Calculus Answers

Prove that a non-decreasing (resp. non-increasing) sequence which is not bounded above (resp. bounded below) diverges to +∞ (resp. to −∞).

Set up a sum of integrals that will give the area of the quadrilateral with vertices at (0,0), (1,1), (0,2), and (2,1), connected in the stated order.

Locate the absolute maximum and minimum for each of the following functions and justify your responses. (Show all work)

Question 1: 𝑓(𝑥) = Cube root of x^2 − 𝑥 on the interval [−1,1] 

Question 2: g(x)=xe^2x on the interval [-2,0]

First and Second Derivative Analysis :For each of the following functions, respond to the given prompts. Show all work that leads to your responses:

Given 𝑓(𝑥) = 3𝑥 3 − 18𝑥 2 − 45𝑥 + 10

a. On what interval(s), is 𝑓(𝑥) increasing? Justify. show all work

b. At what value(s) of 𝑥 does 𝑓(𝑥) have a relative minimum? Justify. show all work

c. On what interval(s), is 𝑓(𝑥) decreasing and concave up? Justify. show all work

A camera is mounted 3.000 feet from a rocket launching pad. The camera needs to swivel as the ro launched to keep it in focus, a) If the rocket is rising vertically at 800 ft/sec when it is 4,000 feet hi fast is the camera-to-rocket distance changing?

 Angela forgot to study for her APUSH exam and it is the day before the exam. She worries that she will fail if she doesn’t study at all, so she decides she will study for at least 2 hours. She estimates her potential score to be 𝑆 = (120𝑡)/( 𝑡+2 )if she studies for 𝑡 hours. But she also knows that the longer she studies, the more tired she will become, and she won’t reach her potential. She estimates that her “fatigue factor” is 𝐹 = (10)/ (𝑡+10). To find her projected grade, 𝐺, she multiplies her potential score by her fatigue factor so that 𝐺 = 𝑆 ⋅ 𝐹. How many hours should Angela study to maximize her grade, 𝐺?

given (x^2)/x^2-4

Find all critical values for 𝑔(𝑥). Show all work and explain

b. Find 𝑔 ′′(𝑥). Then, use your answers from part (a) and the 2nd derivative test to determine if each critical value represents a relative maximum, minimum, or neither. Show all work and explain

c. On what interval(s), if any, is 𝑔(𝑥) concave down. Show all work and explain

For each of the following functions, respond to the given prompts. Show all work that leads to your responses. 

Given: 𝑓(𝑥) = 3𝑥^3 − 18𝑥^2 − 45𝑥 + 10 

a. On what interval(s), is 𝑓(𝑥) increasing? Show all work and explain

b. At what value(s) of 𝑥 does 𝑓(𝑥) have a relative minimum? Show all work and explain

c) On what interval(s), is 𝑓(𝑥) decreasing and concave up? Show all work and explain

Locate the absolute maximum and minimum for each of the following functions and justify your responses. (Show all work)

1) 𝑓(𝑥) = square root of x^2 − 𝑥 on the interval [−1,1]

2) 𝑔(𝑥) = 𝑥𝑒^2𝑥 on the interval [−2,0]

A rectangular box at the top to have volume of 32cubic feet. Find the dimensions of the box requiring least material for it"s construction

Suppose {an}∞n=1 be a sequence of positive real numbers and 0 < x < 1. If an+1 < x·an for every n ∈ N, prove that limn→∞an = 0