1. Find the derivative () ( implicitly. 3 4 4 = 102 0 3 3 + +2 2. Find the derivative () ( implicitly. 3. Find an equation of the tangent line at the given point. If you have a CAS that will graph implicit curves, sketch the curve and the tangent line. 32 2 = 2 3 at (1, 3) 4. Find an equation of the tangent line at the given point. If you have a CAS that will graph implicit curves, sketch the curve and the tangent line. 5. Find the second derivative "() ( . (( + )2 +1 3 = 3 6. Suppose a forest fire spreads in a circle with radius changing at a rate of 5 feet per minute. When the radius reaches 200 feet, at what rate is the area of the burning region increasing? 7. For a small company spending $x thousand per year in advertising, suppose that 4 annual sales in thousands of dollars equal 0 = 80 200.04 . If the current advertising budget is = 40and the budget is increasing at a rate of $1500 per year, find the rate of change of sales. 8. A camera tracks the launch of a vertically ascending spacecraft. The camera is located at ground level 2 miles from the launchpad. (a) If the spacecraft is 3 miles up and traveling at 0.2 mile per second, at what rate is the camera angle (measured from the horizontal) changing? (b) Repeat if the spacecraft is 1 mile up (assume the same velocity). Which rate is higher? Explain in commonsense terms why it is larger. 9. Suppose that you are blowing up a balloon by adding air at the rate of 1 ft3/s. If the balloon maintains a spherical shape, the volume and radius are related by = 4 3. Compare the rate at which the radius is changing when r = 0.01 ft versus 3 when r = 0.1 ft. Discuss how this matches the experience of a person blowing up a balloon. 10. Sand is dumped such that the shape of the sandpile remains a cone with height equal to twice the radius. (a) If the sand is dumped at the constant rate of 20 ft3/s, find the rate at which the radius is increasing when the height reaches 6 feet. (b) Repeat for a sandpile for which the edge of the sandpile forms an angle of 45o with the horizontal