Math 114, Calculus I, Laboratory 12:
Curve Sketching and Optimization
As we have seen, the language of calculus allows us to very precisely describe the shape of the graph of a function. Activities 1, 2 and 3 ask you to explore the application of calculus to curve sketching. As you complete these activities with MapleV, think about the interaction between calculus, which uses language like increasing and concavity to describe graphs, and calculators (or computer software) which produce graphs by plotting a sufficient number of points (?) and connecting them with a smooth curve.
1) Sketch the graph of a function f which satisfies all of the given conditions:
SUBMIT
a sketch of a possible graph for f (Note: more than one graph is possible.)
2) Read the bottom of page 287 and Example 3 on page 290. Do Problem #16 on page 293.
SUBMIT
(a) your estimate for the intervals of increase and decrease, (b) your estimate of the intervals of concavity and (c) printouts of graphs which support your estimates along with a brief explanation of why the printout supports your estimate.
3) Read example 5 on page 292 and do Problem #20 on page 293. Note that the c in these two problems merely represents some constant value that we can modify (as opposed to a critical point.) There is a transitional value of c at which the number of maximum & minimum points and the number of inflection points change. Pay close attention to what happens when c is positive and negative. You might want to try the following commands to get animation:
with(plots): graphs:=[seq (plot(ln(x^2+c/5),x=-3..3,title=convert(c/5.0 ,string)),c=-10..10)]:
display(graphs,insequence=true, view=[-3..3, -6..2]);
SUBMIT
(a) the transitional values of c at which the basic shape of the curve changes and (b) a paragraph (a la Example 5) explaining how you can use the first and second derivatives to deduce the shape of the graph for different values of c. Also submit (c) a single plot showing graphs for the family of functions for various values of c. Label your plot with the values of c for each curve.
4) Suppose you work as a dispatcher for Marten Transport, a local trucking company. Part of your job is to determine the recommended driving speed in order to minimize the total operating cost for the company. The following information about operating costs has been collected:
(i) A truck traveling over a flat interstate at a constant rate of 50 miles per hour gets an (average) fuel mileage of five miles per gallon.
(ii) For each mile per hour increase in the speed, the truck's mileage decreases by one-tenth mile per gallon.
(iii) Diesel fuel costs $1.08 per gallon.
(iv) The fixed cost for use of the truck is $12 per hour.
(v) Marten Transport pays the truck drivers $30 per hour in wages.
SUBMIT
(a) the graph of total operating cost as a function of speed on the interval (50,70) and (b) the constant speed between 50 mph and 70 mph (if such a value exits) that all drivers should maintain on 100 miles of straight interstate for the lowest total operating cost. (c) Explain why your answer gives the most economical total operating cost using the language of calculus.
5) Do Problem #20 on page 307.
SUBMIT
work showing how you determine the most profitable rent to charge.