Problem: Given the rational function \[r(x)\] = \[\frac{1-x}{x-2}\] , explain carefully why as \[x\to {{2}^{-}}\] , \[r(x)\to \infty \] . Be sure to discuss signs and sizes of numbers in your explanation.

3. Fill out the following information about the polynomial \[p(x)\] = \[-2x{{(x+2)}^{3}}(x-4)\] .

1. (1 Point) As \[x\to -\infty \] , \[p(x)\to \] .
2. (1 Point) The y-intercept of \[p(x)\] has coordinates .

c) (1 Point) The zeroes of \[p(x)\] are .

d) (5 Points) Make a rough sketch of \[p(x)\] on the axes below. Don’t worry about the scaling in the y-axis but be very accurate. Be sure to label all points with their (x,y) coordinate s . Also label the behavior of \[p(x)\] near its zeroes using the terminology we learned in class.

4. A function \[f(x)\] undergoes the following transformations to become new function \[g(x)\] :

"Shifted left 3 units, then reflected about the y-axis, then shifted down 2 units."

1. (1 Point) Using function notation, express \[g(x)\] in terms of \[f(x)\] .
   \[g(x)\] =
   Suppose you are given the following additional information about \[f(x)\] :
   \[f(x)\] is a one-to-one function Domain of \[f(x)\] : \[(-\infty ,2)\bigcup (2,\infty )\] Range of \[f(x)\] : \[(-\infty ,1]\]
2. (2 Points) Using interval notation, what are the domain and range of \[g(x)\] ?
   Domain of \[g(x)\] : Range of \[g(x)\] :
3. (1 Point) Using interval notation, what are the domain and range of \[{{f}^{-1}}(x)\] ?
   Domain of \[{{f}^{-1}}(x)\] : Range of \[{{f}^{-1}}(x)\] :
   5. Solve the following equations, show ALL work for FULL credit. Give exact answers. Use natural or common logarithms only if necessary.
   1. (1 Point) \[{{2}^{x}}=14\]
      b) (1 Point) \[{{x}^{2}}=14\]
      
      c) (1 Point) \[3{{(4)}^{t}}=24\]
4. (1 Point) log8 x = \[\frac{-5}{3}\]
   
   e) (1 Point) \[{{e}^{-5m}}=6\]
   f) (2 Points) \[300=50{{\left( 1+\frac{0.03}{4} \right)}^{4t}}\]
   g) (4 Points) \[\log (x-2)+\log (x+1)=1\]
   h) (2 Points) \[2=\frac{8}{3+{{10}^{x}}}\]
   6. Find the domains of the following functions using interval notation.
   Show all work to receive full credit.
   1. (2 Points) \[y=\frac{x-4}{{{x}^{3}}+4x}\]
      b) (2 Points) \[w(t)\] = \[\frac{1}{1-{{e}^{t}}}\]
      c) (4 Points) \[f(x)\] = \[\ln ({{x}^{2}}-5x)\]
      d) (4 Points) \[g(x)\] = \[\sqrt{\frac{3-x}{x+1}}\]
      7. Solve the following formulas for the indicated variable.
      a) (4 Points) \[\ln A=\ln D-rt\] for the variable A. There should be NO logarithms in your final answer!
      b) (4 Points) Solve \[B=p{{t}^{2}}-qt\] for the variable t.
      8. Fill out the following information about the rational function \[f(x)\] = \[\frac{4(x-1)}{{{(x+1)}^{2}}}\] .
      a) (1 Point) \[f(x)\] has a horizontal asymptote of y = .
      b) (1 Point) \[f(x)\] has a vertical asymptote of x = .
      c) (1 Point) \[f(x)\] has a hole at x = .
      d) (1 Point) The y-intercept of \[f(x)\] has coordinates.
5. (1 Point) The x-intercept of \[f(x)\] has coordinates

f) (1 Point) As \[x\to \infty ,f(x)\to \] .

g) (1 Point) As \[x\to -{{1}^{+}},f(x)\to \] .

h) (1 Point) As \[x\to -{{1}^{-}},f(x)\to \] .

1. (1 Point) Use your graphing calculator to find the only local maximum on the graph. State its (x,y) coordinates to one decimal place.
   local maximum at
   j) (5 Points) Make a rough but accurate sketch of \[f(x)\] on the axes below. Be sure to label all points with their (x,y) coordinate s and all asymptotes with their equations .
   
   9. If \[g(x)\] = \[\ln (x)\] , use the Laws of Logarithms to write the following expressions two different ways:
   1. (1 Point) \[g(2a)\] = or

b) (1 Point) 2 \[g(a)\] = or

c) (1 Point) \[g(a+h)-g(a)\] = or

10. There is a corresponding Newton’s Law of Warming for cold objects with an initial temperature of \[{{T}_{0}}\] warming up in a room of constant temperature \[{{T}_{S}}\] . The model for the temperature \[T\] (in °F ) of the warming object as a function of time t in hours is given by \[T(t)\] = \[{{T}_{S}}-D_{0}^{{}}{{e}^{-kt}}\] , where \[{{D}_{0}}\] represents the same difference between the room temperature and the initial temperature of the object.

I take a 20°F turkey out of a freezer on an 80°F day to thaw on the kitchen counter. After an hour the turkey’s temperature reaches 25°F.

a) (5 Points) Find the exact values ( no decimals ) of \[{{T}_{S}}\] , \[{{D}_{0}}\] , and \[k\] for this model.

\[{{T}_{S}}\] =

\[{{D}_{0}}\] =

\[k\] =

b) (3 Points) Cookbooks suggest that to avoid food poisoning you should transfer the thawing bird to a refrigerator when it reaches 30°F. How many hours of thawing can I safely do on the counter? Give your answer to the nearest tenth of an hour.

EXTRA CREDIT: (3 Points) Suppose I put the turkey in a 40°F fridge when the cookbook suggested in part (b). How long must the turkey stay in the fridge for it to completely thaw (ie. reach a temperature of 32°F)? Give your answer to the nearest tenth of an hour.

HINT: the value k is constant for a turkey!

Price: $20.3

Solution: The downloadable solution consists of 11 pages, 930 words and 2 charts.
Deliverable: Word Document