The University of Sydney School of Mathematics and Statistics Tutorial for Week 11 MATH1011: Applications of Calculus Semester 1, 2020 Web Page: www.maths.usyd.edu.au/MATH1011 Lecturer: Clio Cresswell, Brad Roberts Tutorial exercises: 1. After t hours, an organism is producing cells at the rate of 20e t cells per hour. (a) If N ( t ) gives the number of cells produced after time t , estimate the number of cells produced in the first four hours by dividing the interval 0 t 4 into eight equal subintervals and evaluating the sum 8 X i =1 N 0 ( t * i ) × 4 - 0 8 , where each t * i is at the mid-point of the i th subinterval. (b) Find the number of cells produced in the first four hours by evaluating an appro- priate definite integral. 2. An animal population is increasing at a rate of 100+40 t +3 t 2 individuals per year (where t is measured in years). By how much does the animal population increase between the seventh and the tenth years? 3. Suppose that a person is injected with 40 mg of a certain drug, and that the drug is then continuously eliminated from the body, at the rate of 12 . 5e - 0 . 06 t mg per hour after t hours, until it is eliminated completely. (a) Let A ( t ) be the number of mg of drug remaining in the body t hours after the drug was injected into the patient. What does the given information say about the derivative, A 0 ( t ), of A ( t )? (Note that A ( t ) is decreasing .) (b) Find a formula for A ( t ) in terms of t . (c) How much drug has been eliminated from the body after 1 hour? (d) After how many minutes will the person be drug-free? 4. The rate of growth of a bacterial population, P ( t ), is 10(e t + 1), where t is in hours. (a) By how much does the population increase in the first ten hours? (b) Find a formula for P ( t ) if initially the population is 1000. 5. Consider a right-circular cone of radius R and height h . Use integration to find the volume of this cone. 6. Suppose A ( x ) = 5 x 2 describes the area of a cross-section of a solid at x (perpendicular to the x-axis). Find the volume of this solid from x = 0 to x = 1.
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