Real-world applications of local extrema using the First Derivative Test

📚 Quick Study Guide

🔍 The First Derivative Test helps identify local maxima and minima of a function. 📈 If $f'(x)$ changes from positive to negative at $x=c$, then $f(c)$ is a local maximum. 📉 If $f'(x)$ changes from negative to positive at $x=c$, then $f(c)$ is a local minimum. 🧪 Find critical points by setting $f'(x) = 0$ or where $f'(x)$ is undefined. 📊 Use a sign chart of $f'(x)$ to determine intervals of increasing/decreasing behavior. 💡 Local extrema problems often involve optimization: finding the best (maximum or minimum) value of something. 📐 Remember to consider endpoints when finding absolute extrema on a closed interval.

Practice Quiz

1. What real-world problem can be solved by finding the minimum of a function representing cost?
   1. Maximizing revenue
   2. Minimizing material used in production
   3. Calculating velocity
   4. Finding the break-even point
2. A farmer wants to build a rectangular enclosure with the largest possible area using 100 feet of fencing. Which mathematical concept applies?
   1. Finding the derivative
   2. Optimization using local extrema
   3. Calculating the integral
   4. Solving a differential equation
3. An object's height is given by $h(t) = -16t^2 + 64t$. At what time $t$ does the object reach its maximum height?
   1. t = 1
   2. t = 2
   3. t = 3
   4. t = 4
4. In business, what does finding the maximum of a profit function help determine?
   1. The minimum cost
   2. The optimal production level
   3. The break-even point
   4. The depreciation rate
5. A manufacturer wants to minimize the surface area of a cylindrical can with a fixed volume. Which calculus concept is used?
   1. Related rates
   2. Optimization
   3. Implicit differentiation
   4. Integration by parts
6. The daily production cost $C(x)$ of a factory is modeled by a function. Using the First Derivative Test, what are we trying to find by setting $C'(x) = 0$?
   1. The maximum production level
   2. The minimum production level
   3. The production level with maximum cost
   4. The production level with minimum cost
7. A company wants to maximize the volume of a box made by cutting squares from the corners of a rectangular sheet of cardboard and folding up the sides. This is an example of:
   1. Linear programming
   2. A related rates problem
   3. An optimization problem
   4. Implicit differentiation