Calculus Answers

True or False

If the function "f" is odd, then "\\displaystyle{\\int_{-1}^{1}f(x)dx=0}"

Evaluate the limit "\\displaystyle{\\lim\\limits_{\\theta \\to 0} \\dfrac{\\sin{(\\theta^2)}}{\\theta}}" using the l'Hopital's Rule.

Evaluate the limit  "\\displaystyle{\\lim\\limits_{x \\to 0} \\dfrac{e^{x}-x-1}{x^2}}" by using l'Hopital's Rule twice.

(Section 13.3 and Chapter 14) Let D be the region in R 3 p that lies inside the cone z = x 2 + y 2 above the plane z = 1 and below the hemisphere z = p 4 − x 2 − y 2 . (a) Sketch the region D in R 3 .(b) Express the volume of D as a sum of triple integrals, using cylindrical coordinates.

Use a triple integral to determine the volume of the region bounded by z =

p

x

2 + y

2 and z = x

2 + y

2

In 1st octant

A tank having a capacity of 1000 liters, initially contains 400 liters of sugar water having a concen-

tration of 0.2 Kg of sugar for each liter of water. At time zero, sugar water with a concentration of

50 gm of sugar per liter begins pumped into the tank at a rate of 2 liter per minute. Simultaneously,

a drain is opened at the bottom of the tank so that the volume of the sugar-water solution in the

tank reduces 1 liter per minute. Determine the following:

Consider the surface S =  (x, y, z) ∈ R 3 | z = 3 − x 2 − y 2 ; z ≥ 2 . Assume that S is oriented upward and let C be the oriented boundary of S. (a) Sketch the surface S in R 3 . Also show the oriented curve C and the XY-projection of the surface S on your sketch. (2) (b) Let F (x, y, z) = (2y, 3z, 4y). Evaluate the flux integral Z Z S (curl F) · n dS by i. determining curl F and the upward unit normal n of S and using the formula (17.2) on p. 104 of Guide 3 (5) ii. Using Stokes’ Theorem, convert the given flux integral to a line integral. 

Consider the function 𝑦 = tan (𝑥)

a) Show that the first two non-zero terms in the Maclaurin series of 𝑦 are "\ud835\udc65 + 1\/3 \ud835\udc65 ^3" …

b) Use the first two terms of the Maclaurin series of 𝑦 to estimate "tan ( 1\/3 )"