Calculus Answers

A sequence is bounded if and only if it is both bounded above and bounded below

a)    Find the work done by the force field F on a particle that moves along the curve C. F(x, y) = (x² + xy)i + (y – x² y)j               C : x = t, y = 1/t (1 ≤ t ≤ 3)

b)    Use Green’s Theorem to evaluate the integral, assume that the curve C is oriented counterclockwise.  , where C is the triangle with vertices      (0, 0), (3, 3), and (0, 3).

c)     Evaluate the surface integral   , where σ is the part of the cone z =x² + y² that lies between the planes z = 1 and z = 2.

d)    Use the Divergence Theorem to find the flux of F across the surface σ with outward orientation.   F(x, y, z) = z³ i – x³ j + y³ k, where σ is the sphere x² + y² + z² = a².

a)    Identify the iterated integrals which are of the following holding Fubini’s Theorem. Give mathematical reason.

i)   ∫_0^ln3▒∫_0^ln2▒〖e^(x+2y) dydx〗                                    iii)  ∫_3^5▒∫_1^2▒〖1/〖(x+y)〗^2 dydx〗

ii)       ∫_(-1)^1▒∫_(〖-x〗^2)^x▒〖(x^2-y)dydx〗                            iv)  ∫_1^(3⁄2)▒∫_y^(3-y)▒ydxdy

a)    The volume in the first octant bounded by the coordinate planes, the plane y = 4, and the

plane  (x/3) + (z/5) = 1.

b)    Sketch the solid in the first octant that is enclosed by the planes x = 0, z = 0, x = 5,

 z − y = 0, and            z = −2y + 6.

find "y", when "y= sin x - cos x"

1.   Verify the identity 1 - sin² x cot² x = sin² x. Is there more than one way to verify the identity? If so, tell which way you think is easier and why. 

2.John said 𝒔𝒊𝒏 𝒙 + 𝒄𝒐𝒔 𝒙 = 𝟐 has no solution. Do you agree with John? Explain why or why not? 

3.

SKILL CHECK: Verify each identity.  

1.𝒄𝒐𝒕² 𝜽+1 / cot²𝜽=sec²𝜽

2.(𝒄𝒔𝒄^𝟐𝜽 − 𝟏)𝒔𝒊𝒏^𝟐𝜽 ≡ 𝒄𝒐𝒔^𝟐𝜽 

3.8. 𝟏 − 𝒔𝒆𝒄 𝜶 𝒄𝒐𝒔 𝜶 ≡ 𝒕𝒂𝒏 𝜶 𝒄𝒐𝒕 𝜶 − 𝟏 

4.tan A + cot A / sec A csc A =1

5.𝟏 + 𝟐 𝒕𝒂𝒏 ^𝟐𝜽 ≡ 𝒔𝒆𝒄^𝟒𝜽 − 𝒕𝒂𝒏 ^𝟒𝜽 

Trigonometric equation

6.cos x + √𝟑 = - cos x 

7.sin2 x – tan x cos x = 0

8.sin x + √𝟐 = − sin x 

9. 2 cos2 x – 5 cos x = 3

10.√𝟑 csc x + 2 = 0

solve the worded problem

1. On which days of the year are there 10 hours of sunlight in Prescott, Arizona?

2.The tide, or depth of the ocean near the shore, changes throughout the day. The depth of the Bay of Fundy can be modeled by....... where d is the water depth in feet and t is the time in hours. Consider a day in which t = 0 represents 12:00 A.M. At what time(s) is the water depth 3 1 2 feet?

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? Show all work

Given g(x)= (x^2)/(x^2-4)

a) Find all critical values for 𝑔(𝑥).

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

c) On what interval(s), if any, is 𝑔(𝑥) concave down. Justify.

Locate the absolute maximum and minimum for each of the following functions and justify your responses.

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

Question2: 𝑓(𝑥)= 𝑔(𝑥) = 𝑥𝑒^2𝑥 on the interval [−2,0] 

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

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

b) what value(s) of 𝑥 does 𝑓(𝑥) have a relative minimum?

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

y=3x³+2x²+4x+27

find "f'(3)" for"f(x) = x\u00b2 + 10x"