Briggs 6.1: Velocity and Net Change

Transcription

1 Briggs 6.: Velocity and Net Change Problem Consider an object moving along a line. The graph of the velocity, vt), of the object is shown in the figure below. In the figure above, graph the best you can) the position function, st), given that s). Problem Solve the following word problems: a) The velocity function for an object moving along a line east/west is given by vt) t + 4t feet per minute. i) Find the total displacement the object traveled from minutes to 6 minutes assume east is positive).

2 The object s total displacement is given by vt) dt. So we compute: vt) dt t + 4t ) dt [ ] 6 t + t t ) 8 ) So the object s displacement is feet west of its original location. ii) Find the total distance the object traveled from minutes to 6 minutes. First notice that vt) t 4t + ) t )t ). So we can see that vt) > when t <. vt) < when < t 6. Thus, the total distance that the object traveled from minutes to 6 minutes is: vt) dt vt) dt + vt) dt + vt) dt vt) dt t + 4t ) dt [ t + t t ] t + 4t ) dt [ t + t t ) ) ) ) )) + 8 ) 8 ) So, the total distance that the object traveled is 6 feet. iii) Suppose that the object s position minutes into the trip is feet of a placement marker. What is its position relative to the placement marker) at 6 minutes. s6) s) + vt) dt + ) 7. So the object s position at 6 minutes is 7 feet west of the placement marker. ] 6

3 b) Sammy the Snail sets up camp in the median of I-7 and, starting at noon and ending at 6pm, hikes back and forth along the highway. He starts his hike at his campsite. His velocity at time t hours after noon) is given by vt) t )t ) inches per hour. Find the total distance Sammy travelled on his hike. The total distance that Sammy travels is where vt) is positive and negative: vt) dt. The following picture indicates So we compute: vt) dt vt) dt + vt) dt vt) dt + vt) dt + t 7t + ) dt vt) dt vt) dt t 7t + ) dt + t 7t + ) dt [ t 7 ] [ t + t t 7 ] [ t + t + t 7 t + t ) ) ) )) ) 7 )) So Sammy has traveled a distance of inches. ] 6 Problem Assume that the rate of change in dollars per day) of the price of shares of stock in the WeSaySo Company with t in days) is modeled by the equation rt) t + t 6 note that this is technically a discrete function, but prices change so often with stocks that modeling this with a continuous function makes sense). Assume also that the price of a share of stock on day i.e., t ) is $. Answer the following questions: a) Find the rate of change of price at t. r) ) + ) dollars/day. b) Find the price of a share of stock at t.

4 Let pt) denote the price of a share of stock at any time t. Then notice that rt) p t). We will first find pt) for general t, and then substitute t to solve this problem. pt) So, p) $. rs) ds + p) s + s 6 ) ds + [ s + s 6s ] t + t + t 6t ) + 6) + t + t 6t +. c) How fast is the rate of change of price changing at t? r t) 6t +. So, r ) +. d) How much did the price of a share of stock change in the first 6 days i.e., on [, 6])? p6) p) ) $. e) What was the greatest rate of change of price during the first 6 days i.e., on [, 6])? This question wants us to maximize rt) on the closed interval [, 6]. So we need to find all critical points of rt) on [, 6]. r t) 6t + : t Then, using the closed interval method, we simply plug t,, 6 into rt) and check which has the greatest output. r) r) r6) Thus, the greatest rate of change of price is dollars/day when t. f) What was the greatest price of a share of stock during the first 6 days i.e., on [, 6])? This question wants us to maximize pt) on the closed interval [, 6]. So we need to find all critical points of pt) on [, 6]. But note that p t) rt). So critical points of pt) are just roots of rt). rt) t + t 6 t t + ) t )t 7) t, 7 t 4

5 since 7 is not in the interval [, 6]. So we again use the closed interval method: p) p) p6) Thus, the greatest price of the stock is $ when t. Problem 4 Suppose that rt) r e kt with k > ) is the rate at which a nation extracts oil. The current rate of extraction is r) 7 barrels/yr. Also assume that the estimate of the total oil reserve ie, the amount of oil remaining beneath the ground in this country) is 9 barrels. a) Find Qt), the total amount of oil extracted by the nation after t years. Qt) rs) ds Qt) r k rs) ds r e ks ds [ ] t e ks e kt ) r k k 7 e kt ) b) Evaluate lim t Qt) and explain the meaning of this limit. lim Qt) lim t t k 7 e kt ) k 7 ) k 7. c) Find the minimum decay constant k for which the total oil reserves will last forever. For the oil reserves to last forever, we need that lim Qt) 9 t k 7 9 k k.

6 So the minimum value for k is. d) Suppose that the decay constant is half the minimum value found in part c). First note that k. We want to find the value of t such that: 4 Qt) 9 ) 4 7 e 4 t 9 ) e 4 t t ln e 4 t ) ln) t 4 ln) 77.9 years. Problem A bug is moving down the curve y 8, starting at the point A see figure). Its x-coordinate is x increasing at the rate of 6 cm/min. a) At what rate is the bug s y-coordinate changing when x 4? 6

7 We have y 8 x dy dt 8 dx x dt dy dt b) At what rate is the distance between the bug and the origin changing when x 4? The distance between the origin and the bug is given by s x ) + y ) x + y Therefore ds dt x + y x dx ) dy + y dt dt ds xdx/dt) + ydy/dt) dt x + y When x 4 we have y 8/4. Hence ds dt ) c) For what value of x is the bug closest to the origin? Justify your answer. When minimizing distance, it s better to work with the square of the distance instead of distance directly. Finding objective function: distance x + y distance x + 8/x) fx) x + 8/x) Constraint on objective function: Domainf) [, 8] Finding critical points): f x) x + 8 x f x) x 8 x x 8 x x 4 64 x Since we only have one interior point, to verify the absolute minimum occurs at x we examine the sign of f near x : ) x f x) 4 x 64 ) 7

8 Factoring: f x) So the sign of f is the same as the sign of x 8 ). So f x) < for, ) and f x) > for, 8) ) x x 8 ) x + 8 ) Hence there is a local minimum at x. Since there is only one local extrema and f is continuous on [, 8] we have local minimum at x absolute minimum at x 8