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Calculus and Analysis

Differentiation: s = ut + 1/2at^2

This equation represents displacement of a body(s) against time (t) where (u) is the initial velocity and (a) is the acceleration. Differentiate to derive the equation for instantaneous velocity, which would be represented by the gradient of a graph. s = ut + 1/2at^2

Minimization, maximization (Calculus)

An ant is walking around the outside of the cube in "straight" paths (where we define a straight path in this case as one formed by the edges of a cross section created by a plane slicing through the cube). For example, to get from point Q to point R in the picture above on the right, the ant walks along the red path. There are

Comparing Graphs : First and Second Derivatives

X -1.5 -1.0 -0.5 0 0.5 1.0 1.5 f(x) -1 -4 -6 -7 -6 1.0 -7 f'(x) -7 -5 -3 0 3 5 7 Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its derivative f' for selected points x in the closed interv

Graphs : Rectangular and Parametric Forms

Graph x=(t^2+2t+1)^(1/2) y=(t^3+2t^4)/t^2 a. Graph on the interval [0,3] b. Convert to rectangular form. c. Adjust the domain of the rectangular form to agree the parametric form.

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

Word Problem : Intermediate Value Theorem

At 8am on Saturday, a man begins running up the side of a mountain to his weekend campsite. On Sunday at 8am, he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on his way down, he realizes that he passed the same exact place at exactly the same time on Saturday.

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,

Work Done to Pump Water out of a Tank

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 depth of 3 feet. How much work is done in emptying the tank through a hole at the top? (Weight density of water is 62.4 lb/ft^3). *I found the distance to be 6-y and used the disk method to solve

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 in a Graph

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

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.

Surface of Revolution

Y=2*sqrt[x] y=0, x=3 We are using the following formula 2pi intergal of r(x) times the sqrt of 1 + (f'(x))^2.

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.

Differential equations functions

Define: f(x) = (x^2)sin(1/x)+x if x doesn't equal 0 f(x) = 0 if x=0 Prove that the function f:R-> R is differentiable and that f'(0)=1. Also prove that there is no neighbourhood I of 0 such that the function f:I->R is increasing.