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Calculus and Analysis

Vectors

50 Problems Please see the attached files for the fully formatted problems.

Finding Zeros of Functions

Find the zero of the linear function f(x)=3x-12 Find the zeros of f(x)=x^2-2x-3 Find the vertex of f(x)=x^2-2x+4 Find the axis of symmetry of f(x)=x^2-2x+4 Find the zeros and state the multiplicity of each for f(x)=x^2(x+3)(x+1)^4 Find the zeros of f(x)=x^2-8x+12

Derivatves and Rate of Change Word Problems

1.) A deposit of S dollars that earns 100r% annual interest compounded continuously leaves a balance of P = 'Se' power of 'n' or ( 'Se' to the 'n') dollars after "t" years. a) What will an amount of $ 5000 grow to after 15 years at 10% annual interest compounded continuously? b) Determine the rate at which P is growi

Determine whether the given vectors are orthogonal, parallel or neither

Determine whether the given vectors are orthogonal, parallel or neither. <-5,3,7> and <6,-1,2> <4,6> and <-3,2> -i + 2j + 5k and 3i + 4j - k 2i + 6j - 4k and -3i -9j +6k Find a unit vector that is orthogonal to both i+j and i+k. If a = <3,0,-1>, find a vector b such that comp_a b = 2 (component of b in the a direction

Eigenvalues, Eigenfunctions and Sturm-Liouville Problems

1. y'' + k*y = 0 BC: y'(0) = 0 y'(L) = 0 2. y'' + k*y = 0 BC: y(0) = y(&#61552;) y'(0) = y'(&#61552;) 3. y'' + k*y = 0 BC: y(0) = 0 y(&#61552;) +2*y'(&#61552;) = 0 4. y'' + 2*y' + (1+k)*y = 0 BC: y(0) = y(1) =0 Please see the attached file for

Instantaneous rate of change and average rate of change.

1.) Let f(x)=(the integral from 0 to x^2) sint dt. At how many points in the closed interval [0, square root of pi] does the instantaneous rate of change of f equal the average rate of change of f on that interval. keywords: integration, integrates, integrals, integrating, double, triple, multiple

Limits

Please explain how/why: lim x---> - 1 x^2 - 1 / x+1 How would this change if the 1 was positive?

Vectors : Dot Products and Planes and Distance from a Point

1) a = (1, 2, -2); ||b|| = 6. What choice of b will make the dot product a . b the least possible? 2) The plane P pass through the point M (2, 3, 1) and is parallel to vectors u = (1, -1, 4) and v = (2, 1, 0). Find the distance from the point N (3, 2, 4) to the plane P. 3) Find an equation of the plane that passes through (1,

Vectors : Line Equations and Line Segments

1) Find the equation of the line passing through the point (6, 1, 8) in the direction of the vector (-3, 4, -2). Write the equation in vector and coordinate form. 2) Find the equation of the line segment joining points (2, 4, 6) and (1, 7, 8), both in vector and coordinate form. Be sure to specify the set of possible values o

Infinite Limits

Determine the infinite limit #1)lim x-1/x^2(x+2) x->0, #2)lim csc x x->pi -, #3)lim ln(x-5) x->5+, #4)lim 6/x-5 x->5- #4) The slope of the tangent line to the graph of the exponential function y= 2^x at the point (0,1) is limx->0(2^x - 1)/x. Estimate the slope to three decimal places. keywords : find, findi

Limits

Find limit lim x---> - 4 x^2 - 16 / x+4 Please explain keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

Inverse Function

4. A retailer you spoke with in New York City's fashion district imports haute couture from European designers. One of the accommodations which must be considered when importing fashion from other countries is the difference in the size charts. A function that will convert dress sizes in the United States to those in Italy is

Continuous Functions, Primitives and Simpson's Approximation

(a) The function 1/t is continuous on the given interval, therefore integrable. The primitive is log(t). The primitive of exp(t) is exp(t). Therefore the integrals are: (b) The primitive of -cos(x) is -sin(x) anf the primitive of x is x2/2, therefore the integral equals: (c) We apply the so called Simpson's approximation:

Differential Equations

Find the general solution of the following (d^6y/dx^6) -3*(d^4y/dx^4) +3*(d^2y/dx^2) -y=0

Sigma Formulas

Let s (n)= (sigma sign with n as upper bound and i=1 as lower bound) (1+(i/n))^2 * (1/n). Find the limit of s (n) as n approaches positive infinity.

Friction and Equations of Motion

A particle of mass m moves under the influence of constant force f and the friction force &#947;(xdot)n Its equation of motion reads: m(x-double dot) + &#947;(xdot)n = f Find x(t) for n=1 and n=2 if x=0 and x-dot=0 at t=0 Check that your solutions exhibit the correct behavior in the limit. Please see the attac

High School Calculus Problems

4. Find the center, vertices and foci of 4x^2 + 9y^2 -8x +36y +4 =0. Graph, labeling all relevant points including the foci. 5. Find the equation of the hyperbola whose center is the origin, vertex at (2,0) and focus at (3,0). Write the equation of the asymptotes. 6. Find the center, vertices, foci and asymptotes

Domain, range, intercepts

F(x,y)=1/(x^2+y^2) Is the domain (-infinity, 0) union (0, infinity)? Is the range (0, 1]? What are the x and y intercepts and is the z intercept undefined?

Finding X, Y and Z Intercepts of 3-D Graphs

How do you find the x, y, and z-intercepts of a 3-Dimensional graph step by step? I am trying to interpret some graphs and would like this information to assist me in the interpretation. keywords: 3D, 3Dimensional

Simple harmonic oscillation driven by an external force

2. A spring with a 4-kg mass has natural length 1 m and is maintained stretched to a length of 1.3 m by a force of 24.3 N. If the spring is compressed to a length of 0.8 m and then released with zero velocity, find the position of the mass at any time t. 10. As in Exercise 9, consider a spring with mass m, spring constant k,