Question

number 1

94 CHAPTER 2 LIMITS AND DERIVATIVES 2.1 Exercises 1. A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes 30 r (min) 10 15 20 25 V(gal) 694 444 250 111 28 (a) If P is the point (15, 250) on the graph of V. find the slopes of the secant lines PQ when Q is the point on the graph with 5, 10, 20, 25, and 30. (b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines. (c) Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.) 2. A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after r min- utes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. 36 38 40 42 44 r(min) Heartbeats 2530 2661 2806 2948 3080 The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate- after 42 minutes using the secant line between the points with the given values of 1 (a) 36 and -42 (b) 38. and t-42 (c) r40 and 42 (d) 42 and r=44 What are your conclusions? 3 The noint P lies on the curve v r/1

Answer 1

Q ank tlmin V 3 o 25 2 D 10 444 Stope (m) Y 15 250 a)he sope Suaut line luao the paind he 15,250& 5,94) m 614-250-444 5-15 5,2.50) & (o,444) n 444-250 --3&. 10-15 15,250) & (20,11). -27.8 IL-250 20-15 (15,250) & (as,2) M 28-250 22.2 25-15 15, 250) 20,0) m0-250 30-15 -16-6

bTa estuuati th Shope th Aue nud to pne t aage dleps t tun faut uaut l ihe elep h the Serawt lne biluruo the penb (15,250) & (o,444) iA -g8-8 he stope the Auaut lie elurun th Roint (15a50) & (20,1) -27-8 Avceage tue tus slefs -38.9+(-27-8) -32.3 Thus estumald step lue at paut Pl15,25 is33 3 lue CNaw let u use t t slinatt the slae a tauget lui at P falat the a thu taugeut e at P Hu sla

409 STT SUD 4vo 3S0 3co 2o0 150 tob 50 25 30 5 20 tCuain). lne at P the m chauge in Vole d5-320-145 chau lu tine 1t-12-5 4.5 N32 22 V(gal)