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Calculus and Analysis

Linear Functions

Answer the following questions and if required to do calculations please show your work: 1. What is a function? 2. What is a linear function? 3. What form does a linear function take? (I.e., What is the standard mathematical notation of a linear function?) 4. What is the formula for determining the slope of a li

Determining Rate of Change: Example Problem

(a) Suppose the position of an object moving in a straight line is given by s(t)=5t^3 - t^2 +9. Find the instantaneous velocity when t=2. (b) Find the instantaneous rate of change for the function at the given value: f(x)= x^2 + 5x at x=2

Interval Notation

1) Solve the following inequality. Write the answer in interval notation. X/X+3 ≤ -2/5(X-3) 2) Solve the inequality. 3 < |5X+3| < 5 3) a)Find an equation of the line, say y=mx+b, which passes through the point (-3,8) and is perpendicular to the line 3x+3y=33. y=? b)What is the shortest distance from the point (

Calculus Elasticity

Elasticity Research has indicated that the demand for heroin is given by the function q=100p-0.17 a) Find E b) Is the demand for heroin elastic or inelastic? Solve this problem, show all steps used to solve the equation, and explain the answer to receive credit for the solution.

Rate of Change of Number of Vehicles

The average number of vehicles waiting in a line to enter a parking ramp can be modeled by the function f(x) = (x2)/{(2(1-x)} where x is a quantity between 0 and 1 known as the traffic intensity. Find the rate of change of the number of vehicles in line with respect to the traffic intensity for the following values of the inte

Evaluating integrals by parts: Example problems

Please see problems and show step by step solution in detail please. Evaluate the integral (using integration by parts): 1. t(lnt)^2 dt 2. x^3 (SQRT1-x^2) dx 3. sin(lnt)dt 4. ln(1+x^2)dx 5. x^2cos4x dx

Put-Call Parity: One-Year European Call Option

Dynamic energy system stock is currently trading for $33 per share. The stock pays no dividends. A one-year European put option on Dynamic with a strike price of $35 is currently trading for $2.10. If the risk-free interest rate is 10% per year, what is the price of a one-year European call option on Dynamic with a strike price

Riemann integrable problem

Let f be continuous on [a,b] and suppose that f(x)>=0 for all x E [a,b]. Prove that if there exists a point c E [a,b] that f(c)>0, then b int a f> 0. Please refer to attached file for proper formatting.

Parametric Curves...

For the parametric curve given by: x = e^(sqrt(t)) , y = t - ln(t^2) (a) find an equation of the tangent line to this curve at the point corresponding to t=1. (b) For what values of f is the tangent line vertical? (c) For what values of f is the tangent line horizontal?

l'hopital and taylor series questions

Find lim┬(x→0+)⁡〖x^x 〗. (personal note: I think the approach is to use l'Hopital's rule) Prove that e is irrational. (hint: Suppose false, so that e = p/q where p,q ∈ N. Write e = e^1= P_n (1)+ R_n (1), multiple both sides by n! and deduce a contradiction when n ∈ N is sufficiently large). Expand the polynomia

Calculus - Differentiation of the Function

Please see attached. 1. If f(theta) = 10sin^2 (0.4 theta - 0.1) show that f'(theta) = 4 sin (0.8 theta - 0.2) and evaluate f" (pi / 4) correct to 3 decimal places 2. Differentiate the following functions: i. f (t) = (3t-2) sin 2t ii. f (theta) = 2 cos(3 theta - 1) / 3 theta

Fundamental Theorem of Calculus and Uniform Convergence

Suppose f is Reimann integrable on [a,b] and let F(x)= ∫_a^x▒f(t)dt for all x ∈[a,b]. Prove that F is continuous on [a,b] (hint: f must be bounded) Let F(x) = {█(x^2 sin⁡(1/x) if 0<|x|≤1,@0 if x=0)┤ and let f(x) = F'(x) Prove that F'(x) exists for all x ∈[-1,1] Find f(x) for all x ∈[-1,1], and prove t

Argand diagrams

Explain the geometrical relationship between the points in the Argand diagram represented by a complex number z and a + (z-a)e^i*theta where theta is a fixed real number. The values for a=0.76 and b=1.45. Please refer to the attachment for the proper formatting.

Calculus : Cross Product

I Have a question regarding cross products, if you know one of the vectors to be crossed and the result of the cross product how do you find the other vector. For example Vector A is unknown and Vector B=-2 Z and the result is a vector that 5X+8.66Y

autonomous differential equations

A) Check that z(t) = 1 + sqrt(1 + 2t) is a solution of the autonomous differential equation dz/dt = 1/(z-1) with initial condition z(0) =2 b) Estimate z(4) if z obeys the differential equation dz/dt = 1/(z-1) with initial condition z(0) = 2.

Solving a differential equation: Example

Dy/dx= 4x+ 9x^2/(3x^3+1)^3/2 given point: (0,2) (that's 4x plus 9x squared over (3x cubed + 1) to the 3/2 power) Use the differential equation and the given point to find an equation of the function.

Autonomous differential Equation

The rate at which a bacteria population multiplies is proportional to the instantaneous amount of bacteria present at any time. The mathematical model for this dynamics can be formulated as follows: db/dt = kb where b is a function in terms of the time t, b(t) is the number of bacteria at the time t, and k is a constant.

Determining Marginal Cost

1) The marginal cost (dollars) of printing a poster when x posters have been printed is dc/dx = 1/5√x^4 Find the cost of printing posters 18 through 130 The cost of printing posters 18 through 130 is $ _ (Round to the nearest cent)

Calculus Question

Problem: Find the area of the region in the first quadrant bounded by the line y=6x, the line x=4, the curve y=2/x(squared), and the x-axis. (1) Part A: Solve the above problem using any methods, concepts.

Evaluate the sum

I really need assistance on these three calculus problems.... Evaluate the sum: 7 &#8721; k (6k+5) (Simplify your answer) K=1 Evaluate the sum: 6 &#8721; (-13k) K=1 (Simplify your answer) Find the function y(x) satisfying dy/dx=6x-5 and y(5)=0 is y(x) = ? (Simplify your answer)

Find the Radius, Height, and Volume

Please help with these problems, I am stuck so if you could please post your steps, id appreciate it. 1.Use Newton's method to estimate the one real solution of x(exponent 3)+5x+1=0. Start with x0=0 2. If sinh x= 16/63, find cosh x, tanh x, cothx, sech x and csch x. 3. A right triangle whose hypotenus is square root of

Money Circulation Impact

Suppose that each dollar introduced into the economy recirculates as follows: 85% of the original dollar is spent then 85% of that $.85 is spent, and so on. Find: The economic impact (the total amount spent) if $1,000,000 is spent.

Differential Equation Word Problem: A body falling in a relatively dense fluid, oil for example, is acted on by three forces: the weight W due to gravity (acting downwards), a resistence force R and ...

A body falling in a relatively dense fluid, oil for example, is acted on by three forces: the weight W due to gravity (acting downwards), a resistence force R and a bouyant force B (both actin upwards). The wieght W of the object of mass m is mg. The bouyant force B is equal to the weight of the fluid displaced by the object.

Particular Solution of the Differential Equation

See the attached file. 1. Find the particular solution of the differential equation that satisfies the boundary condition. , y(1) = 2 A) B) C) D) E) 2. Find the particular solution of the differential equation that satisfies the boundary condition.

Calculus - Differentiate

I have an exam coming up and I am having problems with these two questions. Please show work and final answer. That way I can rewrite what you do so I understand it. 1. d/dt (x sin x + x^2)/ (x^2 + 1) 2. Find the second derivative if f(x)= sec(4x)

Determining the Limits Involving Infinity

(2) For the function g whose graph is given (see attached), state the following: (a) lim x->∞ g(x) (b) lim x->-∞ g(x) (c) lim x->3 g(x) (d) lim x->0 g(x) (e) lim x->-2+ g(x) (f) The equations of the asymptotes.

Piecewise Function Straight Lines and Parabola

4. Solve: a Solve for x if 2logx = 2 + log25 b Simplify log3 81 - log3 27 + 4log3 v3 c Find the inverse of the function f(x) = e^x + 1 5. This is completing a piecewise function which consists of 2 straight lines and a parabola. Coordinates given are (-6,0) (-3,6)-straight line, (-3,6) (.5, -6) (2,-4) -parabola, and last

Mathematics

1.Find the values of a and b for the polynomial f(x) = 2x^3 + ax^2 - 4x + b, given that f(x) is divisible by x+1 and x-3. Write f(x) in a factorised form. Hint: consider f(-1) and f(3) 2. The population of a country is found to be growing continuously at an annual rate of 1.98% t years after 1 January 1960. The total populat

Statics problems

1. A horizontal circular plate is suspended as shown from three wires that are attached to a support at D and form 30° angles with the vertical. Knowing that the z component of the force exerted by wire BD on the plate is 232.14 N, determine ( a) the tension in wire BD, ( b) the angles ux, uy, and uz that the force exerted at

Find the slope of the function's graph at the given point

1. Find an equation for the line tangent to y= -5-5x^2 at (5,-130) 2. Find the slope of the function's graph at the given point. Then find an equation for the line tangent to graph: f(x) = x^2 + 2, (-2,6) What is the slope of the function's graph at the given point? 3. Using the defenition, calculate the derivative