Calculus Answers

A spring is such that it would be stretched 8 in by a 20 lb weight. Let the weight be attached to

a spring and pulled down 5 in below the equilibrium point. If the weight is started with an upward

velocity of 4 flt/sec, describe the motion. No damping force or impressed force is present.

Find by double integration the area of the region in 𝑥𝑦 plane bounded by the curves 𝑦 = 𝑥

2 and

𝑦 = 4𝑥 − 𝑥

2

.

Find the average value of 𝑓(𝑥, 𝑦) = 𝑥

2𝑦 over the region 𝑅 which is a rectangle with vertices

(−1, 0), (−1, 5), (1, 5), (1, 0).

∬(𝑦 + 𝑥𝑦

−2)𝑑𝐴

𝑅

, 𝑅 = {(𝑥, 𝑦)|0 ≤ 𝑥 ≤ 2, 1 ≤ 𝑦 ≤ 2}

a) Define differentiation and integration in calculus. Also write down the

differences between them.

b) Write down some application of Calculus in CSE.

c) Describe geometrical meaning of definite integral with figure.

Find ∂ϕ

∂x

∂ϕ∂x where ϕ=x

2

+y

2

Show that the curve with parametric equations

x = sin t and y = sin(t + sin t) for 0 ≤ t ≤ 2π

The area A of a circle is increasing at a constant rate of 2cm2s−1. If the area of the circle is given by A = πr2, what is the rate of change of the radius when the radius is 4cm?

The area A of a circle is increasing at a constant rate of 2cm2s−1. If the area of the circle is given by A = πr2, what is the rate of change of the radius when the radius is 4cm?

1. Consider the function y = x2 + 3x + 5. 2x−3

(a) Determine the domain of the function.

(b) Determine the range of the function.

(c) Determine the intercepts of the function.

(d) Find the asymptotes if they exist.

(e) Find the turning points (if they exist) and determine the type of turning points they are.

(f) Manually sketch the graph of the function.