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

Stationary Points points where Copy 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.)

Intervals and Sign Diagrams Copy increasing: decreasing:

Sign Diagrams

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.

Don’t forget the units of measure! Example Copy 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!

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.

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. Copy