Calculus Answers

A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon

Check whether the function f(x,y)={4x^2y/4x^4+y^2 ,(x,y)/=(0,0)
{0,(x,y)=0 is continuous at(0,0)

2.1 Fragments of a dinosaur skeleton are discovered and dated using radioactive decay with
potassium-40. Suppose the half-life of the 14K is 1.5 billion years.
(a) If the bones are known to have 70 million years what is the remaining percentage of
14K in those bones today? Round up to 2 decimal places. [5]
(b) Suppose you have a 500 million year old skeleton, what percentage of 14K will remain
today? Round up to the nearest integer. [5]
(c) If you cannot detect amounts below 1%, what is the maximum age you can date
using 14K? Round up to the next integer.

Evaluate the ∫∫s y dS where S is the portion of the cylinder x2 + y2 = 3 that lies between z = 0 and z = 3.

use the first principle definition to find the derivative of f'(a), where [x] is the greatest integer less than or equal to x.

how to calculate the average rate

Use triple integral to find the volume of region that is below z=8-x^2-y^2 above z=-√4x^2+4y^2 and inside x^2+y^2=4

Use a double integral to derive the area of the region between circles of radius a and b with
α
≤
θ
≤
β
α≤θ≤β
. See the image below for a sketch of the region.

use double integral to calculate area of region that is inside r=6-4Cos Θ

To approximate the value of f
000(xk), the following formulas are used
f
000
k =
1
h
3
[ fk+3 −3 fk+2 +3 fk+1 − fk
]
f
000
k =
1
2h
3
[ fk+2 −2 fk+1 +2 fk−1 − fk−2]
Which formula do you find more accurate, and why?