Exam 1: A Library of Functions

For what value of a is the function $f ( t ) = \left\{ \begin{array} { c } t - a , \quad - 10 < t \leq 3 \\t ^ { 2 } + 10 t , \quad 3 < t < 10\end{array} \right.$ continuous?

If $f(x)=x^{2}-25$ and $g(x)=3 x-4$ , evaluate $\lim _{x \rightarrow 0}$ [ f(x) + g(x)].If the limit does not exist, enter "DNE".

Joe invested $20,000 in the stock market, and Sam invested $20,000.Joe's investment increased in value by 5% per year for 10 years.Sam's investment decreased in value by 10% for 5 years and then increased by 10% for the next 5 years.At the end of the 10 years, whose investment was worth more, Joe's or Sam's?

Write in factored form the equation of the polynomial graphed below.All key features are shown.

The following describes a function that must be invertible: A woman takes a tablet of 10 mg of a common allergy medicine with a half life of approximately 8 hours.L(t)gives the amount of medicine in her system at time t where 0 < t < 24.

If $f(x)=e^{x+2}$ , find and simplify the difference quotient $\frac{f(6+h)-f(6)}{h}$ .

Use algebra to evaluate the exact value of $\lim _{h \rightarrow 0} \frac{(16+h)^{1 / 2}-4}{h}$ .

Delia runs the 3.1 miles from home to the park at 6 mph, jumps on her bike and returns home in 12 minutes. a)Sketch a well-labeled graph of Delia's distance from home as a function of time. b)Find the slope of each segment of the graph and interpret their meaning. c)What does it mean to say that Delia's velocity is inversely proportional to the time she takes for her run/ride?

Suppose a 40 ${ } ^ { \circ } F$ container of water is placed in the freezer overnight.The next morning, it is put on the counter in a 70 ${ } ^ { \circ } F$ room and then at the end of the day heated to the boiling point.What is the range of the function?

Give a possible formula for the function in the following figure:

If the size of a bacteria colony doubles in 8 hours, how many hours will it take for the number of bacteria to be 11 times the original amount? Round to 2 decimal places.

A bar of soap starts out at 100 grams.What is the formula for the quantity S grams of soap remaining after t days if the decrease is 15% per day?

The empirical function W = f(t), given in the graph below, comes from the Wall Street Journal, September 4, 1992.From the graph, determine the domain of the function.

The following functions represent exponential growth or exponential decay.Mark the one(s)that represent exponential growth.

What is the doubling time of quantity Q growing exponentially according to the formula $Q(t)=Q 5^{t}$ ? Round to 2 decimal places.

Solve $6 e^{3 x}=11 e^{8 x}$ for x.Round to 3 decimal places.

Give a formula for the following sinusoidal function as a transformation of $f(x)=\sin (x)$

If the graph of y = f(x)is shrunk vertically by a factor of 1/2, then shifted vertically by 4 units, then stretched vertically by a factor of 2, the resulting graph the same as the original graph.

Suppose that N(t)= 100,000,000 · $2^{t / 10}$ gives the population of a certain country t years after a census was taken.A historian has a collection of documents that are not dated, but do refer to the population of this country at several times.In order to help the historian date these documents, find the inverse function for the function N.Round to whole numbers.

Harley Davidson (ticker symbol HOG)stock prices dropped sharply in late 2008.Series 1 in the graph below shows the actual prices at the end of each week.The trend over time is approximately linear; and the graph of a possible linear model is given by Series 2.Based on the data given, find the linear model and use it to approximate the stock's price on November 30, 2008, assuming the current trend continued.