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# Calculus and Analysis

### Differential equations of all parabolas

Differential Equation (XII) Formation of Differential Equations by Elimination Find the differential equations of all parabolas whose axes are parallel to the axis of y.

### Differential Equations of Exponential Functions

Differential Equation (IX): Formation of Differential Equations by Elimination Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

### Equations of Tangent Lines and Intercepts in Terms of a Variable

Let f(x) = a(7-x^2) where a is not equal 0 (a) find, in terms of a, the equation of the line tangent to the curve at x = -1 (use point slope) (b) find, in terms of a, the y intercept of the tangent line at x = -1 (c) find the x intercept of the tangent line at x=-1

### Prove that the spheres x2 + y2 + z2 = 16 and x2+ (y+5)2 + z2 =9 intersect orthogonally.

Prove that the spheres x2 + y2 + z2 = 16 and x2+ (y+5)2 + z2 =9 intersect orthogonally.

### Differential Equation - Initial Value Problem

11. Consider an electric circuit like that in Example 5 of Section 8.6. Assume the electromotive force is an alternating current generator which produces a voltage V(t) = E sinwt , where E and w are positive constants (w is the Greek letter omega). EXAMPLE 5. Electric Circuits. Figure 8.2(a), page 318, shows an electric circu

### Population Dynamics

I think the solution is probably a simple modification to the predation equations: dN/dt = rN - cNP is the growth rate for the prey population and dP/dt = -dp + gcNP is the growth rate for predator population, where: N= prey density P = predator density d= death rate g = conversion efficiency of prey to predators r = p

### Modeling Data for Linear Functions and Maximizing Profit

1960 88 1970 121 1980 152 1990 205 1997 217 a) Model the data with two linear function. Let the independendt variable represent the number of years after 1960. b) With each function found in part a), predict the amount of maunicipal solid waste in 2005. c) Which of the two models

### Differential Equations : Euler's Method and Initial Conditions

DVc/dt = ( V(t) - Vc ) / ( RC) Supposed V(t) = 2cos(3t). If R = 4 and C = 0.5, use Eulers method to compute values of the solutions with the given inital conditions over the interval 0 <= t <= 10 for Vc(0) = -1

### Sand falls from an overhead hopper to form a right circular cone. If the cone formed has an angle theta find the rate of change of volume with respect to height? If the height is changing at 2cms per minute, what volume of sand is falling from the hopper when the height of the cone is 3 meters? (The volume of a cone is (1/3)pir2h where h is the height and r is the radius.)

Sand falls from an overhead hopper to form a right circular cone. If the cone formed has an angle theta find the rate of change of volume with respect to height? If the height is changing at 2cms per minute, what volume of sand is falling from the hopper when the height of the cone is 3 meters? (The volume of a cone is (1/3)pir2

### Time for object to fall if it is thrown upward, dropped, or thrown downward.

You are standing on a cliff 200 feet high. How long will it take a rock to reach the ground if a) you drop it? b) you throw it downward at an initial velocity of 40 feet per second? c) How far does the rock fall in 2 seconds if you throw it downward with an initial velocity of 40 feet per second?

### Volume of Solid of Revolution : Region y=x^-1/2 on the interval [1,4] is revolved about the x-axis, y-axis and the line y=-2

Find the volume of the solid generated when the region y=x^-1/2 on the interval [1,4] is revolved about a)x-axis b) y-axis c) the line y=-2

### Length of a curve

Suspension bridge. The cable of the suspension bridge shown in the accompanying figure hangs in the shape of a parabola with the equation , where x and y are in meters. What is the height of each tower above the roadway? What is the length of z for the cable bracing the tower?

### Perpendicular Distance from Point to Plane

Show that the perpendicular distance D from the point P0(x0, y0, z0) to the plane ax + by + cz = d is: D = | ax0 + by0 + cz0 &#8722; d | / sqrt(a^2 + b^2 + c^2)

### A corner lot has dimensions 25 by 40 yards

A corner lot has dimensions 25 by 40 yards. The city plans to take a strip of uniform width along the two sides bordering the streets in order to widen these roads. How wide should the strip be if the remainder of the lot is to have an area of 844 square yards?

### Intersection and distance

1. Find the intersection point of the line (x-1)/2=(y+1)/3=z-2 and the plane 2x+y-z=17. 2. Find the distance from point Q(1,-2,3) to the plane 2x-y-z=6. Need steps and solutions.....Thanks!

### Spherical polar coordinates

Please help...this is a revision question very likely to appear in my exam next month but I do not understand it! (See attached file for full problem description)

### Set up the problem by labeling the unknowns, translating the given information into mathematical language, and finding a solution

A radiator contains 10 quarts of fluid, 30% of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the new mixture is 40% antifreeze?

### Finding a particular solution of a differential equation

I already solved the homogeneous portion, and I need help solving the particular solution and of course combining the two to get the entire solution to the differential equation. Not too difficult - see attachment. Please use equation editor if possible. Thank you. --- Given that: dMS/dt = m(MN - MS) - pMS¬

(See attached file for full problem description) --- Find the radius of convergence of the following series...(see attachment for equation) ---

### Differentibility of Functions and Limits

Prove that if f is differentiable at x, then (see attachment) Also, show that for some functions that are NOT differentiable at x, this limit still exists. --- (See attached file for full problem description)

### Finding Integrals Using the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus I need to find the solution of the following problems. Can you explain how? Please see the attached file for the fully formatted problems.

### Derivatives and Application of the Derivative

F(t) = 10,000 / 10 + 50e ^-0.5t HOW do I obtain the derivative? What is the "e" portion of the problem? I know the derivative = 250,000e^-0.5t/ (10+50e ^-0.5t)^2 Please describe in detail the steps taken to arrive at this answer. For example, Why is the top of the equation 250,000e^-0.5t? Why is the bottom (10+50e^-0

### Differential Equations: Solution to Heat Equation

Consider the heat equation delta(u)/delta(t) = (delta^2)(u)/delta(x^2) Show that if u(c, t) = (t^alpha)psi(E) where E = x/sqrt(t) and alpha is a constant, then psi(E) satisfies the ordinary differential equation alpha(psi) = 1/2 E(psi) = psi, where ' = d/dE is independent of t only if alpha = - 1/2. Further, show th

### Study Guide For Final Calculus Exam

This is a study guide for the final exam in Calc 3. Thanks. (See problem set in attached file)

### Differential equation explanation

I need to know how to find a particular solutions to an initial conditions. i know how to find the explicit solution of a differential equation but cannot remember how to find particular solutions. please show your working to help me understand. thanks Problems (also attached): Given the attached information: 1) So how

### Solve and Explain Three Integrals

&#8747; (x^5)/[(1+x^3)^(3/2)] dx &#8747; {square root of [(x+3)/(x+1)]}dx &#8747; (cot^3 v)[(csc v)^(3/2)] dx (I will use the \$ sign for the integral sign) Problem #1: \$ (x^5)/[(1+x^3)^(3/2)] dx the power 3/2 in the denominator is throwing me off greatly, as is the greater power (x^5) in the numerator. attempt 1

### Even, Odd, and Periodic Fuctions : Periodicity and Fundamental Period

Determine whether or not the given function is periodic. Find the fundamental period. (h) sinh x (q) cos(sin x) Please see the attached file for the fully formatted problems.

### Damping and Resonance Problem : Forces and Displacement

A seismograph is a scientific instrument that is used to detect earthquakes. A simple model of a seismograph is shown below. It consists of a particle of mass m to which a pointer is attached. The particle is suspended by a spring of natural length lo and stiffness k and a damper of damping constant r from a platform of height d

### Evaluating the Integral with Limits

Please could you solve the following question showing every stage as simply as possible to get to the correct answer. Where any calculus rules are used could you please explain. Integrate sec^2(3t) - cosec^2(5t) between the limits t = 0 and t = pi/4 Please see attached for a more clear version.

### Second order Diferential equation with Wronskian

Let y1, y2 be twice differentiable functions on an interval (a,b) whose Wronskian is nowhere zero. Show that there... Please see attached.