Calculus 3 and 4
Show all work. Please DON'T submit answers back to me as an attachment. Thank you. Determine whether the function is homogenous. If it is, state the degree: f(x, y)=5x^2 + 2xy
Calculus and Analysis
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Calculus differential equation
Find the order of the differential equation and determine whether it is linear or nonlinear: y^(4) + 3(cos x)y''' + y'=0
Find ∫ e^(-x^2)dx by using Laplace Transform.
Laplace Transform Inverse Laplace Transform To find the value of ∫ e^(-x^2)dx by using Laplace Transform, where the range of integration is
It is a short description for finding the higher order derivatives of the function f(x)= (2^2^x)+1.It represents a Fermat Number when x is a positive integer. Find the higher order derivative of the function f(x) = [2^(2^x)] + 1.
Differential Calculus Fermat Numbers Higher Order Derivatives of f(x) = [2^(2^x)] + 1 The function f(x) = [2^(2^x)] + 1 represents Fermat numbers when x = 1,2,3,... Find the higher order derivative of the function f(x) = [2^(2^x)] +
Minimization of a triangle area
Find the altitude to the base of an isoceles triangle with a side length of 4, such that the area of the triangle is maximal.
Relation between the roots and coefficients of an equation
Theory of Equation Relation between Roots and Coefficients Harmonical Progression Arithmetical Progression Problem
Oscillating Inflow Concentration
Oscillating Inflow Concentration A tank initially contains 10 lb of salt dissolved in 200 gallons of water. Assume that a salt solution flows into the tank at a rate of 3 gal/min and the well-stirred mixture flows out at the same rate. Assume that the inflow concentration oscillates I n time, however, and is given by ci(t
Volume of a Rotating Solid
Let f and g be the functions given by f(x)=e^x and g(x)=ln x. b) Find the volume of the solid generated when the enclosed region of f and g between x = ½ and x = 1, is revolved about the line y = 4. c) Let h be the function given by h(x)=f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval ½ X
Functions : Tangent, Increasing or Decreasing and Area under a Curve
2. Let f be a function defined on the closed interval -3≤x≤4 with f(0) = 3. The graph of f', the derivative of f, consists of one line segment and a semicircle. a) On what intervals, if any, is f increasing? Justify your answer. b) Find the x-coordinate of each point of inflection of the graph of f on t
Differentiation Functions Solved.
See attached file for full problem description. Differentiate the following
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
Finding the higher order derivatives of the function: f(x)=(2^2^x)+1. It represents a Fermat Number when x=1,2,3,...
To find the higher derivatives of the function f(x)=(2^2^x)+1 Differential Calculus Fermat Numbers Higher Order Derivatives of f(x) = [2^(2^x)] + 1 The function f(x) = [2^(2^x)] + 1 represents Fermat
Geometric Series Converging
Determine if the following series converges and if possible give its sum 2/3 + 2/9 + 2/27 + 2/81 + ...
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.
Graph : Convert to Rectangular Form
Let x=cosTheta and y=3sinTheta for0<=Theta<=Pi a.Sketch the graph b.Convert to rectangular 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
Sets : Intermediate Value Theorem
Prove that the interval (0,2)~R. Use the Intermediate Value Theorem.
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
The Slope of the Tangent Line to a Curve
Find the slope of the tangent line to the curve f(x) = 1/sq root of x^2+4 at the point where x=2.
Limitation
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?