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Problem Number 1If f(x) = x^ 2 and g(x) = .43 ln ( x ), sketch graphs of f(x) and g(x) for 0 < x <= 3, showing how you use the appropriate basic points to obtain the shape the graphs of f(x) and g(x). Show how you combine the graphs to obtain the graph of f(x) / g(x).
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Problem Number 2If a(n) is the amount of drug present just before the nth dose, then a(n-1) is the amount present just before dose number n-1.
If the dosage is 600 mg and the proportion removed during a dosage cycle is . 45, then write expressions for the following:
In terms of a(n-1), the drug concentration just after dose number n-1.
In terms of the preceding expression, the drug concentration just before dose number n.
The expression you just wrote, along with the assumed initial condition a(1) = 0 (i.e., just before does #1 the concentration is zero), constituts a recurrence relation for a(n).
Evaluate your recurrence relation for n = 2, 3, 4 and 5.
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Problem Number 3For a certain individual, the expected grade average is a function of the number of weekly hours t spent in concentrated study according to the function
gradeAverage = -. 8 + t / 9.
The number of hours spent is a function t(Q) of the individuals mental health quotient Q, a hypothetical measure related to hours spent by the relation
t(Q) = 47 (1 - e ^ (-. 22 (Q - 60) ) ).
If the student's mental health quotient is an average 100, then what grade average should the student expect?
What grade averages would be expected for mental health quotients of 110, 120 and 130?
What is the upper limit on the expected grade average that can be achieved by this student?
What is the composite function gradeAverage( t(Q) )?
Evaluate your composite function at t = 100, 110, 120 and 130 to check your work.
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Problem Number 4Show how to obtain a table and graph for y = log(y) from a table and graph of y = 10^y.
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Problem Number 5Using the quadratic formula determine the zeros of x^2 - 11 x + 18. Call the zeros z1 and z2.
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Problem Number 6A graph of the Celsius temperature T(t) of a large dead turkey as it approaches room temperature is an exponential function of form T(t) = A b^t + c, where t is clock time in hours. The function has asymptote T = 52, initial temperature T = 0 and approaches twice as close to room temperature in a period of 4.5 hours. A graph of the rate, in millions of microorganisms per minute, at which bacteria reproduce is a quadratic function R(T) of Fahrenheit temperature with vertex at (0, 0) and with the point ( 29, 52).
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Problem Number 7Construct a graph of y = f(x) = 7 * 1.07^x and determine the average slope between x = 4.5 and x = 4.56, and also between x = 8.7 and x = 8.76.
Determine the approximate average value of f(x) on each interval.
Show that the ratio of slope to average value is very nearly the same on both intervals.
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Problem Number 8The temperature of a hot potato in a room at 5 degrees starts out at 50 degrees and after 4 minutes has dropped to 30 degrees. What exponential function models this data, and after how long will the temperature have dropped to 8 degrees?
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Problem Number 9The graph below depicts the length of a spring vs. the weight suspended from it. Construct a good straight-line model for this data and use your model to determine the length of the spring when weight 57 grams is suspended. Determine also the weight required to achieve length 23 cm. The x axis is marked in units of 11 grams and the y axis in units of 26 cm, both starting at (0,0).
Interpret both the x and the y intercepts of this graph.
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Problem Number 10 Write a difference equation for the amount of antibiotic in the body if 6% is removed per hour, and the initial amount is 830 milligrams. Use this equation to find the number of milligrams after 1, 2, 3 and 4 hours.