(a) (i) Use the substitution sin(u) = x + 3 to show that dx 12x + 7 -1 (x+3) V-2x² – 12x+7+c. sin - |-2x² (ii) In part (i), we could have used the substitution cos(u) = x+3 instead, but the form of the antiderivative would be different. Without doing any calculations, explain how this is accounted for in the final result. 1 (b) In a given situation, the rate of change of the variable x is proportional to V16 – t2" (i) Using k as the constant of proportionality, write down a differential equation that describes this situation. (ii) Find the general solution of the differential equation you wrote down in (b)(i). You will need to make use of a (simple!) trigonometric substitution while solving this equation.

Q: Four equations are provided below. Which equation follows from the Power Rule? I. (P)' = pr P 1…

A: Answer is option (a)  Because only equation 3rd follows the power rule .

Q: Differentiate the following: a) y=-x²+x-5 (b) y = 10 + 2 3 (c) y = 2√x - X

A: According to Bartleby guidelines we are supposed to do the first three sub parts of multiple sub…

Q: Find where z = sin (x²) cos (y) COS © 6.x sin (x²) cos (y²) COS 2.x sin (2²) 2x cos 2.x sin (x²) cos…

A:

Q: If f (x, y) = x(x² + y²)-3/2 esin(a²y) find fa(1,0).

A:

Q: 3. Find the derivatives: 3x3-4 a) f (x) = b) y = (x2 + 6x + 2) (+ Vx5 + 3) DO NOT SIMPLIFY 2x²+3 c)…

A: a) Use quotient rule of derivative and find the derivatives of above function.

Q: V sin2(x32 - 1) dx, with u x32 - 1

A: Consider the integral ∫(√x)sin2 (x3/2 - 1) dx.

Q: d (x2+2x)(sin(x)+5) 7. Find: dx Зx+2

A: To find:

Q: For which R is (cos(V), sin(v)) = {(x, y) : x² + y² = 1}. Briefly justify your answer.

A:

Q: Find dy clx In (sin(8x3)) a) y=(√√x2-1) In (sin( b) y = 1+ sin ² (3x)+sin 4 (3x) + sin 6 (3x)+.... +…

A: Since, you have posted multiple questions.As per guidelines we are supposed to answer only first…

Q: f(x) = 5 sin²(x) Hint: Use sin2(x) = -(1 – cos(2x))| %3D %3D n = 1 8.

A:

Q: For sin(u/2), cos(u/2), and tan(u/2) given cos(u)=-sqrt2/3 and sin(u)>0 I got…

A: Given  cos(u)=−23sin(u)>0

Q: Evaluate: 4S d sin(t?)dt dx x 3x²sin(x“) – sin(x²) b. 2x°cos(x°) – 2xcos(x²) a. 3x²cos(x^) – cos(x²)…

A:

Q: ) Write sin(2 sin-(3x)) in terms of x only. (i) 6x/1 – 9x² (ii) 6x (iii) V1 – 9x² - (iv) Not listed

A:

Q: (b) Verify that f(x) = 2 cos(x) sin(1x). f(x) = sin(6x) + sin(7x) = 2 sin = 2 sin 6x + N 2 X COS 6x…

A: Given

Q: If h(x) = In 2x+8 4 x 3 sin x find h'(x).

A: Given :- To find :-

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.