2.2(c) Notes: Polynomial Functions of Higher Degree

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Presentation transcript:

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?

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

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).

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.

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: 2. Zeros: 3. Additional Points: x f(x) x f(x)

Lesson 2: Real-World App – Maximum Volume

Lesson 2: Real-World App – Maximum Volume

Lesson 2: Real-World App – Maximum Volume

Lesson 3: Model It – Tree Growth The growth of a red oak tree is approximated by the function G = -0.003t³ + 0.137t² + 0.458t – 0.839 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.)

Lesson 3: Model It – Tree Growth The growth of a red oak tree is approximated by the function G = -0.003t³ + 0.137t² + 0.458t – 0.839 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.

Lesson 3: Model It – Tree Growth The growth of a red oak tree is approximated by the function G = -0.003t³ + 0.137t² + 0.458t – 0.839 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 = -0.009t² + 0.274t + 0.458. Find the vertex of this parabola. D. Compare the results from parts B and C.

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