MAT 112A (Fall 2021) - Calculus I and Modeling

Week 08: Oct. 11

• Week 08: Oct. 11

  
  

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    Exercise 5.2; 42 plots File
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    5.3Example File
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    #19 Assignment

    Read 5.3.
    5.3; 28, 33, 43, 48, 68, 69, 80, 92ab, 96, 103+find the maximum slope of R.
    Hint on 80:  sketch how c(t) could look and conclude signs for each of the three functions, and incr or decr for c and c'.
    On 92, all letters other than T are treated as positive constants.

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    Skin polygons as in 6.1; 49. URL
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    6.1; 49 solve File
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    myelin sheath URL
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    #20 Assignment

    When finding max and/or min for the remainder of the course, you should show solving derivative = 0, not "maximum" from graph.  (In a very technically detailed or complicated function, may use Wolfram Alpha, TI-89, or equivalent, to solve an equation, or even d( ) to differentiate, if you LABEL that is what you did.)  Of course, assignments and problems may occasionally instruct otherwise.
    Read 6.1.
    6.1; 1, 7, 28, 47, 50+use 2nd deriv test, assuming all constants are positive,
    On 47: Graph only M, not derivatives. Hint: You can use quadratic formula to find the solution.
    6.1; 51&52&53 done together.
    6.2; 9.

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    CircleSquare (Wolfram Mathematica Notebook requires campus license) File
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    Do Dogs Know Calculus? article URL
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    Video of Elvis and Dr. Pennings URL
  • The 6.2 outline of steps for Solving an Applied Extrema Problem is pretty good -- I emphasized step 3 in class.  I would combine steps 4 and 6 into the following advice:
    Alternative ways to decide if a critical number c is a max or min:

    • First Derivative Test  (sign of f' on sides)
    • Second Derivative Test  (f''(c) sign)
    • With domain endpoints, compare f values (at all critical points and endpoints)
    • By graph (showing approximate extrema location that match your exact c)
    • Say "physical reasoning" when only one critical number and physically it must be max (or must be a min).

    Pick any one of these but remember to check one.  (See the CAUTION after the book outline.)

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    #21 Assignment

    Read 6.2
    6.2; 4(non-negative x and y is important; why?), 13, 20, 30, 32, 40, 42.
    Hint on 13: think of energy "cost" by using relative cost/mile where cost/mile on land is 1.

    *Optional Star:  Do two more problems: 6.2; 44 and the following Problem A: 
    Not only dogs know calculus but light does too!  When light hits water it bends because it moves faster in air than in water.  The angle is given by Snell's Law shown on p. 116.  Derive this law by minimizing the total time for light to go from start to finish.  From upper left point to lower right, label constants for the vertical distances through Medium 1 and Medium 2 and the total horizontal distance.  The variable is the horizontal distance x in Medium 1.  Take derivative of total time and set it equal to zero.  Do not solve for x but, instead, interpret the ratios as trigonometric functions and rewrite as Snell's Law.

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    #21 6.2 Optimization Word Problems Solutions File