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# Calculus and Analysis

### Find the absence of limits for z/z-negated

Show that the limit of the function: f(z) = ( z / z-negated)^2 (z-negated -> by this i mean z with a bar above it) as z tends to 0 does not exist. Do this be letting nonzero points z = (x,0) and z = (x,x) approach the origin. note: it's not sufficient to simply consider points z = (x,0) and z = (0,y)

### Differentials Searchlight

A searchlight is 150 feet from a straight wall. As the beam moves along the wall, the angle between the beam and the perpendicular to the wall is increasing at a rate of 2.0 (degrees per second). How fast is the length of the beam increasing when it is 170 feet long?

### The Tangent Line to a Circle

** Please see the attached PDF document for the full problem description **

### Loss of accuracy (Matlab only)

1) Use Taylor expansions to show that f(t+h)-f(t-h)/(2h)=f '(t)+O(h^2) 2) Let f(x)=e^x and t=0. Plot the error (using Matlab) in the approximation of f '(t) by the finite difference in 1) for h=logspace(-1,-16,100). Explain your results 3) Use calculus to show that the optimal h to use in the approximation in 2) - the one

### Solve the Curve Distance using Simpson's Rule

The distances x (in in.) from one end of a barrel plug (with vertical cross section) to its center of mass, as shown in the figure is x=0.9129 integrate x*square root of (0.3-0.1x) (0<=x<=3) Find x with n=12

### Calculus with Parametric Curves

Find the exact length of the curve x = 3*cos(t) - cos(3*t), y = 3*sin(t) - sin(3*t) The curve is plotted in the attachment. How would I integrate the derivatives to get the length of the curve? I think I have to integrate this in several steps, accounting first for the parts below the x-axis, and then for the curve above

### Various Problems in Precalculus

Total of 29 problems that look like these What is the degree of f(x) =3x^6 +4x^2 +x - 1 What is the leading coefficient of f(x) = 3x^6 +x^2 +x - 1 Write a polynomial in simplest form with roots 5i and -5i Solve (x - 1)(x - 2) ? 0 x - 3

### Roots Polynomial Functions

There are total of 14 attached problems and they look like these: Find all possible rational roots of: f(x)=2x ^4 - 5x^3 +8x +4x+7 Find all the real roots of f(x) = x^3 + x^2 -8x -6 Find the number of possible positive and negative real zeros of :f (x) = 3x^3 -4x^2 -5x +1. See the attached file.

### Concept of minimum/maximum for problem solving.

Hello, I have been studying the concepts of minimums and maximum values. However, I have a word problem that is really giving me a hard time and I would appreciate some assitance so I will know how to go about solving such a problem in the future. * A rectangular hole is to be cut in a wall for a vent. If the perimeter of th

### Finding an Equation of the Tangent to the Curve

Find an equation of the tangent to the curve: x = tan (theta), y = sec (theta) at the point (1, sqrt(2)) by two methods: (a) without eliminating the parameter, and (b) by first eliminating the parameter.

### Tangent lines at a point

a)Find the slope of the tangent line to the function y=1/sqrt(2x-1) at the point a, using the definition of a derivative. b) At which point does the tangent to the function have a slope of -1 c) Find an equation of the tangent line at the point in (b)

### Invariance of the divergence of a vector field

9. Show that the operator Del_dot (divergence) is invariant under a coordinate transformation. (Please show complete steps)

### Two Separable Differential Equations

Please find the attached. i) x ̇=-(t+1)x/t ii) x ̇=-t^3/(x+1)^2 a) Find all the solutions of i) and ii). b) for i) and ii) Discuss/Give possible choices for a domain B⊂R^2 such that the IVP with x(t_0 )=x_0 has a unique solution.

### Derivative as an Instantaneous Rate of Change

The value (in thousands of dollars) of a certain car is given by the function V=48/t+3 where t is measured in years. Find a general expression for the instantaneous rate of change of V with respect to t and evaluate this expression when t =3 years. Will the general expression be -48/(t+3)^2, and the rate change -\$1300/year?

### Solving for a Vector

Question: Show that if vector r x (d(vector r)/(dt)) = 0, |vector r| is constant.

### Laplace Transform for Differential Equation

Problem # 2 1. Using "Laplace transform" to solve the differential equation, with f(x) defined in problem 1

### Solutions to First-Order Ordinary Differential Equations

Question 1: (1 points) Solve the following equations for given initial condition. Please represent your answer in the implicit form ( ). Use y(x) instead of y Question 2: (1 points) Integrate the following equations and write a solution for given initial condition. Please represent your answer in the implicit form

### Constant Coefficients and Maple Verification

Could you work out the problem and also add the Maple verification if possible? Please see the full problem description in attached PDF.

### Calculus Compound Interest/Principle

A friend, who is currently 40 years old, wants to know how much money she will have at retirement if she invests \$5,250 every year in a fund that ears 8% per year. a. List the balance on the account for the first three years of the fund if interest is compounded continuously. b. Graph the balance on account against time and

### Calculus: Best Fit Lines and Prediction

See the attached files. Consider the weight of a pet ferret recorded over the first six months of life: Age 0 1 2 3 4 5 6 Weight .69 1.09 1.38 1.61 1.79 1.94 2.08 A. Graph the ferret weights on age and find a mathematical expression that relates weight of a ferret as a function of age. Compute how much weight a ferret

### Calculus: Lower and Upper Estimates

Roger decides to run the marathon. His friend follows in a car and clocks his speed in mph. Roger stops after an hour and a half but his friend keeps the following records: 1. Calculate upper and lower estimates for the distance the two have traveled from the starting gate. 2. If he started at 8am could Roger have walked t

### Metabolizing Drugs

People metabolize drugs at specific rates. For example, caffeine has a half-life of about four hours. Determine a functional relationship between caffeine (in milligrams) in your system and your last cup of java. (Each cup has 80 mgs). a. Graph the function for time= 1, 2, 3, 4, 5 and 6 hours since your last cup. What is th

### Marginal cost function ..

The marginal cost function of a product, in dollars per unit, is Câ??(q)= q^2-50q+700. If fixed costs are \$500, find the total cost to produce 50 items.

### Estimate the total solar PV production between 2000 and 2010.

Homework Set 15: Problem 10 Section 5.4, Problem 11 Section 5.4 Section 5.4: Problem 10 pg. 257 10. World annual natural gas consumption, N, in millions of metric tons of oil equivalent, is approximated by N= 1770 + 53t, where t is in years since 1990. A. How much natural gas was consumed in 1990? In 2010? B. Esti

### Estimate total oil consumption during this 25-year period

Homework Set 13: Problem 8 Section 5.1, Problem 12 Section 5.1 Section 5.1: Problem 8 pg. 239 8. The following table gives world oil consumption, in billions of barrels per year. Estimate total oil consumption during this 25-year period. Year 1980 1985 1990 1995 2000 2005 Oil (bn barrels/year) 22.3 21.3 23.9 24.9 27

### Particle velocity and harmonic motion

You are in your Physics lab and your professor has an experiment set up to help you learn about harmonic motion. The set up is that you have 112.5/pi^2 (approxmately 11.39 kg) on a frictionless surface that is attached to a spring constant 12.5 N/m. You pull the block to 3/pi meters (approximately .955 meters) such that x(0) = 3

### Finding the Limit of Given Functions

Find the limit of the following functions <!--[if !supportLists]-->- <!--[endif]-->Lim ( x --> 0) of [8(x^3+6)/(-4(456)(x^3+6))] <!--[if !supportLists]-->- <!--[endif]-->Lim ( x --> 5) of [10(x-4)(3x-3)/(2x(3x-3))]

### Green's function using Laplace

I would like help with the step's to solve the system, I have three parts: x' + x = f(t) using Green's function Given for the force f(t) = ?(t) with the initial condition x(t = 0) = 0. (a) Using the method of Laplace transformation, determine G(s). (b) Determine G(t). (c) Using the Greenâ??s function G(t), determine t

### Determining parabola/axis interaction points

#1 A parabola is given by the equation: f (x) =-x^2 + 4x + 5 a) Determine the parabola intersection point with the first axis b) Determine the area of the area bounded by the parabola and the first axis #2 The function f (x) and g (x) is given by: f (x) =-x^3 + 1.5x^2 + 6x - 1 and g (x) = 6.5 - x a) Solve th

### Cost Functions: Average, Marginal, and Minimal Average Costs

Please refer to the attached file. For the given cost function C(x)=40000+500x+x2 (the function is 40,000 +500x+x^2) find: a) The cost at the production level 1850 b) The average cost at the production level 1850 c) The marginal cost at the production level 1850 d) The production level that will minimize the average