Q1. Use the Fundamental Theorem Of Calculus and the Newton-Leibnitz rule to answer the following: 1. Determine d •In(x) tan(vt)dt. dx 2. Calculate x10 In xdx.

Q: dy +y = zy dr

A: Given equation is: xdydx+y=xy3

Q: i) dt (1+t)3 j) S(sin?x cos x)dx x+3 k) S: -dx x2+6x-11

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Q: Find the differential of the given functions. a) f(x) = sin(cos(22)) CoS %3D 2x b) g(x) x3 – 7x

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Q: Evaluate the indefinite integral. 2 (t) – 7e' + 8 + 1 + 12 2 sin(t) – 8 cos(t) + sec- dt = +C.

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

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Q: Given f(z) = | t2 – 16 dt 1+ cos²(t) OS At what value of a does the local max of f(x) occur?

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Q: tan x dt+ 1+ t? (1+ f')* 1 -dt=D1 1/e Ve prove LHSis equal t RHS

A: This question is related to integration. We solve this question we divide this question in two part…

Q: Prove that sinh−1 t = ln(t + √t2 + 1 ).

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Q: Find dw/ dt at t = 0 if w = sin (xy + p), x = et, and y = ln (t + 1).

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Q: Differentiate g(t) = (at + 1)2 with respect to t. Assume that a is a positive constant. g'(t) =

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Q: 10. Use differentials to estimate to estimate sin(46°). vZ(n+180) А. D. VZ(T+180) 360 180 В.…

A: Let y = sin (x)

Q: B. If R = xyz’i+÷j – yz*k_find of the differential R, dR.

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Q: Calculate the following definite integrals. -1/2 1 dx "dx dt (t+2)³

A: As per the company guidelines we can solve first three parts. I hope that the given solution will…

Q: Given Lve f(t) dt = In(x) cos(x) + sin(3) :L.e- f(t) dt, find f(x). sin(3)

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Q: If r = sin ( ƒ(t)), ƒ(0) = p/3, and ƒ'(0) = 4, then what is dr/dt at t = 0?

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Q: For the functions w = 8x + 3y, x= cos t, and y = sin t, express dw as a function of t, both by using…

A: The given function is w=8x2+3y2, x=cost and y=sint. We have to express dwdt as a function of t.…

Q: 1. Use the Fundamental Theorem of Calculus to evaluate the following (DO NO 5x -6x dx 1 | cos (x)--…

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Q: Use at least two methods to prove that dr/dt=0 when r=√x2 + y2, x=α cos t, and y=α sin t if α is a…

A: First method   r=x2+y2r=(α cost)2+α sint2            since x=α cost and y=α sint  =α2cos2t+α2sin2t…

Q: (b) If dy/dt = 2, find dx/dt when x = dx dt

A: By differentiating we solve the function

Q: The tangent line to the curve y=f(x) is horizontal at where X = Xo f f(xx) = 00 A O f(xo) = 00 .8 O…

A: The tangent line to the curve y=f(x) is horizontal at where x=x0 if

Q: Find the general antiderivative of the given function. f(x)= sin (7y)+ cos (3y)

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Q: 1) f(t) =t cos(@t) 2) f(t)=tsin(@t) 3) f(t) =t cosh(at)

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Q: Use the Fundamental Theorem of Calculus to calculate the following derivatives: 4 d (a) d.r cos(4t²)…

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Q: 1. Find an antiderivative of f(x) = 16–x². Then, use The Fundamental Theorem of Calculus %3D 4 to…

A: Given : fx = 16-x2

Q: Find the linear approximation at x = 0 to each of f(x) (x + 1)?, g(x) = 1+ sin(2x) and h(x)= e2*.…

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Q: Use the method of first principles to show that d/dt(lnt)=1/t

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Q: Evaluate: L{t f -t e2t Cos t dt}.

A: Evaluate: Lt∫0te2t costtdt First of all we find    L(cost)=ss2+1L(e2tcost) =…

Q: Q1. Use the Fundamental Theorem Of Calculus and the Newton-Leibnitz rule to answer the following: 1.…

A: In the given question we have to solve the question (1) and (2) (1) Determine ddx∫exln(x)tan(t)dt…

Q: Evaluate dt dW at t=0 for the function W = e' – In x , x = t², y= In t

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Q: 3. If x = (sin t+cos t), y = (cos t-sin t), (1 < t < 4), evaluate ip, dt. dt dt

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Q: Given y = 4e* + In x, find the differential dy. %3D

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Q: 2s Prove that L(t sin t) (s²+1)²

A: We can use the concept of Laplace transform in order to solve the given problem.

Q: 6) g(z)= . cos(t) dt

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Q: tanh (22) d) evalvare J seen 2 (2n ) e レッ using a substitution Suitable

A: To evaluate the given integral.

Q: Q1. Use the Fundamental Theorem Of Calculus and the Newton-Leibnitz rule to answer the following: 1.…

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Q: Find the differential of the function. Z = e-9x cos(2rt) dz = dx + dt %3D

A: The total differential of the function  is given by . So, to find , first we need to find the…

Q: 10. Find the indefinite integral. Te cos 3t dt dv u = du = v =

A: Given integration is : ∫e9tcos3t dt   To find the value of integration.

Q: Consider the function g(x) = 5 S cos t dt – x?. Then g' (x) is

A: Given gx=12∫x4x2costdt  -x2

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: Show that g(x) = cos 2 =x+3 ™K(x, t) g(t) dt 0 || sin x cos t, 0≤x≤t Y sin texr

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Q: (e) S x³ cos(x²)dt (hint: use substitution first) VT/2 COS

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Q: Use the second fundamental theorem of calculus along with the chain rule to find d sint dt dx 1 o…

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Q: If r = sin (ƒ(t)), ƒ(0) = pai/3, and ƒ′(0) = 4, then what is dr/dt at t = 0

A: Given:

Q: 15 Evaluate the integral (10 + sin t)" cos t dt using the substitution u = 10+ sin t. + sin t) cos t…

A: We are asked to evaluate the integral.

Q: 18. The equation of the tangent line to 2² sin T # 0 f(x) = 0 x = 0 at r = 0 is 1 (a) y = -x (b) y =…

A: Note:Since you have asked  multiple questions ,we will solve first question for you.To get remaining…

Q: Q2: find dw/dt at given value point for the following function x= cos°(1), y = sin²(1).= =-. 1 at t…

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Q: Using Part I of the Fundamental Theorem of Calculus, 8 d evaluate: dx t7 dt =

A: Given that ddx∫x8t7dt

Q: Q: If In/x2 +y2 =sec(), use implicit differential to proof the following: - dy (²+y*). sec?. tan?…

A: lnx2+y2=secyx

Q: 11. Evaluate the definite integral S" dx. x+a?

A: Topic:- definite integration