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    Basic Calculus

    Various calculus problem set

    Please see the attached file for complete description 1. Write the first five terms of the arithmetic sequence: a1 = 3; d = 3 2. A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of the regions are colored red, two are colored green, and one is colored yellow. If t

    Continuity, Differentiability and the Volume of a Solid

    Please see attached file for full problem description. 1. Decide whether each of the following statements is true or false. If true, explain why. If false, give a counter-example and explain why the counter-example contradicts the statement. a. If then b. If f and g are continuous functions on [a,b], then c. If f

    Cardinality of R and R^2

    How can I show that the cardinality of R and R^2, R=set of Real numbers, is equal. I think by R^2 it is meant R x R, which means an ordered pair, am I right? Is this possible just be showing that the first element in R^2 pair can be matched to R? But this is not necessarily a 'function' by definition, so there are infinite

    Determining Velocity of Objects

    If an object is given an initial velocity straight upward of v0 feet per second from a height of s0 feet , then its altitude's after 1 second is given by the formula: S= -16t2+ v0t + s0. An arrow is shot straight upward with a velocity of 96 feet per second (ft/sec) from an altitude of 6 feet. For how many seconds is t

    Third Prime

    Please help, figured out first two primes but am having allot of problems figuring out the third prime. Also need help with additional problem. Thanks Find lim x ->∞ ( (3x-2) / (3x+2) )^x (3x - 2)^x / (3x+2)^x = lim x ->∞ f'(x)/g'(x) x(3x - 2)^(x-1)*(3)/ x(3x-2)^(x-1)*(3) =(3x - 2)^(x-1)/ (3x

    Answering Objective Questions

    Please choose the correct answer: ∫ 12x(x - 2)^10 dx = x^12 + (24/11)x^11 + C x^12 + (2/11)x^11 + C (12/11)(x - 2)^11 + C (12/11)x(x - 2)^11 + c (x - 20/11)(x - 2)^11 + C (x + 2/11)(x - 2)^11 + C none of these b

    Answers to Objective Questions

    13. Find the area under the curve y = 3e^-x - e^x , from x = 0 to x = ln(3)/2. 3 - 2(2^1/2 ) 4 - 2(3^1/2 ) 1 6 - 2(5^1/2 ) 7 - 2(6^1/2 ) none of these. 14. Calculate, correct to the nearest hundredth, 2.25 ∫(2.73x^2 - 8.41x + 7) dx = 0.5 1.5

    Find equation of line tangent

    Please explain the steps and the solution, thank you: Find an equation of the line tangent to the graph of the curve y = 2^x + ln x at x = 1 (^ means exponent)

    Calculus Problems - secant line joining

    Q#11. The graph of y=f(x)=x^2 + 2x is shown in the graph. i) Find the slope of the secant line joining (2, f(2)) and (4, f(4)). ii) Find the slope of the secant line joining (2, f(2) and (2+h, f(2+h)) _____ + _____ h Q#12. Suppose an object moves along the y axis so that its locations is y=f(x)=x^2+3x at time x(y

    Integral Calculus and Position of an Object

    1. R is the region bounded by the curve y = x^2 -2x and the x axis. Sketch the given region R and then find its area. 2. Using the following integral and the facts that v(t) = x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. Answer the questions a - c. x'

    Mobius Transformations and Conformal Maps

    (1) For j = 1,2 let R_j be the circle of diameter j/2 and center at (j/4)i. Also, let p(z) = 1/z be the inversion map. (a) If G is the region outside R_1 and inside R_2 then prove that p(G) = {z : -2 < Im z < -1}. (b) Prove that e^(pie*z) maps the strip {z : -2 < Im z < -1}onto the upper half-plane H_u. (c) Use the pr

    Exponential Population Growth

    The population P of a city is given by P = 2000e^kt . Let t=0 correspond to the year 1960 and suppose the population in 1950 was 1500. Find the value of k ( to 3 decimal places) and then predict the population in 2000. Please show all steps & graph, solve using derivatives or ln on calc.

    Related Rates

    3.A pulley is suspended 13.5 m above a small bucket of cement on the ground. A rope is put over the pulley. One end of the rope is tied to the bucket and the other end dangles loosely to the ground. A construction worker holds the end of the rope at a constant height (1.5 m) and walks away from beneath the pulley at 1.6 m/s. How

    Differential Equations and Vectors - Circle Equation

    See the attached file. Question (1) First find a general solution of the differential equation dy/dx=3y. Then find a particular solution that satisfies the initial condition that y(1) = 4 Question (2) Solve the initial value problem dy/dx=y^3 , y(0) = 1 Question (3) Find the centre and radius of the circle describe

    Equations of Surfaces

    Surface Equations Problems: 33, 37, 47, 63 See attached file for full problem description.

    Calculus - Numerical Analysis - Modified Eulers method

    Question : Use the Modified Euler Method to approximate the solution to the initial value problem, and compare the results to the actual values. y'=te^3t-2y, 0 <=t<=1 , y(0) = 0 with h = 0.5 actual solution y(t) = (1/5)te^3t-(1/25)e^3t+(1/25)e^-2t For the complete description of the question, please see the attach

    Determine the area of the given region.

    Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

    Velocity of a Ball

    In Exercises 25-30, complete the table using Example 6 as a model. 26. y = 4/(3x^2) In Exercises 53-56, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility

    Volume of a Solid of Revolution

    Find the volume of the solid formed when when the graph of the region bounded by y=e^x, x=0, x=2, and y=0 is revolved about the x-axis.

    Work Done in Compression

    A force of 650 pounds compresses a spring 5 inches from its natural length. Find the work done in compressing the spring 2 additional inches. Answer in inch-pounds.

    Continuity and the Intermediate Value Theorem

    Suppose that f is continuous on [a,b], f(z) < 0, and f (b) > 0. Set z = sup{x: f (t) < 0 for all t contained in [a, x]}. Prove that f (z) = 0. This is key to the proof of the intermediate value theorem. Incorporate the definition of least upper bound into your argument.

    Uniformly Continuous Definition

    Let E be a nonempty subset of R and f:E-->R. State the definition f is uniformly continuous on E. Prove f(x) =x^2 is uniformly continuous on the interval[0,1]. keywords: uniform continuity

    Finding the Optimum

    The purpose of this project is to find the values of x and y that will yield the optimum (maximum or minimum) value of a system and the optimum value of a system using algebraic and graphical methods. Follow these ten steps (see mp1.jpg) to determine the optimum value of z and the values of x and y that yield the optimum valu

    Rouche's formula

    (a) How many roots of the equation z^4 &#8722; 6z + 3 = 0 have their modulus between 1 and 2? (b) Find the number of the roots of the equation z^6 &#8722; 5z^4 + 8z &#8722; 1 = 0 in the annulus {z : 1 < |z| < 2}