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Brad Froehle

Math 1A PDP Worksheet

October 6, 2009

You should work on the following problems in groups of 3. Try to get through as many as you can, but you aren't expected to finish everything. Instead, you should make sure everyone in your group knows how to solve all the problems, and not just the answers. Logarithms Find y for each function or curve, using logarithmic differentiation if appropriate. 1. y = xx 2. y = (tan x)1/x 3. y = ln(x2 + y 2 ) 4. y = sin x ln(5x) 5. y = log2 x Exponential Growth and Decay Note: if you have calculators, feel free to use them on these problems. 1. A thermometer is taken from a room, where the temperature is 20 C, to outside, where the temperature is 5 C. After one minute, the thermometer reads 12 C. (a) What will the thermometer read 1 minute later? (b) When will the thermometer read 6 C? 2. A sample of tritium-3 decays to 94.5% of its original amount after 1 year. (a) What is the half-life of tritium-3? (b) How long will it take to decay to 20% of its original amount? 3. Suppose you have a balance of $500 on a credit card with a 20% interest rate. Assuming you make no payments and there are no late fees, how much do you owe after 1 year if the interest is compounded (a) annually? (b) monthly? (c) daily? (d) continuously?

Related Rates Solve the following related rates problems . 1. Suppose you are on top of a 10ft ladder that is leaning against a wall, with the base of the ladder 3ft away from the wall. If your nemesis kicks out the bottom of the ladder so that it travls at a constant 2ft/s, how fast are you falling when you are 1 ft off the ground? How about after 2 seconds? 2. While working as a boat valet for a restaurant on one of the local lakes one of your jobs is to pull customers' boats into the dock using a rope tied to the front of their boat. If you hold the rope 1m above the edge of the dock, and take in rope at a rate of 1m/s, how fast is the boat approaching the dock when it is 8m away? 3. In Search and Rescue operations for missing persons, one of the most important things is establishing a Search Area, which is a circle centered at the last known location of the person and radius equal to the maximum distance they could have travelled in the time they've been missing . Suppose you're looking for a hiker who can walk at a maximum of 4ft/s. At what rate is the area of your Search Area increasing after the person has been missing for 1 hour? For 2 hours? For 1 day? 4. While working for a party supply store, your job is to fill baloons from a helium tank. If the helium tank can output gas at .5 ft3 /s, at what rate is the volume of a spherical baloon increasing after 5 seconds? At what rate is its surface area changing when the volume is 2 ft3 ? 5. A slide at the new playground is shaped like the hyperbola y = 1/x. If a kid's vertical speed is a constant 1 m/s, what is his horizontal speed when x = 2?

A good strategy is as follows: Begin by drawing a big picture. Label distances that are changing in time with variable names. Write an equation (or equations) that relate the changing variables. Differentiate with respect to time. Plug in the given values, and solve for your unknown. No, I'm not making this up; we Minnesotans really do do this. In truth, the situation is a little more complicated. Natural features (rivers, cliffs, roving barbarian hordes) can all narrow it down.

6. Radar guns (as used by baseball, police officers, etc) work by calculating the rate of change of the distance between the gun and the object being tracked. Since the object may not be coming straight at the radar gun, the speed of the object and the speed registered by the gun may be different. Let's explore: (a) A police officer is sitting at the side of the road, pointing his radar gun at the far lane of traffic, which is 20ft away. Now suppose you are driving your car down the road at 150 ft/s (approx 100 mi/h). What will the cop's radar gun read when you are 500 ft down the road? 100 ft? 10 ft? 0 ft? (b) Is it possible for the radar gun to ever register a speed faster than what you are actually driving? Why or why not? (c) Now suppose you're the cop and your radar gun reads 90ft/s for a car 500 ft down the road (again, still in that far lane which is 20ft away)? If the speed limit is 100ft/s (approx. 70 mi/h), can you give this driver a ticket? What if the car is only 50 ft away? 10ft? 7. Suppose you are 5ft tall and are a photographer taking pictures of the International Hot Air Balloon Festival which is held each year in Albuquerque, NM. From a position 200 ft away from the takeoff point, your favorite baloon starts rising at a constant rate of 10 ft/s. (a) If you track the baloon the whole way up, at what rate must you be changing the angle between your camera and the ground after 5 seconds? (b) In order to keep the baloon in focus, you also need to know at what rate the distance between you and the baloon is changing. What is this rate after 5 seconds? 8. Suppose you are a stage actor, and there is a bright light mounted on the ground at the edge of the stage pointed at the curtain which is 50 ft from the light. If you are 5 ft tall and walk at 5ft/s in a straight line from the light to the curtain, at what rate is the height of your shadow changing when you are 10 ft from the curtain? At what rate is the angle from the light to the top of your shadow changing? 9. At noon, ship A is 150km west of ship B. Ship A is sailing east at 35km/h and ship B is sailing north at 25km/h. How fast is the distance between the ships changing at 4pm? 10. On a hot August day you order a snowcone, but sadly are unable to eat it before it melts. By some great cosmic coincidence, the liquid left over from melted ice is exactly enough to fill the paper cone part, which is 3in tall and has a radius of 1in. Using a straw, you decide to sip out the sugary colored water a rate of .5 in3 /s. How fast is the height of the liquid in the cone decreasing after 5 seconds? 1 Recall that the volume of a cone is 3 r2 h 11. A lighthouse is on a small island 3km from the nearest point on shore. If P is that nearest point and the coastline is a straight line and the lighthouse rotates at 4 rev/min, how fast is the beam of light moving accross the shore when it is 1km from P ? 12. The minute hand on a watch is 8mm long and the hour hand is 4mm long. How fast is the distance between the tips of the two hands changing at 1 o'clock?

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