MATH 2070 (J. Hanna) Extreme Points & Saddle Points Name: ___________________________________________ LA 8.1 ________ / 10 1. 1 On the basis of past sales, a company that sells imported blends of coffee and tea estimates that next year's revenue from the sale of c thousand cans of coffee and t thousand boxes of tea is given by the function R with output in thousands of dollars. Some contours of R are shown on the graph. a. Mark the critical point on the graph with an X. b. Identify and classify the critical point by completing the following sentence. A _________________________ (classify the point) occurs when ______________ cans of coffee and ________________ boxes of tea are sold for a revenue of _________________. 2. The following graph is a graph of weekly costs (in thousands $) for a certain company when x hundred labor hours are used and y thousand refrigerators are produced. a. Mark the critical point on the graph. thousand refrigerators b. Classify the critical point as a relative maximum, relative minimum or a saddle point. 200 300 400 500 c. Interpret the critical point (3 coordinates!) in the context of the problem. zone p p, zone hundred labor hours 3. The table below gives some values of the function G(m, p) that gives the cost, in cents, of a phone call that lasts m minutes when the call is made from zone p. m, length of call (in minutes) Circle the two critical points within the body of the table. 3 5 7 Write each point in mathematical notation and classify the 1 97 99 98 point as a relative maximum, relative minimum or saddle 2 point. 109 110 109 3 96 97 96 4 71 72 73 5 46 47 48 6 32 36 33 7 43 45 44 Explain why G(3,6) = 32 is not a relative minimum. MATH 2070 (J. Hanna) Extreme Points & Saddle Points Partial Chapter 7 Review 4. The table below shows some values of R(z, w), the U.S. Postal Service parcel post rate (in dollars), for packages shipped to zones z and packages weighing up to but not exceeding w pounds. w, Weight (pounds) 1 2 3 2 3.42 3.42 3.45 4 4.05 4.05 4.51 6 4.33 4.33 5.01 8 4.60 4.60 5.44 10 4.83 4.83 5.82 12 5.03 5.03 6.16 14 5.23 5.23 6.48 16 5.40 5.40 6.76 a. Draw and label the $11.00 contour curve. z, zones 4 3.45 5.33 6.20 6.92 7.58 8.17 8.69 9.17 5 3.45 5.80 7.70 8.96 10.01 10.92 11.72 12.42 6 3.45 5.95 8.03 9.60 11.00 12.40 13.80 14.86 7 3.45 6.00 8.25 10.74 13.24 15.15 17.05 18.20 8 3.45 6.05 8.84 12.53 15.26 16.98 18.57 20.05 b. Find a cubic cross-sectional model that could be used to estimate the charge for a package sent to zone 9 and weighing 8 pounds. Define the model completely. c. Use your unrounded equation to estimate the charge to send an 8-pound package to zone 9. d. Find d R( z,8) when z = 9. dz 5. Consider the function A( w, r ) 15 w ln( r ) r w 16 r 2 w3 e 3 r . a. A r b. A w c. dr dw