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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

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

1. Give the definition of limit in three forms: ε?δ , 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

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)

Derivatives of Functions Simplified

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)


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

Derivatives Functions Simplified

Find the derivative of the function and simplify. f(x)= x^3(5 - 3x^2) f(x)= x^2 + x - 1/x^2 - 1 f(x)= (5x^2 + 2)^3 g(x)= x sqrt(x^2 + 1) given f(x)= 3x^2 + 7x +1, find f''(x)

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

Use the definition of the limit to find the derivative.

Use the definition of the limit to find the derivative of the function f(x) = 7x + 3 Find the slope of the graph of F at the indicated point f(x) = sqrt(x) + 2; (9,5) use the derivative to find the equation of the tangent line to the graph of f at the indicated point. f(x) = x^2 + 3/x; (1,4)

Derivatives (4 Problems)

Find the derivative of the functions. y= (2x - 7)^3 h(x)= (6x - x^3)^2 f(t)= sqrt(t + 1) f(x)= x^3(x - 4)^2

Derivatives at a Point (3 Problems)

Find the value of the derivative of the function at the indicated point. f(x)= 1/3 (2x^3 - 4) point (0, -4/3) h(x)= x/(x - 5) point (6,6) f(t)= (2t^2 - 3)/3t point (2, 5/6)

Rate of Change : Exponential Growth

4.) Ever wonder why bacteria are so hard to contain ? A certain coloney of bacteria has an initinal population of 10,000. After t hours, the coloney has grown to a number P(t) given by P(t) = 10000(1+.86t + t^2). a.) Find the growth rate (rate of change) of the population P with respect to time t. [FInd P'(t)] b.) Fi