1) By using the fundemantal theorem of calculus to find f(x) where d. Vi+3 dr and use f(t) =/? +3 dx f

Q: 1. If f(x) cos(t) dt, then f'(x) =

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Q: Find constants c1 and c2 such that F(x) = c1x sin x + c2 cos x is an antiderivative of f (x) = x cos…

A: To find constant c1 and c2 such that Fx=c1 xsinx+c2cosx is an antiderivative of fx=x cosx. Solution:…

Q: If f is continuous and f(x)dx=10 on the interval [0,4] find xf(x^2) dx on the interval [0,2]

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Q: If F(x)=sinx+6x2−4 is an antiderivative of f(x) then f(x)=

A: If F(x) is antiderivative of f(x) , then f(x) is derivative of F(x) .

Q: 2. If f (x) = Sn* et? 0, dt, find f'(x) V2t+sect

A: The solution is given as

Q: 9. Evaluate f, L" x²y²z dz dy dx ху Hw

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Q: Suppose that f(x) has a continuous derivative, (T/2 f(-т/2) — 0, f(T/2) = T/4 and dx = 3. %3D 7/2…

A: Given that:

Q: Apply the Fundamental Theorem of Calculus to get the following derivative. d In (t² + 1) dt dx 1

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Q: If f(æ) = S Tm dt for x > 1, 1 | In t find f'(2).

A: Please refer the attached image for complete solution.

Q: Q2: Find dyidx for the following function:- (1) y = In(x + 4)-xtan (4) y =x*+1) (2) xy + x*y = x +…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 4. Find if y is a function of r defined implicitly by da 2*y + sin a = xe" + In

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Q: Let f(x)= 9 sin x/ 2 sin x + 4 cos x Then f'(x)= The equation of the tangent line to y=f(x) at a =…

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Q: Use Part 2 of the Fundamental Theorem of Calculus and the following formula to find the derivative.…

A: The given formula is  ddx∫ag(x)f(t) dt=f(g(x))g'(x) We have to find the value of  ddx∫2sin…

Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

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Q: dy Prove that dx of x2 +2x is f'(x)=2x + 2

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Q: If; of t = x then at t=1 is (Note that these functions in N) dx

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Q: Q2: Expand f(x,y) = [sin (x y³)]² about (5, 1).

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Q: dy + ydx %3D 3x²y?dx,

A: Given differential equation is bernaulli's differential equation.,  and we can solve it by reducing…

Q: 2. Show that the function u = p(xy) + VTÝ v(y/x) provides the equation „Pu dx² 0. dy²

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Q: Find the slope of the tangent line to the graph of x²-Iny=0 at x=2 O None of these O 4e4 O e4 O 4

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Q: Calculate the derivative e10x de L9 In (t)dt using Part 2 of the Fundamental Theorem of Calculus.…

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Q: 1. Find the derivative (dy/dx) of the following functions: In a) y = 3x7- 7x³ +21x2 In b) y = (x2 +…

A: As per the guidelines, we are supposed to answer only one question at once. Kindly repost the…

Q: Find the derivative of the given function at P. (3, 3, 3) in the direction of u = 3i + 4j f (x, y,…

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Q: d Use the Fundamental Theorem of Calculus to find the derivative. dx dt 1+ vE

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Q: 9 Evaluate the integral f 2x (x² + 1)° dx using the substitution = x² + 1. и — с 2.x (x? + 1)° dx +C

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Q: If A(x)= f(t)dt is increasing on [0,10], then f(x) must be increasing on [0,10].

A: Given,A(x)=∫0x f(t) dt

Q: 1. If F(x)= = cos a sin r (√t-1)' dt, find F'(x).

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Q: Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)= ].…

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Q: 5. Given F(x) = cos (t) dt, determine F'(x). 2

A: Using leibniz rule  Explanation is given below....

Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

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: If f =x-e" and F=x'yi-2y'zj + xy'z'k, find: div(F x grad(yf )) at (4,1,-2)

A: The function                     Then, we have to find

Q: Let f(x, y) = ln (3x + y2). Show that fxy = fyx

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

A: Given  C'x=2x2-3x   ...1C0=4000   ......2

Q: Show  that the function Z =1/2(eysinx - e-ySin(x) satisfies the La Place’s equation  =0

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Q: If f(x, y, z) = In(x³ + yl² + 2®) + V (x12 + y6 + 23), then find fr, fy and f; at (1, 1, 1).

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Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

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Q: Let f (x) = 2x* – 3x3 + 5x + 1. Find the value of at x = 1. %3D df-1 - dx

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Q: Use Part 2 of the Fundamental Theorem of Calculus to find the derivatives. d/dx ∫_0^x▒〖e^√t dt〗…

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Q: ** If g(x) = S [e* - In(2t + 1)]at , find g"(x) dx ex. (2x + 1)2 A ex – In(2x + 1) 4 ex + (2x + 1)2…

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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: The derivative of y with respect to given a = 1 is Onone of the choices O y = yx³-1 O V = 0 = = 9 *…

A: I have stated the used formula in the solution

Q: if f(x)=integral t^3+5t^2+6, from 0 to x. what is f''(x)=?

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Q: (3) If | f(x) dx exists and c is a real constant, prove that f(2) dz = / f(x + c) dz. | {(x+c) Vel

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: 10 Evaluate the integral f 2x (x2 + 4) " dx using the substitution u = x² + 4. 2x (x² + 4)*" dx =|…

A: The given integral ∫2xx2+410dx

Q: Prove that fxy fy x for the following: %3D f = e + x Lny + y Lnx f = xy 2 3 + xy4 f = Ln ( 2x + 34)

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.