Ellipse Problem: Recall that the area of an ellipse is A = ab, where a and b are the

(xy2) = ?

Name the technique you used in Problem Q1.

Name a property of derivatives you used in Problem Q1.

Name another property of derivatives you used in Problem Q1.

If velocity and acceleration are both negative, is the object speeding up or slowing down?

If dy/dx is negative, then y is getting ?.

If y = x cos x, then = ?.

If y = ln x, then dy/dt = ?.

If y = xex, then = ?.

If f(x) = sin x, then ( ) = A. 1 B. 0.5 C. 0 D. 0.5 E. 1

Bacteria Spreading Problem: Bacteria are growing in a circular colony one bacterium thick. The bacteria are growing at a constant rate, thus making the area of the colony increase at a constant rate of 12 mm2/h. Find an equation expressing the rate of change of the radius as a function of the radius, r, in millimeters, of the colony. Plot dr/dt as a function of r. How fast is r changing when it equals 3 mm? Describe the way dr/dt changes with the radius of the circle.

Balloon Problem: Phil blows up a spherical balloon. He recalls that the volume is (4/3) r3. Find dV/dt as a function of r and dr/dt. To make the radius increase at 2 cm/s, how fast must Phil blow air into the balloon when r = 3? When r = 6? Plot the graph of dV/dt as a function of r under these conditions. Sketch the result and interpret the graph.

Ellipse Problem: Recall that the area of an ellipse is A = ab, where a and b are the lengths of the semiaxes (Figure 4-9d). Suppose that an ellipse is changing size but always keeps the same proportions, a = 2b. At what rate is the length of the major axis changing when b = 12 cm and the area is decreasing at 144 cm2/s? Figure 4-9d

Kinetic Energy Problem: The kinetic energy of a moving object equals half the product of its mass and the square of its velocity. K = mV2 As a spaceship is fired into orbit, all three of these quantities vary. Suppose that the kinetic energy of a particular spaceship is increasing at a constant rate of 100,000 units per second and that mass is decreasing at 20 kg/s because its rockets are consuming fuel. At what rate is the spaceships velocity changing when its mass is 5000 kg and it is traveling at 10 km/s?

Base Runner Problem: Milt Famey hits a line drive to center field. As he rounds second base, he heads directly for third, running at 20 ft/s (Figure 4-9e). Write an equation expressing the rate of change of his distance from home plate as a function of his displacement from third base. Plot the graph in a suitable domain. How fast is this distance changing when he is halfway to third? When he is at third? Is the latter answer reasonable? Explain. Figure 4-9e

Tugboat Problem: A tugboat moves a ship up to the dock by pushing its stern at a rate of 3 m/s (Figure 4-9f). The ship is 200 m long. Its bow remains in contact with the dock and its stern remains in contact with the pier. At what rate is the bow moving along the dock when the stern is 120 m from the dock? Plot the graph of this rate as a function of the distance between the stern and the dock. Figure 4-9f

Luke and Leias Trash Compactor Problem: Luke and Leia are trapped inside a trash compactor on the Death Star (Figure 4-9g). The side walls are moving apart at 0.1 m/s, but the end walls are moving together at 0.3 m/s. The volume of liquid inside the compactor is 20 m3, a constant. a. Write an equation expressing the rate of change of depth of the liquid in terms of the width and length of the region inside the compactor. b. When the side walls are 5 m long and the end walls are 2 m long, is the depth of liquid increasing or decreasing? At what rate?

Darth Vaders Problem: Darth Vaders spaceship is approaching the origin along the positive y-axis at 50 km/s. Meanwhile, Han Solos spaceship is moving away from the origin along the positive x-axis at 80 km/s. When Darth is at y = 1200 km and Han is at x = 500 km, is the distance between them increasing or decreasing? At what rate?

Barn Ladder Problem: A 20-ft-long ladder in a barn is constructed so that it can be pushed up against the wall when it is not in use. The top of the ladder slides in a track on the wall, and the bottom is free to roll across the floor on wheels (Figure 4-9h). To make the ladder easier to move, a counterweight is attached to the top of the ladder by a rope over a pulley. As the ladder goes away from the wall, the counterweight goes up and vice versa. Figure 4-9h a. Write an equation expressing the velocity of the counterweight as a function of the distance the bottom of the ladder is from the wall and the velocity at which the bottom of the ladder moves away from the wall. b. Find the velocity of the counterweight when the bottom is 4 ft from the wall and is being pushed toward the wall at 3 ft/s. c. If the ladder drops all the way to the floor with its bottom moving at 2 ft/s and its top still touching the wall, how fast is the counterweight moving when the top just hits the floor? Surprising?

Rectangle Problem: A rectangle of length L and width W has a constant area of 1200 in.2. The length changes at a rate of dL/dt inches per minute. a. Find dW/dt in terms of W and dL/dt. b. At a particular instant, the length is increasing at 6 in./min and the width is decreasing at 2 in./min. Find the dimensions of the rectangle at this instant. c. At the instant in part b, is the length of the diagonal of the rectangle increasing or decreasing? At what rate?

Conical Water Tank Problem: The water tank shown in Figure 4-9i has diameter 6 m and depth 5 m . Figure 4-9i a. If the water is 3 m deep, and is rising at 5 m/h, at what rate is the volume changing? b. If the water is pumped out at 2 m3/h, at what rate is the depth changing i. When the water is 4 m deep? ii. When the last drop is pumped out? c. If water flows out under the action of gravity, the rate of change in the volume is directly proportional to the square root of the waters depth. Suppose that its volume V is decreasing at 0.5 m3/h when its depth is 4 m.

Cone of Light Problem: A spotlight shines on the wall, forming a cone of light in the air (Figure 4-9j). The light is being moved closer to the wall, making the cones altitude decrease by 6 ft/min. At the same time, the light is being refocused, making the radius increase by 7 ft/min. At the instant when the altitude is 3 ft and the radius is 8 ft, is the volume of the cone increasing or decreasing? How fast?

Planetary Motion Problem: On August 27, 2003, Mars was at its closest position to Earth. It then receded at an increasing rate (Figure 4-9k). In this problem you will analyze the rate at which the distance between the two planets changes. Assume that the orbits of the two planets are both circular and are both in the same plane. The radius of Earths orbit is 93 million mi and the radius of Mars orbit is 141 million mi. Assume also that the speed each planet moves along its orbit is constant. (This would be true if the orbits were exactly circular.) Mars orbits Figure 4-9k the Sun once each 687 Earth-days. The Earth, of course, orbits once each 365 Earth-days. Answer the following questions. a. What are the angular velocities of Earth and Mars about the Sun in radians per day? What is their relative angular velocity? That is, what is d /dt? b. What is the period of the planets relative motion? On what day and date were the two planets next at their closest position? c. Write an equation expressing the distance, D, between the planets as a function of . d. At what rate is D changing today? Convert your answer to miles per hour. e. Will D be changing its fastest when the planets are 90 degrees apart? If so, prove it. If not, find the angle at which D is changing fastest. Convert the answer to degrees. f. Plot the graph of D versus time for at least two periods of the planets relative motion. Is the graph a sinusoid?

SketchpadTM Project: Use dynamic geometry software such as The Geometers Sketchpad to construct a segment AB that connects the origin of a coordinate system to a point on the graph of y = e0.4x, as shown in Figure 4-9l. Display the length of . Then drag point B on the graph. Describe what happens to the length of as B moves from negative values of x to positive values of x. If you drag B in such a way that x increases at 2 units/s, at what rate is the length of changing at the instant x = 5? At x = 2? At what value of x does the length of AB stop decreasing and start increasing? Explain how you arrived at your answer. Figure 4-9l