Graphing and Applications
Unit Resources:
Assignments, past Quizzes and Exams:
EXAM - Curves and Applications
Assignments, past Quizzes and Exams:
EXAM - Curves and Applications
- 2023 Exam and Answer Keys
- Applications:
- 2023: Quiz and Answer Key
- 2021: Summary and Answer Key
- Curve Sketching Quiz
- 2021: Summary and Answer Key
- Checks for Understanding
- Problems 1: Original and Answer key
- Problems 2: Original and Answer key
- Problems 3 (old exam): Original and Answer key
- Curve Sketching (Graphing) 1: Original and Answer Key
- Curve Sketching (Graphing) 2: Original and Answer Key
APPLICATIONS: (PROBLEM SOLVING)
Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989: Chapter 3
- Solve problems using properties of functions.
- Solve problems using rates.
- General Process:
- We need to match the number of variables we have with the number of equations we are working with.
- We need to have a function that we are maximizing or minimizing.
- To find the maximum or minimum values of the function we solve for dy/dx=0.
Maximum and Minimum Problems
- Many problems involve two variables.
- Create a function in one variable to solve the problem.
Geometric Shape Lesson 1 [Notes Geometric 1]
Geometric Shape Lesson 2 [Notes Geometric 2]
- Geometric Lesson 2 HW: Answer Key
- Video Answers [2. Isos Triangle] [4. Cylinder]
Velocity and Acceleration
- The first derivative of 'displacement' is velocity.
- The first derivative of 'velocity' is acceleration (so acceleration is the second derivative of displacement).
- Examples (Notes) [video Q1] [video Q2]
Applications of Derivatives: Check for Understanding Problem 1 and Answer Key
Rate Problems
Use as Review:
- We need to know what rate we are solving for - are we taking the derivative with respect to a measurement or with respect to time?
- We need to match the number variables to the number of rates.
- If we have too many variables for the number of rates:
- We will need to substitute related values leaving only the variables that match our rates.
- Or we may need to find a rate in one context and use it in a related context to solve.
- Examples (Notes)
- Videos: [Example 1] [Example 4] [Examples 5 & 6] [Q8] [Q9] [Q10] [Q13] [Q14] [Q15]
- Textbook Questions: page 146 [correct answer Q8: decreasing at 26.32 cm^2/s] [video Q17]
- Similar Triangles skills [video]
- Check For Understanding Problems 2: Original and Answer key
Use as Review:
- Applications 2019Quiz 1: [Quiz 1 (2019)] [Quiz 1 Key] [getting started video Q1] [cylinder video Q2]
- Applications 2019Quiz 2: [Quiz 2 2019] [Quiz 2 Key]
- Problem Unit Assignment (old exam): [Original] and Answer key
GRAPHING:
Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989: Chapter 4 and Chapter 5
Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989: Chapter 4 and Chapter 5
- Use properties of slope and concavity to determine
- regions of increase and/or decrease for function.
- the extreme value of a function.
- Use limits and other function properties to sketch functions.
The First Derivative Rule:
Notes and Examples: Original [Video Ex1] [Video Ex2] [Video Ex3ab] [Video Ex3cd] [Video Ex4]
- When a function changes from a region of increase to a region of decrease, the function passes through a maximum value.
- When a function changes from a region of decrease to a region of increase, the function passes through a minimum value.
- The value of the function (maximum or minimum) is "f(x)".
Notes and Examples: Original [Video Ex1] [Video Ex2] [Video Ex3ab] [Video Ex3cd] [Video Ex4]
- Text Questions [1c, 3ac, 4ac] [video]
- Examples
[Intro Video] - what the first derivative and the second derivative tell us about the function
The Second Derivative Test:
We use two concepts:
If we have a region of concave down at a point of zero slope, that point will be a maximum value.
Notes and Examples: Original
Curve Sketching (Graphing) 1: Original and Answer Key
The Second Derivative Test:
We use two concepts:
- We need to know where we have zero slopes (this is where we have a maximum value or a minimum value).
- We need to know if the function is concave up (positive second derivative value) or concave down (negative second derivative value).
If we have a region of concave down at a point of zero slope, that point will be a maximum value.
Notes and Examples: Original
- [Video 1: Ex 2abc]
- [Video 2: Examples "abc"] we should give out prizes for finding my 'calculated mistakes'... the second derivative in b) is 6x not 4x, luckily my mistake does not change the zero for the 2nd derivative, so the point of inflection stays as (0,1).
- Examples (Notes)
Curve Sketching (Graphing) 1: Original and Answer Key
Vertical and Horizontal Asymptotes - Rational Functions
- Vertical Asymptotes are determined by the value that causes dividing by zero.
- Horizontal Asymptotes are determined by the limit to infinity.
Limits to Infinity - polynomial functions
- We need to express polynomial functions as products of infinity to justify that we are at positive infinity or negative infinity (y-values) as we move towards infinity (x-value).
Even and Odd Functions
- Even functions have symmetry to the y-axis. Every term in the polynomial function has an even degree.
- Odd functions have symmetry to the origin (rotate 180 about origin). Every term in the polynomial function has an odd degree.
Limits to Infinity (Polynomials) and Odd/Even Notes: Original
Curve Sketching (Graphing) 2: Original and Answer Key [Graphing video]
Curve Sketching Summary Assignment: Curves 2020 [Hints] Curves Key 2020