One of the following 3 questions will appear (word for word as it is below) on the exam. You should prepare your own response to these questions and expect to spend about 10 minutes responding to this question on the exam. You may consult other class members or other students currently enrolled in math 170. You may not request help from your instructor, tutors, or people who have had calculus before this semester.

1. At t = 0 minutes, a beverage bottle containing 12 ounces of 72 degree Fahrenheit liquid is placed in a 40 degree Fahrenheit refrigerator. After t minutes, the temperature (in degrees Fahrenheit) of the bottle is given by , where a, b and care positive constants, with 0 < b< 1. In addition, 2 points on the graph of Temp(t)are

• what the constants a and crepresent and their numeric values;
• the value of the constant b;
• a determination of the time t , when the temperature of the liquid is 42 degrees Fahrenheit.

3. Give a detailed discussion of exponential functions, including graphical, numerical, verbal and algebraic properties that such functions possess, and their comparisons to polynomial or other functions.

The remaining questions involve most of the important concepts which you should understand for exam 1. If you can do these problems well and quickly, you are well prepared for the exam. Several questions which cover similar concepts will appear on exam 1, but the actual exam questions may look entirely different than the ones included here.

1. Let t denote time, measured in hours, starting at noon. The table below gives the temperature (in degrees Fahrenheit), over a 24 hour period, as a function f of time. Use the table to answer the following questions.

a) What is the value of f(14) and state in words what this value represents.

b) Was it colder at t = 4 or t = 16?

c) At what approximate time(s) of day was the temperature 3 degrees Fahrenheit?

d) Was the temperature rising or falling at t = 14?

e) What is the average rate of change in temperature from noon to midnight?

f) Estimate (as best you can) the instantaneous rate of change in temperature at 2 pm.
 
 

2. A bug starts out 10 feet from a light, flies closer to the light, then farther away, then closer than before, then farther away. Finally the bug hits the bulb and flies off. Sketch a possible graph of the distance of the bug from the light as a function of time.

3. How do you decide if a table of values for a function f(x) can be closely approximated by a linear formula f(x) = m*x + b?

4. Shown below is a graph of the function f(x), as x runs from 0 to 5. Use the plot to answer the following questions.

a) What is f(2)?

b) For what value of x does f(x) = 2?

c) For which values of x is f(x) decreasing?

d) Draw in the tangent line to f(x) at x = 2 and then estimate the slope of this tangent line.

e) What (approximately) is the instantaneous rate of change of f(x) at x = 2?

f) For which values of x is f(x) < -3?

g) For x values between 0 and 5, how many solutions does f(x) = 0 have? What are they?

h) Sketch a graph of -f(x) - 3.

5. Use parametric equations to sketch a graph of P(t) = 5*0.6^t, and its inverse, for values of t from 0 to 5.

6. Find (accurate to 3 decimal places) all solutions (values of w) to 8*w^3 = 1.2^w.

7. Find possible formulas for each of the functions represented below.

a) P(t) is the size of a population (after t years) which starts at 250 and decreases by 4% each year.

b) The functions f(x) and g(x) given by the following table:

c) The functions h(w) (thick) and k(w) (thin) plotted below.

8. NOTE: Graphs of k(w) and h(w) are shown just above (in problem 7). Let f(w) be the piecewise function defined by

a) Calculate f( -1), f(0), f(1), f(2), and f(3).

b) Estimate , , , and 

9. Find an x-range, centered at x = 0, so that the graph of f(x) = on your x-range lies between y = -1 and y = -0.8.

10. Estimate each of the following limits: , , .

11. If f(x) = 12 - 3*2.4^x, find a formula for the inverse of f(x). For what value of x does f(x) = -10? What is ?

12. If f(x) = (2^x)* cos(x - 0.3) + 0.5, find the average rate of change of f(x) from x = 1 to b, where b is each of the values 1.2, 0.95, 1.0003 and .999997. Estimate the equation of the tangent line to f(x) at x = 1, and the instantaneous rate of change of f(x) at x = 1. Finally, on a well selected viewing window, plot both f(x) and its tangent line for x values from 0 to 2.

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