Calculus Answers

Express the following in Sigma notation:

0 + 1 + 2/4 +3/9 + 4/16 + 5/25...

2 + 4/3 + 6/6 + 8/9 + 10/12 + 12/15...

Determine whether each of the following converges or diverges
i)∞
∑ 1/k(k-1)(k-6)
k=10

ii)∞
∑ sin k/ k^2
k=1

iii) ∞
∑ 1/2k^2+k
k=1


iv)∞
∑ 1/(ℓn k)^k
k=2

v) ∞
∑ k^k/ k!
k=1

vi) ∞
∑ (4k-5/2k+1)^k
k=2

calculate

i)lim x-tan x/ x-sin x
x→0

ii) lim 3x^2-2x-1/x^2-x
x→1

iii) lim e^x-1/ ℓn(x+1)
x→0

iv) lim x^x
x→0+

Find the Taylor’s Polynomial of degree n about x = c and the remainder for the function f given

i) f(x)=√(3-x) and c=1

ii) f(x)=2x^4 + 3x^3 + 4x^2 + 5x + 6, n=3 and c=1

iii) f(x)=cos x and c=0

Determine the convergence of:
∞
i) ∫ 2x / 1+x^2 dx
-∞

∞
ii) ∫ x^-3 dx
1

A vertical container with base area measuring 14 cm by 17 cm is being filled with identical pieces of candy, each with a volume of 50.0 mm3 and a mass of 0.0200 g. Assume that the volume of the empty spaces between the candies is negligible. If the height of the candies in the container increases at the rate of 0.28 cm/s, at what rate does the mass of the candies in the container increase?

I = integral x^2 ln x dx

Find the area of thebounded plane region R lying between curves y=x^2 - 2x and y= 4-x^2

Find the area of the region bounded by the graph of curve y-x=6 , y-x^3=0 and 2y+x=0

Find the area of the curve from the following equations:
7x - 9y + 63 = 0
2x + y - 2 = 0
11x + 9y -99 = 0