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# Derivatives

### Derivatives, Second Derivatives and Profit Function - Lemonade Stand

A. Write a function for your profits for each price you charge. This is done by multiplying (P-.5) times your function (y= -100x + 250). I.e. if your function is Cups Sold = 1000 - 100P, your profit function would be (P - .5)*(1000 - 100P). B. Calculate the first derivative of your profit function, and create another table

### Derivatives, Revenue Function, Maximizing Profit - Lemonade Stand

Data: regression equation: y= -100x + 250 regression coefficient: r= -1 X Y Predicted value 0.25 225 225 0.5 200 200 0.75 175 175 1 150 150 1.25 125 125 1.5 100 100 1.75 75 75 2 50 50 2.25 25 25 2.5

### Definition of a Limit and Derivative, Product Rule, Tangent Line

1. Give the definition of limit in three forms: &#949;?&#948; , graphical, and in your own words. 2. Define the derivative. List what you consider to be the five most useful rules concerning derivatives. 3. Give an argument for the product rule. 4. What is the tangent line approximation to a function? 5. What is the Taylor p

### Rate of Change, Derivatives & Product and Quotient Rule

See the attached file. 71. The local game commission decides to stock a lake with bass. To do this 200 bass are introduced into the lake. The population of the bass is approximated by P(t) = 20 (10 + 7t)/(1 + 0.02 t) where t is time in months. Compute P(t) and P'(t) and interpret each. 57. The monthly sales of a new compute

### Derivatives : Average Cost, Marginal Cost and Minimum Cost

If it costs Acme Manufacturing C dollars per hour to operate its golf ball division, and an analyst has determined that C is related to the number of golf balls produced per hour, x, by the equation C = 0.009x squared - 1.8x + 100. What number of balls per hour should Acme produce to minimize the cost per hour of manufacturing t

### Derivatives to calculate volume and area

Volume. An open box is to be made from a six inch by six inch square piece of material by cutting equal squares from the corners and turning up the sides. Find the volume of the largest box that can be made Area. A rectangular page is to contain 36 square inches of print . The margins at the top and bottom and on each si

### 14 Derivative Problems : Word Problems, Velocity, Acceleration and Rate-of-Change

Find 3rd derivative f(x)= 3/16x^2 Find the indicated value f(x)= 9-x^2 value f''(-sq rt 5) Find f'''(x) f''(x)=2x-2/x Find the second derivative and solve the equationf''(x)=0 f(x)=x/x^2+1 The velocity of an object in meters per second is v(t)=36-t, 0<t<6 Find the velocity and acceleration of the

### Application Problem Involving Derivatives

A container with a rectangular base, rectangular sides and no top is to have a volume of 2 subic meters. The width of the base is to be 1 meter. When cut to size, material costs \$20 per square meter for the base and \$15 per square meter for the sides. What is the cost of the least expensive container?

### Definition of the Derivative, Product and Quotient Rules

1. Differentiate from first principles( for x radians): a) sin x b) cos x 2. Products and quotients For a function, f(x), which can be expressed as a product or quotient of other functions, u(x) and v(x), there exist a) the product rule, f(x) = u(x) ? v(x),

### Derivatives (5 Problems)

Find the derivative of the function and simplify. y= (3x^2 + 7) (x^2 - 2x) s= (4 - 1/t^2)(t^2 -3t) f(x)= x^2 + x - 1/ x^2 -1 f(x)=^3 sqrt(x^2 -1) g(x)= sqrt (x^6 - 12x^3 + 9)

### Differentation

The weekly demand and cost functions for a product are p= 1.89 - 0.0083x and c= 21+ 0.65x write the profit function for this product. find the marginal cost of the function. C= 475 + 5.25x^2/3 find the marginal revenue function. R= x(5+ 10/ sqrt(x) Find the marginal profit function. P= 1/1

### Rate of Change and Derivatives Applied to a Falling Rock and Profit Margin

A rock is dropped from a tower on the Brooklyn Bridge, 276 feet above the east river. Let t represent the time in seconds. a.) write a model for the position of the function (assume air resistance is negligible.) b.) find the average velocity during the first 2 seconds. c.) find the instantious velocity when t=2 and t

### Apply the Chain Rule to Finding Partial Derivatives

A. Use the chain rule to find dz/ds and dz/dt as functions of x, y, s and t B. Find the numerical values of dz/ds and dz/dt when (s,t) = (2, -2). Suppose z = x2 sin y, x =... (Please see the attached file for the fully formatted problem).

### Apply the chain rule to a partial derivative

Use the chain rule to find dw/dt as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite e5t as x. dw/dt = Suppose: w = x/y + y/z ... (Please see the attached file for the fully formatted problem.) Note: Use exp() for the exponential function. Your answer should be an expression in

### Show that the composite function G (z) = g (2z - 2 + i) is analytic in the half plane

Show that the composite function G (z) = g (2z - 2 + i) is analytic in the half plane x > 1, with derivative .... see attachment

### Verifying Entire Functions

Apply the given theorem to verify that each of these functions is entire: (a) f (z) = 3x + y + i (3y - x) (b) f (z) = sin x cosh y + i cos x sinh y (c) f (z) = e-y sin x - i e-y cos x (d) f (z) = (z2 - 2) e-x e-iy. See attached file for proper formatting.

### Change of Coordinates Lagrangian

Consider a Lagrangian system, with configuration space R^n, given by (x^1, ... x^n); and Lagrangian L(x', ..., x^n; v^1, ... v^n). Now consider a new system of coordinates, (y^1,... ^n), for this same system, so the y's are functions of the x's; and, inverting, the x's are also functions of the y's. Find the Lagrangian in the y-

### Lennard-Jones Potential : First, Second, and Third Derivatives

Please see the attachment. I am trying to take the first, second, and third derivatives of this equation, so I can utilize them to determine relationship to force and modulus, and thermal expansion coefficient. Please help me take the derivatives properly. Thanks!

### Word Problems : Derivatives and Rate of Change

41) Suppose that the average yearly cost per item for producing x items of a business product is C(x)=10+(100/x) . if the current production is x=10 and production is increasing at a rate of 2 items per year, find the rate of change of the average cost. 45) Suppose a 6ft tall person is 12 ft away from a 18-ft tall lamppost. i

### Max, min, inflection pts

If f(x) = x^4 - 4x^3 + 10 find the relative extrema of the function and the points of inflection of its graph. Also, sketch the graph of the function. a. The x-value(s) of the relative minima of function f: ________________ b. The x-value(s) of the relative maxima of function f: _______________ c. The x-value(s) of poi

### Derivatives : First Principles and Leibniz Notation (Five Problems)

Please assist me with the attached problems, relating to first principles and Leibniz Notation.

### Gompertz Equation

2. The Gompertz equation y'(t) = y[a-b*ln(y)] is an important model for avascular tumor growth. In the avascular growth phase, tumor cells obtain nutrients directly from the surrounding tissue. (The transition from avascular to vascular growth is marked by the onset of angiogenesis, the formation of blood vessels, which are

### Derivatives and Rate of Change : Calculate the rate a shadow is moving up a wall.

I need to determine how fast a shadow is moving up a wall. Given the heigth of the wall the height if the object that cast the shadow. The length of the wire the object moves on, and the height of the light that casts the shadow. I have worked out the first sections in an Excel 2000 spreadsheet but I need a push in the right di

### Lagrangian multipliers

1) Who can explain me Lagrangian multipliers with drawings scheme etc... 1)I just can't imagine what is happening in space with Lagrangian multipliers. 2) I did this problem but here also I can't understand it, because I can't understand what is happening in space! could you explain it with drawings and schemes please : the

### Evaluation the radius of curvature

Curvature (III) Differential Calculus Evaluate the radius of curvature at any point (x,y) for the curves : (a) xy = c2 (b) y = (1/2)a(ex/a +

### Question about Limits and Derivatives

Please show me all of your work so that I can understand how to do the problems correctly. Please double check all of your answers to that you are sure everything is correct. Thanks. 1. Find the limit L. Then use the definition to prove that the limit is L. . 2. Find the limit: . 3. Calculate the derivative of .

### Derivatives

Solve the following two equations. In each case, determine dy/dx: a.)y=xcos(2x^2) Is this right? y'=x(-sin)(2x^2)(4x) =-4x^2sin(2x^2) b.)y=xe^-x^2 Is this right? y'=-xe^-x^2+1(e^-x) =-xe^-x^2+e^-x

### 4724

Category: Business > Management Subject: Management Science Details: 1. Which of the following statements are true for f(x) = x2? (Chapter 10) a. f(x) is a concave function b. f(x) </= 20 is a convex set c. f(x) >/= 5 is a convex set d. none of the above 2. Which of the following statements are true for f(x) =

### Differentiation and Rate of Change Word Problems

1) Differentiate the equations a) y=8/5xsquared b) y=4cosX - 3ex 2)The formula C=60=t3/12 This equation refers to a machine in a workshop. This machine costs £C to lease each week according to the formula and t is the number of hours per week worked by the machine. The rate of increase of cost during the week is given by

### Differentiation : Critical Point - Find Maximum Value

A manufacturer produces cardboard boxes that are open at the top and sealed at the base. The base is rectangular and its length is double its width. Let x denote the width in metres. the surface area of each such box is fixed to be 3 square metres. The manufacturer wishes to determine the height h and the base width x, in metres