Section 3.7
1. Graphs of the position functions of two particle are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.
(c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle, Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount Δr. How can you approximate the resulting change in area Δr if Δr is small?
2.
(c) Show that the rate of change of the volume of sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true.
3. A particle is moving along a hyperbola xy = 8. As it reaches the point (4, 2), the y-coordinate is decreasing at a rate of 3 cm/s. How fast is the x-cooydinate of the point changing at that instant?
4. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
5. A trough is 10 ft long and its end have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being. filled with water at a rate of 12 ft3/min, how fast is the water level rising When the water is 6 inches deep'?
6, A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every two minutes. How fast is a rider rising when his seat is 16 rn above ground level?
7. The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance beiween the tips of the hands changing at one o'clock?