- Math Major/Minor lunch this Friday:
When: Friday 11/06/15
Where: MP 4th Floor Center Atrium
Time: 11:30-1:30
- For example, suppose you need to compute $\sqrt{15.96}$. Now the
easy way is to use your calculator, but the cowboy way would be to use
the linearization! Let's have a go at that one.
- We know the square root of 16, which is close: our answer should be near 4.
- Draw a picture! The picture will demonstrate two different types of quantities that we discuss in this chapter: differentials $dy$, and increments $\Delta y$.
- We use the tangent line for our approximation, and, when we're done, we can ask
- is our answer an over- or an under-estimate?
- Differentials versus increments -- death match!
- Examples
- #2, p. 187
- #5
- #32
One of the main purposes of calculus is to help us to locate extrema -- maxes and mins -- places where a function takes its largest or smallest values.
For example, many farmers are extremely quantitative, and focused on optimizing their strategies for feeding, selling, etc.
- Why do
sheep farmers care about maxima and minima?
- Lamb Dashboard
- Slaughter weights:
Table 3. Target slaughter weights for ewe and wether lambs produced from sire and dam breeds of varying mature weights (lb) Ewe breed mature wt (lb) Sire breed mature weight (lb)240230220210200190180170160150140130120240120.0 117.5 115.0 112.5 110.0 107.5 105.0 102.5 100.0 97.5 95.0 92.5 90.0 230117.5 115.0 112.5 110.0 107.5 105.0 102.5 100.0 97.5 95.0 92.5 90.0 87.5 220115.0 112.5 110.0 107.5 105.0 102.5 100.0 97.5 95.0 92.5 90.0 87.5 85.0 210112.5 110.0 107.5 105.0 102.5 100.0 97.5 95.0 92.5 90.0 87.5 85.0 82.5 200110.0 107.5 105.0 102.5 100.0 97.5 95.0 92.5 90.0 87.5 85.0 82.5 80.0 190107.5 105.0 102.5 100.0 97.5 95.0 92.5 90.0 87.5 85.0 82.5 80.0 77.5 180105.0 102.5 100.0 97.5 95.0 92.5 90.0 87.5 85.0 82.5 80.0 77.5 75.0 170102.5 100.0 97.5 95.0 92.5 90.0 87.5 85.0 82.5 80.0 77.5 75.0 72.5 160100.0 97.5 95.0 92.5 90.0 87.5 85.0 82.5 80.0 77.5 75.0 72.5 70.0 15097.5 95.0 92.5 90.0 87.5 85.0 82.5 80.0 77.5 75.0 72.5 70.0 67.5 14095.0 92.5 90.0 87.5 85.0 82.5 80.0 77.5 75.0 72.5 70.0 67.5 65.0 13092.5 90.0 87.5 85.0 82.5 80.0 77.5 75.0 72.5 70.0 67.5 65.0 62.5 12090.0 87.5 85.0 82.5 80.0 77.5 75.0 72.5 70.0 67.5 65.0 62.5 60.0 Target slaughter weight = ((sire breed mature wt. + ewe breed mature wt.)/2) x .50
The mathematician realizes that this table is unnecessary -- that the formula at the bottom tells the story: the weight is linearly related to ewe and sire weight, being the average of the two.
- Lamb Dashboard
- Our author begins by distinguishing between global extrema and
local extrema:
- What do you imagine is the difference?
- Why do you suppose that we care?
- Take a moment and sketch a continuous function on the
interval [-2,2]. Make it look kind of interesting -- maybe even
ugly.
Now, answer these questions:
- does your function have a global maximum and a global minimum?
- does your function have local extrema which are not global extrema?
- If your function does have extrema, how would you characterize the derivative of the function at those extrema?
- Now I want you to sketch a function defined on the interval [-2,2] that
does not have any global extrema. What is true about your function?
- Now I want you to sketch a smooth function on [-2,2] that has a
value of x=a where $f'(a)=0$ but no
extremum there. What is true about your function?
- Extreme Value Theorem
- Fermat's Theorem
- Find the critical numbers for f (x values at which the derivative is zero or doesn't exist).
- Find the values of f at the critical numbers of f in (a,b).
- Find the values of f at the endpoints of the interval (at a and b).
- The largest and smallest of the values from steps 1 and 2 are the absolute or global extrema.
- #8, p. 204
- #22
- #31 (and let's decide whether they're local maxes or mins)
- #45
- #63