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

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

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

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  1. 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! MAT 121

  2. Recall: Determining Limits • Calculate: How calculate limits? Try subbing in value! Simplify if possible and return to (a). Use a numerical approach: Sub in values closer and closer to the limiting input value. Look at a graph of the function and visualize the limit. Play the S & S Game: Identify SIZE and SIGN MAT 121

  3. Vertical Asymptotes • Use Results: MAT 121

  4. Horizontal Asymptotes • Use Results: MAT 121

  5. Horizontal and Vertical Asymptotes Example: Determine all horizontal and vertical asymptotes for the function MAT 121

  6. Absolute Extremes 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

  7. Absolute Extremes For f(x) = 3x4 + 4x3 + 1, determine its absolute extreme values over the interval [−2,1]. 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. 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? MAT 121

  8. Absolute Extremes Example: Determine all horizontal and vertical asymptotes for the function MAT 121

  9. Putting It All Together: Curve Sketching Example: Sketch a graph of each function. Do not show graph on your calculator! MAT 121

  10. Putting It All Together: Curve Sketching MAT 121

  11. Assignments WebAssign 4.2 due tonight. 4.3 due Monday night. Quiz #4 now open! MAT 121

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