# 4 math 2211 calculus of one variable i study guide

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MATH-2211: Calculus of One Variable I Study Guide for Test 3 (out of 4) Practice Problems 1. A certain type of bacteria, given a favorable growth medium, triples in population every 8 hours. Given that there were 100 bacteria to start with, how many bacteria will there be in a day and a half? (Note: Give an exact answer.) 2. Radio-isotopes of different elements have different half-lives. Magnesium-27 has a half-life of 9 . 45 minutes. What percentage of a given initial amount of magnesium-27 will remain after an hour? (Note: Round your answer to two decimal places.) 3. Because of a virus introduced into a bacterial culture, the population of the bacteria decreased from 3000 to 1000 in 50 minutes. Assuming exponential decay, determine how many bacteria will remain after 2 hours from the beginning of the experiment. (Note: First give an exact answer, then round it to the nearest integer.) 4. A particle is moving in the circular orbit x 2 + y 2 = 61. As it passes through the point (6 , 5), its y -coordinate is decreasing at the rate of 5 units per second. At what rate is the x -coordinate changing? 5. The volume of the spherical balloon V is increasing at a constant rate of 16 cubic feet per minute. How fast is the radius r increasing when the radius is exactly 5 feet? How fast is the surface area A increasing at that instant? 6. The length of a rectangle is increasing at a rate of 3 cm / s and its width is increasing at a rate of 9 cm / s. When the length is 7 cm and the width is 5 cm, how fast is the area of the rectangle increasing? 7. A cylindrical tank with radius 7 m is being filled with water at a rate of 4 m 3 / min. How fast is the height of the water increasing? 8. Find the absolute maximum and absolute minimum values of the given function on the given interval. (a) f ( x ) = 3 sin x - cos x , on x [0 , 2 π ] (b) f ( x ) = 3 x - 6 arctan x , on x [0 , 4] (c) f ( x ) = 7 - 54 x + 2 x 3 , on x [0 , 4] (d) f ( x ) = ln( x 2 + 2 x + 4), on x [ - 2 , 2] (e) f ( x ) = 2 x + cos x , on x [0 , 2 π ] (f) f ( x ) = x + 2 cos x , on x [0 , 2 π ] (g) f ( x ) = x 2 ( x + 5) 3 , on x [ - 3 , 1] (h) f ( x ) = 2 x x 2 + 9 , on x [ - 4 , 2] – 5–
MATH-2211: Calculus of One Variable I Study Guide for Test 3 (out of 4) 9. Determine whether the given function satisfies the conditions of Rolle’s Theorem on the given interval. If so, find all numbers c that satisfy the conclusion of the theorem. (a) f ( x ) = 2 x 2 - 4 x + 7, on x [ - 1 , 3] (b) f ( x ) = x 3 - 12 x 2 +45 x - 34, on x [3 , 6] (c) f ( x ) = tan 2 ( πx ), on x 3 4 , 5 4 (d) f ( x ) = x 2 + 1 x 2 , on x [ - 1 , 1] (e) f ( x ) = 2 cos(3 x ), on x [0 , π ] (f) f ( x ) = 3 cos(2 x ), on x [0 , π ]