Mathematics Homework Help

NEED ALL STEP FORE EACH QUSESTIONS

(1) Find parametric equations for the path of a particle travelling clockwise around the circle centered

at (2, 3) with radius 5 starting at the point (2, ;2). Use a parameter t, 0 ≤ t ≤ 2π.

(2) Find dy/dx and d2y/dx2 in terms of t for the curve

x = t2 + 1

y = et t 1

For what values of t is the curve concave up?

(3) Consider the curve given by

x = 1 + sin t

y = et + t2

from t = 0 to t = 5.

(a) Write an integral representing the length of the above curve. You do not need to evaluate or

simplify the integral.

(b) Suppose we rotate the above curve about the line x = =2. Write an integral representing the

surface area of the resulting surface of revolution. You do not need to evaluate or simplify the

integral.

(4) Find a rectangular equation for the following curve given in polar coordinates.

r2 sin 2θ = 6

(5) Consider the curve given in polar coordinates

r = 2 + cos 3θ

(a) Sketch the above curve

(b) Find the equation of the tangent line to the curve at θ = π

6

(c) Find the area enclosed by the above curve.

(6) Consider the curve given in polar coordinates

r = eθ

(a) For what values of θ,0 ≤ θ ≤ 2π, does this curve have a horizontal tangent line?

(b) For what values of θ,0 ≤ θ ≤ 2π, does this curve have a vertical tangent line?

(7) Consider the curve

r = cos 2θ

(a) Draw a graph of this curve.

(b) Write an integral representing the lenth of one loop of this curve.