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# Derivatives

### 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

### Derivative : What are the steps?

What are the steps for finding the derivative? 2x + 512/x

### Word Problem : Minimize Using a Derivative

The math department is planning to build a park for calculus students along the riverbank. The park is to be rectangular with an area of 512 square yards and is to be fenced off on the three sides not adjacent to the river (draw a picture) a.) What is the least amount of fencing required for this job? b.) How long and

### Change in Revenue: Derivative Problem

The Happy Hound Haven Company estimates that the revenue (in dollars) from the sale of x doghouses is given by R(x)= 625+.03x+.0001x^2. Approximate the change in revenue from the sale of one more doghouse when 1000 doghouse are sold. (Make sure to do this using derivatives). Am I correct? The derivative is R'(x) = 0.03+0.00

### Derivatives : Rate-of-Change Word Problem

For several weeks , campus security has been recording the speed of trafic flowing past a certain intersection on campus. The data suggests that between 1:00 and 6:00 pm on a normal weekday, the speed of the traffic at the intersection is approximately S(t)=t^3-10.5t^2+30t+20 miles per hour, where t is the number of hours past

### Find the extrema

Find the extrema of the function f(x)=x^3-x^2-x+2 on the closed interval [0,2] Find two numbers whose product is 192 and the sum of the first plus three times the second is a minimum. (there should be a primary and a secondary equation)

### Application of Derivatives Maximum Revenues

Find the number of units x that produce a maximum revinue R. R=800x-0.2x^2 R=48x^2-0.2x^3 Find the number of unites x that produce the minimum average cost per unit C. C=1.25x^2+25x+8000 C=0.001x^3+5x+250 find the amounts of advertising that maximizes the profit P. (s and p are measeured in thousands

### 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

### Line Integral & Partial Derivatives of a Circle

Let and let C be the circle , . A. Compute Note: Your answer should be an expression of x and y; e.g. "3xy - y" B. Compute Note: Your answer should be an expression of x and y; e.g. "3xy - y" C. Compute Note: Your answer should be a number Please see the attached file for the fully formatted proble

### Derivative of an Integral Function

Find the derivative of the function g(x)=the integral as tan(x) goes to x^2 of dt/(sqrt(7+t^4)); g'(x)=___

### 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 of Composite Functions: Differential of an Inverse

Problem: If y = tan^-1[(1+x)/(1-x)], find dy/dx. See the attached file.

### Derivatives of Composite Functions: Differential Coefficient

Problem: Find the differential coefficient of sec(tan-1 x) with respect to x.

### Implicit differentiation function

Use implicit differentiation to find dy/dx x^2 + 9xy + y^2=0 y^2 + x^2 - 6y - 2x - 5 =0

### Multiple Derivatives (3 Problems)

Given f(x)= sqrt x, find f''''(x) given f(x)= x^2 + 3/x, find f''(x) given f'''(x)= 20x^4 - 2/x^3, find f'''''(x).

### Finding Derivatives of Equations

G(t)= t/ (1 - t)^3 f(x)= (x^2 + 1/x)^5 f(x)= [(x - 2)(x +4)]^2 f(s)= s^3(s^2 - 1)^5/2 g(x)= (3x + 1)^2/ (x^2 + 1)^2

### 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)

### 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

### Velocity & Acceleration: Position of the Function of a Particle

The position of the function of a particle is given by s= 1/t^2 + 2t + 1 where s is the height in feet and t is the time in seconds. Find the velocity and acceleration functions. use implicit differentiation to find dy/dx x^2 + 3xy +y^3 = 10.

### 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)

### Derivatives and Rate of Change of Bacteria Population

G(t)= 3t^2 / sqrt (t^2 + 2t - 1) The number N of bacteria in a culture after t days is modeled by: N= 400 [ 1- 3/ (t^2 + 2)^2] Complete the table. What can you conclude? t 0 1 2 3 4 dN/dt _ _ _ _ _.

### Derivatives (3 Problems)

Find the derivative of the functions. f(x)= x(3x - 9)^3 y= x sqrt(2x + 3) y= t^2 sqrt (t - 2)

Y= 4x^3/2 /x