Application Of Derivatives

Step 1: Recall the condition for increasing function A function is said to be increasing in an interval if:
Step 2: Find the derivative of
Step 3: Check the sign of for For : Since is positive for all positive values of , we have:
Step 4: Conclusion Since for all , the function is an increasing function when . Graphical Interpretation: The function is a parabola opening upwards with vertex at . For , as increases, increases continuously.

Step 1: Identify given information Let be the radius of the air bubble. We need to find when cm.
Step 2: Formula for volume of a sphere The volume of a spherical bubble is:
Step 3: Differentiate with respect to time
Step 4: Substitute the given values When cm and :
Step 5: Final answer Interpretation: When the radius is 1 cm, the volume of the air bubble is increasing at the rate of cubic centimeters per second (approximately ).