Lesson 3 Optimization with Exponential Functions Optimizing Exponential

Lesson 3 – Optimization with Exponential Functions Optimizing Exponential Business Models

Optimizing Exponential Functions Many scenarios use exponential models to represent relationships. Finding a maximum or minimum still involves setting the 1 st derivative equal to zero. To find a max. or min. , set

Optimizing Exponential Functions 1. The effectiveness of studying for an exam is described by: where E represents effectiveness based on studying for t hours. If a student has a maximum of 30 hours available for studying, how long should they study for maximum effectiveness? Differentiate:

Optimizing Exponential Functions 1. Find critical number(s): Check end points :

Optimizing Exponential Business Models A consultant determines that the proportion of people who will respond to an ad for a new product, after it has been running for t days, is described by: 2. The ad can reach up to 10 000 people, and each ad response results in revenue of $0. 75 (on average). The cost of running the ad is $30, 000 to start plus $5, 000 per day. a) What is , and what does it represent?

Optimizing Exponential Business Models b) What percentage of potential customers have responded to the ad after 7 days? Write the revenue function, cost function, and profit function after t days. What is the profit after 7 days? Revenue: c)

Optimizing Exponential Business Models Cost: Profit: After 7 days: d) For how many days should the ad run to maximize profit? Assume a maximum ad budget of $200, 000. Interval for t :

Optimizing Exponential Business Models Find critical numbers: Check end points:

Practice: p. 245– 247 #1 (use Desmos), 2, 4– 6, 8*, 9– 13 *For #8, when drug is introduced after 60 minutes, there already 4096 bacteria (use this as “initial value” in a function of the form )