Calculus Answers

Plant leaf epidermal cells are thin and flat, and can be approximated as having a cylindrical shape with a radius r and a thickness h
The thickness of these cells h is about the same, but the radius r varies from leaf to leaf.
(a) Find expressions for a plant leaf epidermal cell’s absorption and consumption of a nutrient (as for the spherical cell in Chapter 1). Assume the absorption rate per unit surface area is k1 and the consumption rate per unit volume is k2.

Differentiate from the first principle the cube root of x plus cube root of x.

Differentiate cube root of x plus cube root of x

Given the following function:

F (X) = -93 - 31X2 - 19X

1. What is the value of the critical point for this function?

2. Is this critical point a maximum, minimum or is it indeterminate?

Suppose that F:R3→R3 is an inverse square force field F(x, y, z) =(c)/(x^2+y^2+z^2)^3/2 [x;y;z]. Find the work done by F in moving an object from a point P1 to a point P2 in terms of the respective distances from these points to the origin.

Use Green’s Theorem to find the area inside x^2/3+y^2/3= 1.

In this question, we are going to derive the Surveyor’s Formula for finding the area of a polygon in terms of its coordinates.(a) If C is the line segment from (x1, y1) to (x2, y2) and ω=−ydx+xdy is a differential 1-form, show that ∫Cω=x1y2−x2y1 (b) If the vertices of a polygon (in counterclockwise order) are (x1, y1),(x2, y2), . . .(xn, yn),show that the area of the polygon is A=12[(x1y2−x2y1) + (x2y3−x3y2) +. . .(xn−1yn−xnyn−1) + (xny1+x1yn)] (c) Find the area of a pentagon with vertices (0,0),(2,1),(1,3),(0,2),(−1,1).

A fluid motion is given by ~v = (y sin z - sin x)ˆi + (x sin z+ 2 yz)ˆj + (xy cos z + y
2
)
ˆk. Is
the motion irrotational? If so, find the potential function of ~v.

air is being pumped into a spherical balloon at a rate of 10cm^3/min. Determine the rate at which the radius of the balloon is increasing when the radius of the balloon is 20 cm

Air is being pumped into a spherical ballon at a rate of 10cm^3/min. Determine the rate at which the radius of the ballon is increasing when the radius of the ballon is 20cm