Calculus Answers

Suppose that the following summation notation represents a geometric series: sum j=1 ^ n a If the ratio of this series is r, which of the following represent the sum of the series? Check all that apply

A helicopter takes off from a point 80 m away from an observer located on theground, and rises vertically at 2 m/s. At what rate is elevation angle of theobserver’s line of sight to the helicopter changing when the helicopter is 60 mabove the ground In the answer of this question there is (when t=30 cos=4/5) i did not got this point sir can you please explain it .it will really help me

Y= x . X^1/2 - 2/x^2 . Find the first derivative

The owner of a large house decided to spend some money making it more secure. An expert analysed the situation and said that for $5000 the security rating, R, of the property would rise from its current score of 30 points to 100 points. Every $500 spent after that would lift the rating by 5 points.
The owner feels that the more security devices he pays for the more tedious he is going to find it to enter and leave his property with all that he will have to remember lock/unlock. He feels there is an owner convenience rating, C, which will go down by 2 points, from an initial 100 points, for every $500 he spends over the $5000 that he accepts is necessary.
As both the security rating and the convenience rating interest him he decides to multiply them together to form the secure but not inconvenient rating, Z.
Clearly showing your use of calculus, determine how much the owner should spend on security to maximise Z.

Evaluate the following integral if D is the region bounded by xy=a, xy=b, xy^1.4=c and xy^1.4=d, where 0< a < b and 0< c < d. ∫ ∫ D x^8y^10 dA. Note: D should be at the bottom of the second integral, it is not part of the function.

Set up a triple integral that represents the volume of a cylinder of radius R and height h in (a) Cartesian coordinates, (b) cylindrical coordinates and (c) spherical coordinates.

Evaluate the following integral if n is an integer and D is the region bounded by x^2+y^2=a^2 and x^2+y^2=b^2 where 0< a < b. ∫ ∫ D (1) / [(x^2+y^2)^(n/2)] dA. What happens to your answer in the limit a→0^+? Note: D should be at the bottom of the second interval, it is not part of the function.

Suppose f(0)=2, f’(0)=7, f(1)=5, f’(1)=8, g(1)=1, g’(1)=-9 and H(x)=f(ln(x))+ln(g(x)). Find dH/dx, x=1
What is H’(1)?

Suppose g(x)=ln(ln(ln(f(x)))), f(9)=A and f’(9)=B Find the derivative dg/dx, f=9
What is g’(9)?

what is the domain of 1/x and domain of x^2-x-6