-.) a) Show: The substitution u(x) = sin(r) +y(x)+2 makes the differential equa- tion y'(r) + cos(a) = 5 (sin(z) +y(x) +2)“/5 %3D (*) separable. b) Soluo the

Q: f" (x) = cos(x), f(0) = -2, f' (0) = -2, f" (0) = 6.

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Q: Let y = sec (3x)V 361 cos (3x) + 646 + 289 sec (3x) , 0 <x< a²y Find at x= 2 12 dx Hint : Before…

A: SOLUTION-

Q: 1 lim h-0 2h sin(x²)dx = %3D -2h O cos(1) sin(1)

A: Using Fundamental theorem of calculus: ∫-2h0sin(x2)dx=sin(0)-sin((-2h)2)=0-sin(4h2)=-sin(4h2)

Q: 2. (a) This is correct. (b) This is correct. (c) This is incorrect. Because the differentiation is…

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Q: (a)Deduce the 1$, 20d and 3rd derivatives of y = cos²x.

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

Q: lim tanx i. x→ 0 lim In(1+ x)] 11. x→ 0 lim sin ax i1. where a and b are constants. r0sinbxr

A: We are only allow to to do first question and three subpart of question So if you need rest of…

Q: 1- csc x ... Differentiate: f(x) = sec x f' (x) = csc² a – sin æ ... f' (x) = – sin a – sec? x ...…

A: We have to find derivative

Q: (a) If (In a] = , evaluate dx d [In x²] dx

A: Since you have asked multiple questions in single request so we  will be answering only first…

Q: Compute the derivative of f(x) = ( sin(2x + 1))/°. 4 (sin(2x+ 3 1) -1/3 • cos (2x + 1) 4 -1/3…

A: Given

Q: (1) If y = sin(tan-x), prove that %3D (1+ x²)?y" + 2x(1 + x²)y' +y = 0.

A: Since you have asked multiple questions in single request.so we will answering only first question.

Q: Find g'(0) if g(0) = cos 0 sin 0. [A] cos²0 [B] cos²0 + 1 [C] cos²0 sin²0 - - [D] sin²0 [E] sin²0 +…

A: As we know that; The Product rule: f·g'=f'·g+f·g' Use the common derivatives.…

Q: Given that y(x)= sin( In 2x ). Which of the following formula is CORRECT to find y'(x). v(e) = (In…

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Q: ACTIVITY 1 : Practice ! Find the derivative as indicated : 1) у %3 = 3 cos x + 2 sin x ; find y' 3)…

A: given  we need to find the derivative of the following

Q: The derivative of f (#) cos(-2 a +a - a2 - 2 a +1) is O f (a) = (8 a° – 3 a? +2 a + 2) sin (-8 a + 3…

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Q: After applying variation of parameters to y " +y = sec(x), the following equations are obtained c{…

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Q: sinx+5* dy for y=In dx e3Xtan(x³) O a. Some other answer cosx+5*In5 3x²sec²(x³) - 3 tan(x³) b. dy dx…

A: We have given y=ln(sinx+5xe3xtan(x3)). Let us simplify right hand side using logarithmic…

Q: post office sells 4-cent sheep stamps and 2-cent sunset stamps. Cole bought 5 stamps spent exactly…

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Q: a2f Solve the second derivative forf (x,y,z) = e 2+y²) əx2 sin (x² + sin (x² +y²' sin -2e x2 sin…

A: We have to find the second derivative ∂2f∂x2 for fx,y,z=esinx2+y2z.

Q: 1. (a) Find L{t*e*t – e' cos v7t} (b) Find L{sin 2t sin 5t}. [ Recall sin Asin B = (cos(A – B) –…

A: Since you have asked multiple questions, we will solve the first two questions for you. If you want…

Q: a) Using the definition of derivative, calculate the derivative of the function of f(x) = /3x +1 (b)…

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Q: 1. Calculation of derivatives: Show that cos' z = – sin z, sin' z = cos z by using definitions (a)…

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Q: A rental truck costs $45.45 for a day plus 94¢ per mile. (a) If x is the number of miles driven,…

A: Given that, a rental truck costs $45.45 for a day plus 94c per mile.

Q: (2x) Sinx 2. Prove that (1) Jy₂ (x)=√√(x) =_ (11) J-y₂(x) = √(3/1²) Cosx

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Q: 2. Find g'(x) for g(x) = x cos-l-V1- r²+ez³ cos z. [Simplify answer as much as possible.]

A: topic -derivative

Q: (a) Calculate the differential df of the function f = tan(x). Use it to find an = π/4, approximation…

A: Part (a):First, we need to calculate the differential df of the function f = tan(x). The derivative…