Calculus Answers

Let D1 be the region in the XY-plane enclosed by the triangle with vertices (0, 0),(1, 1),(2, 0). Let D2 be the semi-circular region of radius 1 with the center at (1, 0) lying below the x-axis. Let D = D1 ∪ D2.

a) Argue whether D is an elementary region.

b) Compute the area of D.

c) Without using Green’s theorem, evaluate "\\intop"_C(xydx + x²dy), where C denotes the boundary of D oriented clockwise.

d) Use Green’s theorem to evaluate the line integral given in part c).

A mathematical biologist created a model to administer medicine reaction (measured in change of blood pressure or temperature) with the model given by 

2

R= m^2 (C/2- M/3)

where c is a positive constant and m is the amount of medicine absorbed into the blood. The sensitivity to the medication is defined to be the rate of change of reaction R with respect to the amount of medicine absorbed in the blood.

Adiabatic law for the expansion of air is

PV^1.4=C,P(pressure) in lb/in^2 ,V(volume)in cubic inches, and C(constant). At a specific instant, the

pressure is 40 lb/in^2 and is increasing at the rate of 8 lb/in^2each second. If

C=5/16

, what is the rate of change of the volume at this instant?

all the cube root of iota (i) in a complex number (C) are z1= cosπ/2 + i sinπ/2 , z2= cosπ/6 +i sinπ/6 and z3= cos5π/6 + i sin5π/6 .

Verify if the statement is true or false. Give reason for your answer in the form of a short proof or a counterexample.

Using Epsilon and Delta definition, show that limit of x approches to 2 for 3x-5 is equal to 1.

𝑦(𝑥)=𝑐1𝑥+𝑐2+𝑥2−1

During the first 40 s of a rocket flight, the rocket is propelled straight up so that in t seconds it reaches a height of s=0.3t³ ft. (a) How high does the rocket travel in 40 s? (b) What is the average velocity of the rocket during the first 40 s? (c) What is the average velocity of the rocket during the first 1000 ft of its flight? (d) What is the instantaneous velocity of the rocket at the end of 40 s?

Suppose you want to buy a tablet that costs P 9,000.00

a. How long would it take you to raise the money if you could save:

a.1. Ten pesos a week?

a.2 Twenty pesos a week?

a.3 Fifty pesos a week?

a.4 One hundred pesos a week?

a.5 Two hundred pesos a week?

c. Write an equation that best represent the situation.

d. Explain what will happen as the weekly savings approaches zero.

d.1 What do you think is the possible lowest weekly savings?

d.2 What do you think is the possible highest weekly savings?

the radius of the planet Mars is measured to be 3350 km , with a measurement error of plus or minus 100 km. What is the percent error when calculating the volume of the planet? Round your answer to 2 decimal places

Find the volume generated by revolving the given region about the given axis.

(a) The region bounded by y = x⁴ , x = 1 and y = 0 about Y axis.

(b) The triangle with vertices (1, 1),(1, 2)(2, 2) about X axis.

(c) The region in the first quadrant bounded by x = y − y³ , x = 1 and y = 1 about X axis.