1. 2.2(c) Notes: Polynomial Functions of Higher Degree
Date:
2.2(c) Notes: Polynomial Functions of Higher Degree
Lesson Objective: Use the Intermediate Value Theorem to approximate zeros and solve real-world application problems.
You will need: calculator, colored pens
Real-World App: What is the maximum volume you can make from a 24” square piece of board?

2. Lesson 1: Intermediate Value Theorem
2.2(a) & (b) Recap:
Test Intervals:
Tips on Graphing Polynomials: 1. 2. 3.

3. Lesson 1: Intermediate Value Theorem
2.2(a) & (b) Recap:
Test Intervals: Intervals between the zeros used to test the sign
Tips on Graphing Polynomials:
LCT.
Find Zeros.
Plot a few additional points based on Turning Points.

4. Lesson 1: Intermediate Value Theorem
Intermediate Value Theorem , IVT: A polynomi-al function f will take on every value between f(a) and f(b) in the interval [a, b]. Therefore, if there is a change in direction (negative to posi-tive or vice versa) between f(a) and f(b), then there will be a real zero between f(a) and f(b).

5. Lesson 1: Intermediate Value Theorem
Intermediate Value Theorem , IVT: A polynomi-al function f will take on every value between f(a) and f(b) in the interval [a, b]. Therefore, if there is a change in direction (negative to posi-tive or vice versa) between f(a) and f(b), then there will be a real zero between f(a) and f(b).
Approximating Zeros:
Use the IVT to find change in direction.
Use increments of tenths to pinpoint zero.

6. Lesson 1: Intermediate Value Theorem
Use the IVT to approximate the real zeros of f(x) = 3x³ – 5x + 2 to the nearest hundredth. Then graph the function.
1. LCT: Zeros:
3. Additional Points: x
f(x) x
f(x)

7. Lesson 2: Real-World App – Maximum Volume

8. Lesson 2: Real-World App – Maximum Volume

9. Lesson 2: Real-World App – Maximum Volume

10. Lesson 3: Model It – Tree Growth
The growth of a red oak tree is approximated by the function G = t³ t² t – where G is the height of the tree (in feet) and t (2 ≤ t ≤ 34) is its age (in years).
A. Use a graphing utility to graph the function. (Hint: Use a viewing window -10 ≤ x ≤ 45 and -5 ≤ y ≤ 60.)

11. Lesson 3: Model It – Tree Growth
The growth of a red oak tree is approximated by the function G = t³ t² t – where G is the height of the tree (in feet) and t (2 ≤ t ≤ 34) is its age (in years).
B. Estimate the age of the tree when it is grow-ing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year.

12. Lesson 3: Model It – Tree Growth
The growth of a red oak tree is approximated by the function G = t³ t² t – where G is the height of the tree (in feet) and t (2 ≤ t ≤ 34) is its age (in years).
C. Using calculus, the point of diminishing returns can also be found by finding the vertex of the parabola given by y = t² t Find the vertex of this parabola.
D. Compare the results from parts B and C.

13. 2.2(c): DIGI Yes or No
Use the IVT to approximate the real zeros of 1.
f(x) = -2x³ – 3x² + 3