Calculus 12: Evaluation 3 Outline and Review

Transcription

1 Calculus 12: Evaluation 3 Outline and Review You should be able to: 1. Differentiate various types of functions including trigonometric, exponential and logarithmic functions, 2. Solve various related rates problems 3. Solve various motion application questions 4. Determine critical points, intervals for which a function is increasing, decreasing, concave up and concave down. Review Questions: 1. A ladder 26 feet long leans against a vertical wall. The foot of the ladder is drawn away from the wall at a rate of 4 feet per second. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 10 feet from the wall? 2. A conical tank has a radius at the top of 5 feet and a height of 10 feet. Water runs into the tank at a 3 ft constant rate of 2. How fast is the water level rising when it is 6 feet deep? min 3. A major league baseball diamond is a square with side 90 ft. A batter hits the ball and runs towards first base with a speed of 24 ft. sec a) At what rate is her distance from second base decreasing when she is halfway to first base? b) At what rate is her distance from third base increasing at the same moment? 4. Two carts, A and B, are connected by a rope 39 ft. long that passes over a pulley P (see the figure). The point Q is on the floor 12 feet directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of the instant when cart A is 5 ft. from Q? 2 ft. How fast is cart B moving towards Q at sec 8 ft. At what rate is the angle sec between the string and the horizontal decreasing when 200 ft. of string have been let out? 5. A kite 100 ft. above the ground moves horizontally at a speed of 6. Gravel is being dumped from a conveyor belt at a rate of 30 ft 3 min. coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft. high? and its 7. A lighthouse is on a small island 3 km away from the nearest point P on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? Horton High School Page 1

2 Calculus 12: Evaluation 3 Outline and Review 8. A 1.6 m tall woman is walking away from a street light which is at the top of a 5 m pole with a speed of 2 m/sec along a straight path. (a) How fast is the tip of her shadow moving when she is 15m from the pole? (b) How fast is her shadow lengthening at that point? 9. A car is approaching an intersection from the east at a speed of 12 m/s while a truck is moving away from the intersection from the north at a rate of 15 m/s. How fast is the distance between them decreasing when the car is 30 m east of the intersection and the truck is 40m north of the intersection? 10. An airplane at an altitude of 10,000 ft is flying at a constant speed on a line that will take it directly over an observer on the ground. If, at a given instant, the observer notes that the angle of elevation of the airplane is 60 degrees and is increasing at a rate of 1 degree per second, find the speed of the airplane. 11. A girl starts at a point A and runs east at a rate of 10 ft/s. One minute later, another girl starts at A and runs north at a rate of 8 ft/s. At what rate is the distance between them changing 1 minute after the second girl starts? 12. The ends of a water trough 8 feet long are equilateral triangles whose sides are 2 ft long. If water is pumped into the trough at a rate of 5 ft 3 /min, find the rate at which the water level is rising when the depth of the water is 8 inches. 13. Differentiate each of the following expressions with respect x. Isolate dy/dx. a) 3 y x 4x 2x b) 2 3 y (2x 5) ( x 3) c) 5 3x 2 y d) y sin (4x) 2 x 2 e) 2 2 xy 3y x f ) 2 y tan( x 3) g) 2 3 sin ( x) y 5x h) y x cos(3x) i) 3x ln(3x) y 2xe j) y 2x 4 k) 2x 5 4 y log 4 (sin(5x)) l) y tan(2x) 14 Two planes at the same altitude converge on a point at a right angle. Plane A is flying at 600 mph and plane B is flying at 450 mph. At what rate is the distance between them decreasing when plane A is 200 miles from the point and plane B is 150 miles from the point. 15. A particle P moves on the number line shown below. The diagram to the right of the number line shows the position, in metres to the right of zero, of the particle as a function of time 0 t 8 seconds a) When is P moving to the right? b) When is P have velocity of zero? c) When is P speeding up? d) When is P moving toward the origin. 5x 2 2x 16. Determine the interval for which is decreasing. 17. Determine the interval for which y 3ln( x 2 4) is concave up. y e Horton High School Page 2

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