Calculus Answers

Verify that y = e2xsinx is a solution to the differential equation d2y/dx2 - 4dy/dx+5y=0

Suppose a particle P is moving in the plane so that its coordinates are given by P(x,y), where x = 4cos2t, y = 7sin2t. (i) By finding a,b ∈ R such that x2 /a2 + y2 /b2 = 1, show that P is travelling on an elliptical path. (ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t). (iii) How fast is the distance between P and the origin changing when t = π/8?

Verify that the given family of functions solves the differential equation.
i. dy/dt = (1 - 2t)y² , y = 1/( C - t + t²)
0
ⁱⁱ· dy/dt = y²sin(t) ,y = 1/(C + cos(t))

Evaluate the integral ∫ (2xᶟ- 4x- 8) / (x⁴- xᶟ + 4x²- 4x) dx

Verify that y = e² ͯ sin x is a solution to the differential equation
d²y/dx² - 4 dy/dx + 5y = 0

Find the volume of the solid generated by revolving the region bounded by the curves
y = x² and y = 4x - x² about the line y = 6.

Differentiate the following functions with respect to x:
(i) ln(1 + sin² x)
(ii) x ͯ

Evaluate the integral ∫¹ₒ [x²⁄(√4-x²)ᶟ]dx

Verify that the given family of functions solves the differential equation.
(i) dy⁄dx= (1-2t)y², y=⅟(C – t + t²)
(ii) dy⁄dx= y² sin t, y=⅟(C + cos t)