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Calculus and Analysis

Proportionality and Rate of Change Word Problem

A container has the shape of an open right circular cone. The height of the container is 10cm and the diameter of the opening is 10cm. Water in the container is evaporating so that its depth h is changing at the constant rate of -3/10 cm/hr. Show that the rate of change of the volume of water in the container due to evaporati

Parametric Equations : Circle and Ellipse

Create parametric equations for: a. A circle of radius 2 centered at (3,1). b. An elipse with a horizontal major axis of length 5 and a verticle minor axis of length 4.

Maximum and Minimum Distance Word Problem

Johnny Steamboat wants to sail from his island home to town in order to purchase a book of carpet samples. His home island is 7 miles from the nearest point on the shore. The town is 35 miles downshore and one mile inland. If he can run his steamboat at 12 mph and catch a cab as soon as he reaches the coast that will drive 60


Let X(n)=Sum{1/(n+i), i=1->n}, find the limit of X(n) as n tends to infinity

The Convergence of Darbox Sums and Riemann Sums

1. Let k >= 1 be an integer, and define Cn = SIGMA (1/(n+i)) as i=1 to kn (a)Prove that {Cn} converges by showing it is monotonic and bounded. (b)Evaluate LIMIT (Cn) as n approach to the infinity

Work done to pump water out of tank

Would like second opinion or other way to solve problem, hopefully using Disk method - OTA #103642 answered last time. Perhaps OTA #103642 could send me email regarding this formula? Problem - A tank is in the shape of an inverted cone (pointy at the top) 6 feet high and 8 feet across at the base. The tank is filled to a de

Finding a Centroid of a Region

Find the coordinates of the centroid of the region bounded by the curves y=3-x and y=-x^2+2x+3. *I first found m or area, by rho(Integral 0 to 3)[(-x^(2) +2x +3) - (3-x)]dx and the result was 9rho/2. Second, I found Mx, by rho (Integral 0 to 3) [(-x^2 +2x +3)+(3-x)/2][(-x^2 +2x +3)-(3-x)]dx and the result was 54rho/5,

The Length of a Curve

Using the curve y=2x^(3/2) +3 (3/2 is the power on x) from x=0 to x=4 : A.) estimate the length of the curve by computing the straight line distance between the endpoints. *I found the answer to be sq root of 272 or approx. 16.4924. Sound correct? B.) Compute exact length of the curve. *I'm using S = The Integral

Volume of Revolution about X-axis

Given the region bounded by y=x^2 and y=-4x+12 and y=0 , find the volume of the solid generated by rotating this region about the x axis. I found a volume of 477pi/5 or approx. 300, does this sound right?

Area of Region

Find the area of the region bounded by the following graphs: y=x^2 ; y=8-x^2 ; and y=x^2+6. I first figured out the area of (8-x^2) minus (x^2) using integral from -2 to 2 and found 64/3 . Then I figured out the area of (8-x^2) minus (x^2+6) using integral -1 to 1 and found 4. I then just subtracted these two regions fr

Lower Estimate, Upper Estimate?

Please see the attached file for the fully formatted problems. 14) A power plant generates electricity by burning oil. Pollutants produced as a result of the burning process are removed by scrubbers in the smokestacks, Over time, the scrubbers become less efficient and eventually they must be replaced when the amount of pol


Find the linearization, L(x) of f(x) at x = -7 f(x) = sqrt(x^2 + 15).

Solving Equations : Newton's Method

Please see the attached file for the fully formatted problem. The equation x2 - 3x + 1 = 0 has a solution for x>= 0. Give the third approximationby using Newton's method. Your first approximation is to be 1.

Taylor Approximation Related Problem

Please see the attached file for the fully formatted problems. Questions pertain to Second order Taylor approximations and integrals for two first order differential equations.


Prove that if x>0, then 1+x/2-x^2/8<(1+x)^(1/2)<1+x/2

Analysis/total differentials

Evaluate f(1,2) and f(1.05, 2.1) for the function Æ'(x,y)=x/y. a) calculate &#916;z b) use the total differential dz to approximate &#916;z

Chain rule

You are given the function w=yz/x, where x=&#952;^2, y=r+&#952; and z=r-&#952;. Find &#8706;w/&#8706;&#952;. a) using the appropriate chain rule b) converting w to a function of r,&#952; before differentiating. Which of the above is quicker?


A metal plate is located in an xy-plane such that the temperature T at (x,y) is inversely proportional to the distance from the origin, and the temperature at point P(3,4) is 100 (i.e. the temperature at any point (x,y) is described by the function T(x,y) = 500/(x^2 + y^2)^1/2 a) in what direction does

Proof: Upper and lower limits

Please see the attached file for the fully formatted problems. Let be a sequence of real numbers. We define and I'm having trouble with the following three proofs: 1) Show that 2) Show that if the limit of only exists when , then . 3) Show that if , then the limit exists, and .

8 calculus problems

Please answer eight (8) calculus problems. Please show as much works as possible for every problem. The problems are posted in the following website:

Mass and centroid of a Plane Lamina

Please see the attached file for the fully formatted problem. Find the mass and centroid of a plane lamina with the given shape and density delta, the region bounded by y = x2 and x = y2 delta(x,y) = x2 + y2.

Word problem derivatives

A ball is dropped from the top of a building which is 1000 feet tall. GIVEN (s(t)=-16t^2+v(initial)t+s(initial)) A. Write the position and velocity functions for the ball. B. Find the instantaneous velocity went t = 2 seconds. C. How long does it take the ball to reach the ground. Please solve using calculus (derivativ