Applied Calculus (MAT 121)Dr. Day Friday, Feb 24, 2012

Return to “the limit”As the inputs approach some value, is

there a recognizable pattern of approachment in the output values?

Vertical Asymptotes: Denominator Issues?

Horizontal Asymptotes: x∞, x−∞

Absolute Extreme Values

Putting it All Together: Sketch the Graph of a Function!

Friday, February 24, 2012

MAT 121

MAT 121

Recall: Determining Limits

Calculate:

Friday, February 24, 2012

How calculate limits?(a) Try subbing in value!(b) Simplify if possible and return to (a).(c)Use a numerical approach: Sub in

values closer and closer to the limiting input value.

(d)Look at a graph of the function and visualize the limit.

(e) Play the S & S Game: Identify SIZE and SIGN

MAT 121

Vertical Asymptotes

Use Results:

Friday, February 24, 2012

MAT 121

Horizontal Asymptotes

Use Results:

Friday, February 24, 2012

MAT 121

Horizontal and Vertical Asymptotes

Example: Determine all horizontal and vertical

asymptotes for the function

Friday, February 24, 2012

MAT 121

Absolute Extremes

Friday, February 24, 2012

We have determined relative max and relative min values using first derivatives. We now focus on identifying the MOST EXTREME values among those RELATIVE possibilities.

Often, this involves LIMITING THE DOMAIN, and, therefore, considering what occurs at the END POINTS. Here are examples.

MAT 121

Absolute Extremes

Friday, February 24, 2012

(A)For f(x) = 3x4 + 4x3 + 1, determine its absolute extreme values over the interval [−2,1].

(B)If g(x) = x – 2√x: (i) determine its absolute extreme values over the interval [0,9]. (ii) determine its absolute extreme values for all x.

(C)For 2006, operating rate of factories, mines, and utilities in a certain region of the country, expressed as a % of full capacity, is given by the function shown below, where 0 ≤ t ≤ 250 is measured in days. On which of the first 250 days of 2006 was the rate the highest?

€

C(t) = 80 +1200t

t 2 + 40,000

MAT 121

Absolute Extremes

Example: Determine all horizontal and vertical

asymptotes for the function

Friday, February 24, 2012

MAT 121

Putting It All Together: Curve Sketching

Example: Sketch a graph of each function. Do not show

graph on your calculator!

Friday, February 24, 2012

MAT 121

Putting It All Together: Curve Sketching

Friday, February 24, 2012

MAT 121

WebAssign4.2 due tonight.4.3 due Monday night.Quiz #4 now open!

Assignments

Friday, February 24, 2012