1. Unit 5: Introduction to Differential Calculus
21A Increasing and Decreasing Functions, 21B Stationary Points, 21C Rates of Change, 21D Optimization
Unit 5: Introduction to Differential Calculus

2. Stationary Points points where
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points where
each is a local maximum, local minimum, or a horizontal inflection
local maximum: point at the top of a “hill”
local minimum: point at the bottom of a “valley”
horizontal inflection: stationary point at neither a hill or valley
a function has at most one global maximum (or minimum) but may have multiple local maximums (or minimums)
a max. (or min.) value is the y-coordinate of a global max. (or min.)

3. Intervals and Sign Diagrams
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increasing:
decreasing:

4. Sign Diagrams

5. Example
Sketch a graph by determining end behavior and x-intercepts. Find the x-coordinates of the stationary points using the derivative. Create a sign diagram.

6. Don’t forget the units of measure!
Example
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The volume of air V (m3) in a hot air balloon after t minutes is given by
Find
the initial volume
the volume at
the rate of increase in volume at
Determine the time at which the balloon contains a maximum volume of air
Don’t forget the units of measure!

7. Optimization optimum solution: maximum or minimum value of a variable
often are local or global maximums or minimums
May need to write an equation. Use as few variables as possible. Don’t forget to consider what restrictions the domain has.
Sketch using a sign diagram and key points. Find stationary points by solving when the first derivative is equal to zero. Intercepts often indicate boundaries to the domain and range.
Determine the optimum solution.

8. Practice
p. 589: 1(b)(d)(g), 2(c)(e) p. 593: 1(b)(c)(d), 3, 5(c)(g), 7, 10 p. 596: 5 p. 597: 1(a), 2, 4 p. 601: 2, 11 Read and follow all instructions. List the page and problem numbers alongside your work and answers in your notes. Use the back of the book to check your answers.
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