A Dash of Limits - Arizona State University

A Dash of Limits - Arizona State University

Simple Rules for Differentiation Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and differences. Rules Power Rule a

a 1 f ( x ) x f ( x ) ax

For the function , for all arbitrary constants a. Rules Sums and Differences Rule If both f and g are differentiable at x, then f g f g the sum and the difference are differentiable at x and the derivatives are as follows.

F (x ) f (x ) g(x ) has a derivative F (x ) f (x ) g(x ) Rules Sums and Differences Rule If both f and g are differentiable at x, then f g f g the sum and the difference are differentiable at x and the derivatives are as follows.

G (x ) f (x ) g(x ) has a derivative G (x ) f (x ) g(x ) Example 1 Use the simple rules of derivatives to find the derivative of f (x ) x 6 Example 2

Use the simple rules of derivatives to find the derivative of D( p) 10p 3 2 Example 3 Use the simple rules of derivatives to find the derivative of 6

y 4 x Example 4 Use the simple rules of derivatives to find the derivative of 3 y 6x 15x 2 Example 5 Use the simple rules of derivatives to find the

derivative of 5 p(t ) 12t 6 t t 4 Example 6 Find the slope of the tangent line to the graph of the 3 x = 9. Then find the function y x 4 5x at 2 equation of the tangent line.

Example 7 Find all value(s) of x where the tangent line to the function below is horizontal. 3 2 f (x ) x 5x 6x 3 Example 8 Assume that a demand equation is given by

q 5000 100p Find the marginal revenue for the following levels (values of q). (Hint: Solve the demand equation for p and use the revenue equation R(q) = qp .) a. q = 1000 units b. q = 2500 units c. q = 3000 units Example 9-1 An analyst has found that a companys costs and revenues in dollars for one product are given by the functions

C (x ) 2x and 2 x R (x ) 6x 1000 respectively, where x is the number of items produced. a. Find the marginal cost function. Example 9-2

An analyst has found that a companys costs and revenues in dollars for one product are given by the functions C (x ) 2x and 2 x R (x ) 6x 1000 respectively, where x is the number of items produced.

b. Find the marginal revenue Example 9-3 An analyst has found that a companys costs and revenues in dollars for one product are given by the functions C (x ) 2x and 2 x R (x ) 6x 1000

respectively, where x is the number of items produced. c. Using the fact that profit is the difference between revenue and costs, find the marginal profit function. Example 9-4 An analyst has found that a companys costs and revenues in dollars for one product are given by the functions C (x ) 2x and

2 x R (x ) 6x 1000 respectively, where x is the number of items produced. d. What value of x makes the marginal profit equal 0? Example 9-5 An analyst has found that a companys costs and revenues in dollars for one

product are given by the functions C (x ) 2x and 2 x R (x ) 6x 1000 respectively, where x is the number of items produced. e. Find the profit when the marginal profit is

Example 10-1 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2 M (t ) 3.044t 379.6t 14274.5t 139433 where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.

a. 1920 Example 10-2 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2 M (t ) 3.044t 379.6t 14274.5t 139433 where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find

the rate of change of money in circulation in the following years. b. 1960 Example 10-3 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2 M (t ) 3.044t 379.6t 14274.5t 139433 where t represents the number of years

since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years. c. 1980 Example 10-4 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2

M (t ) 3.044t 379.6t 14274.5t 139433 where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years. d. 2000 Example 10-5 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3

2 M (t ) 3.044t 379.6t 14274.5t 139433 where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years. e. What do your answers to parts a-d tell you about the amount of money in circulation in those years?

Recently Viewed Presentations

  • WELCOME TO LAKE HAVASU HIGH SCHOOL ATHLETIC PARTICIPATION

    WELCOME TO LAKE HAVASU HIGH SCHOOL ATHLETIC PARTICIPATION

    (such as doctor's appointment) on a Friday, the day of a game or the day after an away game the athletic administration must be notified 24 hours prior to the absence and the Athletic Director, Athletic Administrative Assistant or administration...
  • PRESENTATION CHIETA PARTNERSHIP CONFERENCE OF 26 NOVEMBER 2015:

    PRESENTATION CHIETA PARTNERSHIP CONFERENCE OF 26 NOVEMBER 2015:

    Shareholder- own the ships and define its purpose. Governing Board- steers the ship and appoint key crew,set the direction,monitor and ensure corrective and pro-active action is taken where needed. Management - we row . CHIETA, The Catalyst for Enhanced Skills,...
  • Office of the

    Office of the

    Dr. Ash Bullard of the Department of Fisheries recently received $132,000 from BP through the Gulf of Mexico Alliance's Gulf Research Initiative to enable him to conduct time-sensitive sampling and related studies. Five teams responded to the recent RFP from...
  • Present Value Calculations

    Present Value Calculations

    Costs of process equipment scale according to the "six-tenths rule" C2/C1 = (Q2/Q1)0.6 See, for example: "Cost and Optimization Engineering" by F.C. Jelen and J.H. Black, McGraw-Hill, 1983. Other Items Working Capital Raw materials and supplies inventory Finished goods in...
  • BMS 631: Lecture 3 - Purdue University

    BMS 631: Lecture 3 - Purdue University

    BMS 631: Lecture 4 Fluorescence J. Paul Robinson, PhD SVM Professor of Cytomics & Professor of Biomedical Engineering Purdue University www.cyto.purdue.edu Notice: The materials in this presentation are copyrighted materials. If you want to use any of these slides, you...
  • Assessment Arrangements update Purpose of this session:  Update

    Assessment Arrangements update Purpose of this session: Update

    We will also continue to use past papers for revision. Within key stages, we will continue to assess what pupils understand and can do in a way that best suits our school. We will report . the pupil's scaled score...
  • You Need: Pencil, Maya Angelo Poem, Argumentative Letter ...

    You Need: Pencil, Maya Angelo Poem, Argumentative Letter ...

    poem, "Life Doesn't Frighten Me At All". The speaker of. the poem was unafraid, confident, and willing to face. the challenges before them. Like the speaker of the poem, people do not have to fear their fears, but be confident...
  • Town of Queen Creek Fire Department

    Town of Queen Creek Fire Department

    Long frontage with Queen Creek Wash. Direct connection to the wash trail. Alternative "B" ...