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Assignment 1--The derivative as scientific concept, and some limit problems.
Please read sections 3.1, 3.2, and 3.3.
Please do problems 3.1.{5,10,17,33,34,36,37,38}
For each problem below, answer in plain English. I suggest you write your
answers on a hardcopy of this worksheet.
Problem 1: Let p be the position (in meters) of a particle,
and let t be time (in seconds). What does it mean to say that the derivative of
p with respect to t equals 4?
Problem 2: Let I be the daily intake (in grams) of a certain drug,
and let S be the resulting patient survivability
(in expected years of life remaining). What does it mean to say that
the derivative of S with respect to I equals -0.5? Do you recommend this
drug?
Problem 3: Let x be the number of stitches required to repair
performed on a just-damaged garment, and let y be the number of stitches required to repair the same garment if it is long neglected
(instead of receiving x stitches). What does it mean to say that the derivative of y with respect to x is -9.0?
Problem 4: One evening, you decide to drive from Warm Valley to Cold Peak. Let t be time in hours, x be position in miles, and y be the temperature of the air.
- Translate: The derivative of x with respect to t is 60
- Translate: The derivative of y with respect to x is -0.1
- What therefore is the value of dy/dt, and what does that value mean?
- State explicitly how you determined dy/dt from the other two derivatives.
Problem 5: Shivering is a way of converting stored biochemical energy
to heat to protect against cold weather. Temperature T, of course, is naturally
measured in degrees. Severity of shivering could be measured by the number (M) of
muscle groups recruited to shiver.
For a cold person, should dM/dT be positive, negative, or zero? Explain.
What about for a very warm person?
Problem 6: Let E=f(x) be the elevation in feet of the Mississippi river x miles from its source. What are the units of dE/dx? What can you say about the sign of dE/dx? Using
figures (available online) for St. Louis, MO and Memphis, TN, estimate dE/dx.
Problem 7: The time (in minutes) T=f(C) required for a chemical reaction depends on the quantity (in grams) C of catalyst present.
- Translate: f(5)=9
- Translate: f '(5)=-1
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