Aim: Mean value theorem. HW: p 253 # 37, 39, 43 p 272 # 7, 8 p 308 # 5, 6

Transcription

1 Mr. Apostle 12/14/16 Do Now: Aim: Mean value theorem HW: p 253 # 37, 39, 43 p 272 # 7, 8 p 308 # 5, 6 test 12/21 Determine all x values where f has a relative extrema. Identify each as a local max or min: 1

2 The Mean Value Theorem if f is continuous on the Closed Interval [a,b] and differentiable on the open interval (a,b) there exists a number c in (a, b) such that f '(c) = (f(b) f(a))/(b a) in other words the instantaneous rate of change is equal to the average rate of change f(x) = x 2 on [0,2], find an x value where the MVT holds 2

3 if f(x) = 5 4/x find a 'c' on the interval [1,4] for which the MVT holds. (where f '(c) = (f(b) f(a))/(b a) 3

4 How do we solve Mean Value Theorem Questions? 1) 2) 3) 4

5 Find where f(x) = (x +1)/x satisfies the MVT on the interval [1/2, 2] 5

6 Existence Theorems: Mean value theorem: Intermediate Value theorem: 6

7 If a car goes 352 ft in 8 sec, what can we say about the speed of the car during the trip? 7

8 When can we NOT use the Mean Value Theorem? 8

9 What is concavity and the second derivative test? A function is concave up on an interval if f ' is increasing A function is concave down on an interval if f ' is decreasing 9

10 Practice: 10

11 Looking at a graph of f ' to find max, mins and concavity of f: 11

12 Let's look at line motion problems again x(t) = (t 6)(t 2) 3 a) Find the values for t for which the particle is at rest b) Find any values of t for which the particle changes direction c) Find the interval over which the particle moves to the left 12

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14 Optimization How do we optimize certain situations? For example, what is the largest rectangle in terms of area which can be drawn between the x axis and a sine graph? What two numbers which add to 20 will give us the largest possible product? 14

15 How do we solve these kinds of problems? Like in the real world, functions are rarely going to be given to us Step 1: Understand the problem Read the question carefully, identify information we know, figure out what we are being asked to fin Step 2: Develop a mathematical model Draw pictures, write a function in terms of one variable Step 3: Sketch a Graph If applicable draw a graph and figure out what values in the domain may make sense Step 4: Identify Critical points and endpoints find f ' =0 or undefined Step 5: Solve the Mathematical Model What is the value which optimizes our function? If we are maximizing it will be a max if we are minimizing it will be a min Step 6: Interpret Solution Put your answer in the context of the question. Translate the mathematical answer into a real world answer 15

16 What two numbers which add to 20 will give us the largest possible product? 16

17 This is how I think about these problems 1) What variable are we trying to maximize? 2) Can I write an equation where this variable is alone on one side of the equal sign? 3) Find the maximum and minimum value of the function 4) Do I want to use the minimum or maximum 5) Answer the question 17

18 what is the largest rectangle in terms of area which can be drawn between the x axis and a sine graph? 18

19 If you are creating a open topped box by cutting corners out of a sheet of metal which measures 20 inches by 25 inches and folding up the remaining edges to create the sides of the box. What is the largest volume we can make the box have? 19

20 You have been asked to design a cylindrical can whose volume is 1000 cm 3. You goal is to use the least amount of material to make the can so that you can save money. What should the height and radius of the can be? 20

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