z*i – 11xz'j – 6xy°k. 5.7 div F= curl F=

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Q: (2) Prove that for any integer L () NTX / sin? dx = L. L -L

A: ∫ cos(ax) dx = 1a sin(ax) +C

Q: b) Use L'Hôpital's Rule to evaluate the limit L. e2x lim X→ ∞ x²

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Q: Find curl (2z i + 3x j + 6y k). curl(2z 2z i + 3x j+ 6y k) = ( i )i + ( i j + ( i )k

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Q: P(x) Let P(x) and Q (x) be polynomials with the degree of Q (x) greater than the degree of P (x) and…

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Q: A swimming pool is circular with a diameter of 40 feet. Depth is constant along east-west lines and…

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Q: If 1 f(x) 1 .3 + 1 + x 4 X° + 3 then find f'(0) +

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Q: Describe the graph of the equation. r= 5 cos t i - 18 sin 1 j +k O An ellipse in the planex = 1,…

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Q: 2. Compute Ja F · ds for F c(t) = (t , In(1+ t²), e*) for 0 <t < 1 VV, where V(r, y, z) = 4x² In(1+…

A: Find your answer below

Q: k(u) is a degree 3 polynomial with k(0)=-21.24 and zeros 4 and 3i. Find an equation for k with only…

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Q: Describe the graph of the equation. r = 4 cos t i – 14 sin t j + k O An ellipse in the plane z = 1,…

A: Here, the given parametric equation is: r=4 cos t i-14 sin t j+k⇒r(x(t),y(t),z(t))=(4cos t, -14 sin…

Q: Evaluate the integral. (Use C for the constant of integration.) 2 cos (a) da sin(a)

A: Given query is to find the value of the integral.

Q: For the curve given by r(t) = (5 sin(t), –2t, – 5 cos(t)), Find the unit tangent T(t) = (| Find the…

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Q: Write the expression as a single Integral -3 1. Sf f(x)dx + S, f(x)dx = -3 - 3. S f(x)dx + S, f(x)dx…

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Q: Consider the following statements. 1-If B is a symmetric matrix of 9x9 so the elements b32 and bz3…

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Q: Find curl (5z i + 6x j + 3y k). curl (5z i + 6x j + 3y k) = ( i )i + ( i i )k

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Q: Recall the geometric series 1/(1 − x) = 1 + x + x^2 + x^3 + x^4 + . . . for |x| < 1. Determine the…

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Q: 6 + (-3) + 0 + ..+ 411 = 97 (a* + 5) - 0 k = 1

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Q: 16. Find the volume of the region that lies under the graph of the paraboloid z = over the rectangle…

A: Here, to find the volume of the paraboloid z=x2+y2+2 over the rectangle R=(x,y)∈ℝ2|-1≤x≤2, 0≤y≤2

Q: 2. Find the area of the parallelogram whose vertices are given by: P: (1,2, –1), P2 = (4, 2, -3), P3…

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Q: Evaluate the integral: (5si + 9s²j + 10k) ds Answer : i+ j+ k

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Q: 8. Relationship Between Continuity and Differentiability: (a) Justification needed): If lim…

A: To discuss the  relationship between continuity and differentiability.

Q: dy Compute the derivative y dx y = 5x4 – x-4 (Express numbers in exact form. Use symbolic notation…

A: Given,y=5x4-x-4  Formula: If, y=xn then, y'=nxn-1

Q: Calculus solve by integration by parts

A: Our Aim is to evaluate the integral by integration by partsI=∫xx+13/2 dx  -(i)

Q: Find curl (3z i + 4x j + 6y k). curl (3z i + 4x j + 6y k) = ( i )i + ( i )j + ( i )k

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Q: Given L f(x) dx = 16 and i g(x) dx = 8, Evaluate the following I [2f(x) - 3g(x)] dx

A: Given query is to find the value of the indicated integral.

Q: Describe the graph of the equation. r = 15 sin 5t i + 15 cos 51 j O Itis a circle of radius 15 in…

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Q: Consider the functions. f(u) = u³ + u g(x) = cos (x) Find the following function and derivatives.…

A: The given functions,fu=u8+ug(x)=cos(x)

Q: 8. The ideal gas law PV volume, and temperature of a confined gas, where n is the number of moles of…

A: We need to prove the statement.

Q: Find V· (V x F), if F(x, y, z) = 7e"*i+ 9xe"j – 8e"²k. V· (V × F) =

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Q: 17-20 Express the limit as a definite integral on the given interval. 17. lim sin x, Ax, t, [0, 7]

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Find div F and curl F if F(x,y, z) = x*i – 7j + yzk. - div F= curl F=

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Q: The function ƒ (x) = |x| is continuous at every real number. %3D False True

A: Let's find.

Q: ||r'(t) x r"(t)|| Use the formula k(t) to find k(t). 13 r(t) = t’i+t5j K(t) =

A: We have to find k(t)