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CHAPTER 5!

CHAPTER 5!. Heather Anna Greg. OVERVIEW. Increasing/Decreasing Concavity Stationary Points Inflection Points Minimum/Maximum Mean-Value Theorem Rolle’s Theorem Rectilinear Motion Absolute Min/Max Optimization (Applied Min/Max). Increasing/Decreasing.

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CHAPTER 5!

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  1. CHAPTER 5! Heather Anna Greg

  2. OVERVIEW • Increasing/Decreasing • Concavity • Stationary Points • Inflection Points • Minimum/Maximum • Mean-Value Theorem • Rolle’sTheorem • Rectilinear Motion • Absolute Min/Max • Optimization (Applied Min/Max)

  3. Increasing/Decreasing • When the graph is increasing the slope/derivative is positive. • When the graph is decreasing the slope/derivative is negative. • When the graph is constant or at a min/max then the slope/derivative is zero. • Stationary points: the zero’s of the 1st derivative Ex. increasing (2, ∞) decreasing (-∞, 2) 2

  4. Concavity • If the 2nd derivative is positive then it is concave up. • If the 2nd derivative is negative then it is concave down. • Concave Up (C.Up): it is like a cup and it can hold water. • Concave Down (C.Down): it is like an umbrella and it repels water. • Inflection Point: the point were the concavity changes or the Mid Point. Ex: C. Down (-∞, 1) 1 C. up (-1,∞)

  5. Relative Min/Max • The zero’s of the first derivative can be the relative extremities • Negative to Positive is a Minimum • Positive to Negative is a Maximum

  6. Practice Problems • Ex: Increasing:_____________ Decreasing:_____________ Concave U:_____________ Concave D: _____________ Stat. Points:_____________ Inflect. Points :_____________ Relative Max:_____________ Relative Min: ______________

  7. Ex: cosx, [0, 2π) Increasing:_____________ Decreasing:_____________ Concave U:_____________ Concave D: _____________ Stat. Points:_____________ Inflect. Points :_____________ Relative Max:_____________ Relative Min: ______________

  8. Mean- Value Theorem • Between any 2 points on the graph of a differentiable function there is at least one place where the tangent line and the secant line are parallel.

  9. Rolle’s Theorem • If is continuous and it is differentiable (not a vertical line or and absolute value and (a)=0 and (b) =0 then there is at least one point between a and b where =0. • If there are 2 x-intercepts there is guaranteed to be a min or a max between them. • Does not apply to asymptotes, holes, absolute value and corner points.

  10. Rectilinear Motion • Position s(t) Velocity s’(t) Acceleration s’’(t) • Ex:

  11. Describe how the graph below changes with time…

  12. Absolute Min/Max • Extreme Value Theorem • If a function is continuous on a finite closed interval [a,b] then it has an absolute max and an absolute min on [a,b] at the critical points or the endpoints • Ex: Find the absolute max and min values of on the interval [1,5] and determine where these values occur (1, 23) ~ absolute min (2,28) (3,27) (5,55) ~ absolute max 0= 6 0=x -3 0= x-2 1 3 5 2

  13. Optimization • Procedure • Draw and Label a Picture • Find a formula for the quantity that is to be maximized or minimized • Eliminate variables by expressing the quantity to be maximized or minimized as a function of one variable • Find the interval of possible values based upon physical restrictions (Domain) • Use techniques from 5.4 to find the max or min

  14. A garden is to be laid out in a rectangular area and protected by a chicken wire fence. What is the largest possible area of the garden if only 100 running feet of chicken wire is available for the fence?

  15. An open box is to be made from a 16 inch by 30 inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain the box with the largest volume? What is the greatest volume that can be obtained?

  16. The figure below shows an offshore oil well located at point w that is 3km from the closet point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 5 km from A by piping it on a straight line under water from W to some shore point between A and B and then on to B via pipe alone the shoreline . If the cost of laying pipe is $800,000/km under water and $400,000/km over land, how far from point A should the point P be located to minimize the cost of laying the pipe?

  17. Extra • Use the graph of y=f’(x) to fill in the blank with a <,>, or=. (Assume f’’(2) is a minimum value for this graph) • f(0)____f(1) • f(1) ____f(2) • 0 ____f’(0) • f’(1) ____ 0 • f’’(0) ___0 • 0 _____f’’(2)

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