Calculus I

1
Calculus I

• Section 4.7
• Optimization Problems

2
Warm-up

• Verify that the function satisfies the hypothesis
  of the Mean Value Theorem (MVT) on the given
  interval. Then find all values of c that satisfy
  the conclusion of the MVT.

3
Mean Value Theorem

• Let f be a function that satisfies the following
  hypotheses
• f is continuous on a, b
• f is differentiable on (a, b)
• Then there is a number c in (a, b) such that

4
Optimization

• One of the places calculus has wide application
  is in the field of optimization. From minimizing
  cost or maximizing profit in business to finding
  the minimum time for a dog to reach its ball, we
  can use the derivative to help us.

5
Stewarts Steps

• Understand the problem
• Draw a diagram
• Introduce notation
• Beat your head against the wall
• Repeat step 4
• Use the material from sections 4.1-4.3 to solve.

6
Important!

• Do yourself a big favor and work through all of
  the examples in this section prior to starting
  the homework.

7
Classic Example

• A company wants to produce a flower box with an
  open top from a square piece of material, 3 ft
  wide, by cutting by cutting a square out of each
  of the 4 corners. Find the largest volume the box
  can have.

8
Another Example

• A company wants to produce a flower box with an
  open top and a square bottom that will hold
  32,000 cm3 of soil. Find the dimensions that
  minimize the amount of material used.

9
Same as Light?

• Two vertical poles PQ and ST are secured by a
  rope from the top of the first pole to a point R
  on the ground between the poles and then the top
  of the second pole. Show that the shortest length
  of such a rope occurs when q1 q2

10
See Elvis Run

• Please get out the Do Dogs Know Calculus
  handout. Verify

11
See Elvis Run

• Do the Statistical analysis of the data. Well
  run with the authors assumption that the 4
  points in the upper right are outliers and can be
  removed from the data and fit the line to the
  remaining data.

12
Quiz on Friday

• There will be an Optimath quiz on Friday.

13
Another

• Find the point on the line 6x y 9 that is
  closest to the point (3, 1).
• What is the equation of the line that passes
  through these 2 points?