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A curve f(x,y) is defined by sin(3y)+3ye^(-2x)+2x^2 = 5. Find dy/dx

In questions where we have a function of x and y equal to a constant, we need to find dy/dx indirectly.We use the formula (df/dx) + (df/dy)(dy/dx) = 0So all we do is differentiate each term in the functio...

Answered by Lewie W. • Maths tutor

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Consider the closed curve between 0 <= theta < 2pi given by r(theta) = 6 + alpha sin theta, where alpha is some real constant strictly between 0 and 6. The area in this closed curve is 97pi/2. Calculate the value of alpha.

Student uses the definition of area [A = 1/2 integral r(theta)^2 d theta], and proceeds using standard integration techniques to give a quadratic solvable for alpha. [alpha^2 = 25] Thus, alpha = 5.

Answered by Graham C. • Maths tutor

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The curve C has equation 2yx^2 + 2x + 4y - cos(πy) = 45. Using implicit differentiation, find dy/dx in terms of x and y

2x²y + 2x + 4y - cos(πy) = 45Applying implicit differentiation:4xy + 2x²(dy/dx) + 2 + 4(dy/dx) + πsin(πy)(dy/dx) = 0Moving all (dy/dx) terms to one side:2x² (dy/dx) + 4(dy...

Answered by Prahlad M. • Maths tutor

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Find the integral of [ 2x^4 - (4/sqrt(x) ) + 3 ], giving each term in its simplest form

We begin by rewriting it in a more workable form: 2x⁴ - 4x^-1/2 + 3. Indices are easier to integrate than fractions.Now, we integrate each term separately. The first term is 2x^4...

Answered by Prahlad M. • Maths tutor

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Express the following in partial fractions: (x^2+4x+10)/(x+3)(x+4)(x+5)

(x^2+4x+10)/(x+3)(x+4)(x+1) = A/(x+3) + B/(x+4) + C/(x+1) [A(x+4)(x+1) + B(x+3)(x+1) + C(x+3)(x+4)]/(x+3)(x+4)(x+1) = (x^2+4x+10)/(x+3)(x+4)(x+1) [A(x+4)(x+1) + B(x+3)(x+1) + C(x+3)(x+4)] = (x^2+4x+10) A(...

Answered by Alessandro S. • Maths tutor

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