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D08 - Advanced Applications
​(including Optimization & Implicit Relations)

Overview

D08 Standard Document

Objectives & Resources

1. Understand that the derivative of a function has multiple interpretations and applications, including those that involve instantaneous rates of change. [EU 2.3]
2. Be able to solve a variety of problems, including rectilinear motion, related rates, optimization, and other advanced problems. [LO 2.3C]
     a. Motion Problems: Know that the derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration. [EK 2.3C1]

MOTION PROBLEMS

Picture
EXPLORATION
Picture
PRACTICE
     TEXTBOOK​
​          4.2 (page 209): 30.
     b. Related Rates Problems: Know that the derivative can be used to solve related rates problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known. [EK 2.3C2]

RELATED RATES PROBLEMS

Picture
EXPLORATIONS
     Falling Ladder Activity
     Cone Simulation

​     Cube Simulation Growth
​     Cube Simulation Shrink
Picture
PRACTICE
     TEXTBOOK Lesson 4.6: Rates & Related Rates
     Exercises 4.6: #1, #3, #4, #7, #13, #15.

Picture
PRACTICE
  • Practice: Related Rates Intro
  • Practice: Related Rates (multiple rates)
  • VIDEO: Related rates (approaching cars)
  • VIDEO: Related rates: Falling Ladder
  • Practice: Related Rates (Pythagorean Theorem)
  • VIDEO: Related rates: water pouring into a cone​
  • Practice: Related rates (advanced)
  • VIDEO: Related rates: shadow
  • VIDEO: Related rates: balloon​​
     c. Optimization Problems: Know that the derivative can be used to solve optimization problems, that is, finding a maximum or minimum value of a function over a given interval. [EK 2.3C3]

OPTIMIZATION PROBLEMS

Optimization Problems usually involve finding "extrema." This link provides a good description of local/relative maximum/minimum values as well as global/absolute maximum and minimum values.
QFT - Finding Extrema - Summary of Questions (Blue)

​
QFT - Finding Extrema - Summary of Questions (Period 4)
Optimization Sample Problem: Maximizing Volume of Open-Top Box (Rectangular Solid) Made from Letter Size Sheet of Paper
Picture
PRACTICE
​   TEXTBOOK
​     4.3 (p218): 2, 5, 6, 7, 10, 11.
     4.1 (p199): 1, 2, 3, 4, 9, 16. REVIEW
     4.2 (p209): 1, 2, 3, 10, 17. REVIEW
Picture
PRACTICE
  • ​Optimization Practice
     d. Other Applied Problems: Be able to solve problems involving rates of change in applied contexts by knowing that the derivative can be used to express information about rates of change in these contexts. [LO 2.3D and EK 2.3D1]

3. Make sense of problems and persevere in solving them. You will be able to ... ​

     a. Understand the problem, find a way to tackle it, and work until it is done.
     b. Analyze givens, constraints, relationships, and goals.
     c. Make conjectures about the form and meaning of the solution, and plan a solution pathway rather than simply jumping into a solution attempt.
     d. Check your answers to problems using common sense and alternative methods.

SAMPLE CONTENT ASSESSMENT ITEMS

Problem Set 1
KEY Problem Set 1
File Size: 1968 kb
File Type: pdf
Download File

Problem Set 2
File Size: 431 kb
File Type: pdf
Download File

KEY Problem Set 2
File Size: 2226 kb
File Type: pdf
Download File

D07 Review Problems
File Size: 1170 kb
File Type: pdf
Download File

KEY D07 Review Problems
File Size: 1022 kb
File Type: pdf
Download File

The Khan Academy and Textbook problems above are also good examples of possible assessment items.
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