# D complete the following conjecture where a b and c

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d) Complete the following conjecture, where a , b , and c are real numbers: If f ( x ) is a continuous function on [ a, b ] , then b a f ( x ) dx = b + c a + c dx .
140 The AP CALCULUS PROBLEM BOOK 5.5 The Logistic Curve 1187. The graph of a function of the form P ( t ) = M 1 + Ce rMt , where M , r , and C are constants, is called a logistic curve. Graph the function y ( x ) = 8 1 + 10 e 0 . 9 x in the window 1 x 10, 1 y 9. What value does y approach as x → ∞ ? What appears to be the y -value of the point where dy/dt is changing the fastest? 1188. The solution to the di ff erential equation dP dt = r ( M P ) P is a logistic curve, where C is determined by the initial condition. Can the values found in the previous problem be found without solving the di ff erential equation? In other words, in the equation dP/dt = 0 . 001(100 P ) P , what does P approach as x → ∞ ? What appears to be the P -value of the point where dP/dt is changing the fastest? 1189. A 2000 gallon tank can support no more than 150 guppies. Six guppies are introduced into the tank. Assume that the rate of growth of the population is dP/dt = 0 . 0015(150 P ) P , where t is in weeks. Find a formula for the guppy population in terms of t ; then, determine how long it will take for the guppy population to be 100. 1190. A certain wild animal preserve can support no more than 250 gorillas. In 1970, 28 gorillas were known to be in the preserve. Assume that the rate of growth of population is dP/dt = 0 . 0004(250 P ) P , where t is in years. Find a formula for the gorilla population in terms of t ; then, determine how long it will take for the gorilla population to reach the carrying capacity of the preserve. What is the gorilla population when the rate of change of the population is maximized? 1191. Solve the di ff erential equation dP/dt = kP 2 for constant k , with initial condition P (0) = P 0 . Prove that the graph of the solution has a vertical asymptote at a positive value of t . What is that value of t ? (This value is called the catastrophic solution .) 1192. Given a di ff erential equation of the form ay ′′ + by + y = 0, find constants a and b so that both y = e x and y = e 2 x are solutions. 1193 (AP) . At each point ( x, y ) on a certain curve, the slope of the curve is 3 x 2 y . If the curve contains the point (0 , 8), then its equation is A) y = 8 e x 3 B) y = x 3 + 8 C) y = e x 3 + 7 D) y = ln( x + 1) + 8 E) y 2 = x 2 + 8 The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificied his life. —Earnest Renan
CHAPTER 5. APPLICATIONS OF INTEGRALS 141 5.6 Differential Equations, Part Two 1194. You are driving along the highway at a steady 60 mph (88 ft/sec) when you see an accident ahead and slam on the brakes. What constant decceleration is required to stop your car in 242 ft? 1195. The rate of change in the number of bacteria in a culture is proportional to the number present. AP Biology students at Rockdale discovered that there were 3000 bacteria initially, and 90,000 bacteria after two hours.