Final Exam Review
1 . The distance d (measured in feet) between Sylvia and her house is modeled by the formula d = t2 + 3t + 1 where t represents the number of seconds since Sylvia started walking. Find Sylvia's average speed for the time period from t = 2 to t = 7 seconds. Explain what average speed means in the context of this problem.
2. Suppose that the function h relates a dairy farmer's cost (measured in dollars) as a function of the number of gallons of milk produced. Explain what each of the following represents.
(a) h(18)
(b) h(b) = 25
(c) 7h(9)
(d) h(4.3) - h(6.4)
3. Given the function f defined by f (x) = 3x + 2, define f-1 the inverse function of f. Then evaluate f(f-1(x)) and f-1(f(x)).
4. A plastic children's pool is originally filled with 200 gallons of water. The pool springs a leak, and the water level starts dropping by 1 gallon every half hour. Define a function which gives the volume of water in the pool as a function of time since the leak started. Make sure to define your variables.
5. Suppose you put 5 pennies on the first square of a chessboard. On the second square of the chessboard, you place triple the number of pennies on the first square. If you continue this pattern, how many pennies will you place on the fifth square? How many pennies will you place on the fiftieth square? Define a function that describes the number of pennies on a given square of the chessboard. Let n represent the number of the square and P = f (n) represent the number of pennies on the given square.
6. In 1998, the population of Podunk, IA was 219. Then Podunk experienced a burst of growth, and by 2007 its population was 843. Assume the population of Podunk grew exponentially over this period. Determine the population's 9-year growth factor and percent change, and its 1-year growth factor and percent change. Then define a function that gives the population of Podunk in terms of the number of years that have elapsed since 1998.
7. Find the difference quotient, (f(x + h) -f(x))/h, for the following functions.
(a) f (x) = 12x + 6.5
(b) f (x) = 6x2 + 7x - 11
Part -2:
1. Solve each of the following for x.
(a) log2(2 + x) + log2(7) = 3
(b) ln(3) 2 ln(x) ln(5x) = ln(x + 9)
2. Rewrite each of the following as sums and differences of logarithms.
(a) log7 (4/y)
(b) log2 (12x4)
(c) log5 (10x3/y5)
3. Consider the following table.
| x |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
| f(x) |
9 |
2 |
6 |
-4 |
-5 |
-8 |
-9 |
| g(x) |
3 |
0 |
3 |
2 |
-3 |
-1 |
-5 |
a) Is f-1 a function? Explain why or why not.
b) Is g-1 a function? Explain why or why not.
c) Evaluate the following expressions:
i) f-1(-9)
ii) (g(2))
iii) f-1(0))
iv) g(f-1(2))
4. You are going to invest $5000. One account advertises 5% APR compounded monthly, and another account advertises 4% APR compounded continuously. For each of the accounts, how long will it take for your investment to grow to $9000? (Write your answers in exact form.) Which account should you choose for your investment?
5. Explain how the average rate of change of a function is related to the concavity of its graph. Use your calculator to sketch the graph of y = x3 + 10x2 +1. Mark the point where this function's average rate of change switches from decreasing to increasing.
6. Suppose a circle of radius 6 km is centered at the origin, and an angle of measure 2/3 radians is swept out counterclockwise from the positive x-axis. Let (x, y) be the point where the terminal ray of the angle intersects the circle. Find the coordinates of (x, y) both in radius lengths and in kilometers.
7. Evaluate the following limits:
a) lim 3/(x - 7)
x→7+
b) lim 3/(x - 7)
x→7-
c) lim x2 - 2/(x2 + 3x + 2)
x → ∞
d) lim x2/(x+5)
X→ ∞
e) lim (2x2 + 7)/(5x4 -10)
X→ ∞
8. Evaluate the following without a calculator.
a) cos-1 (cos (5Π/6))
b) arcsin (sin (5Π/6)
c) tan-1 (tan (5Π/6))
9. A Ferris wheel has a radius of 52 feet and the horizontal diameter is located 58 feet above the ground. Define a function g to represent the distance (in feet) of a Ferris wheel bucket above the ground as a function of the number of minutes, t, since the Ferris wheel began rotating from the 3 o'clock position. Assume the Ferris wheel rotates at a constant rate of 1 radian per minute. Sketch the graph of g. Be sure to label the axes.