Exam 4: Exponential Functions

The table below shows $v$ , the dollar value of a share of a certain stock, as a function of $t$ , the time (in weeks) since the initial offering of the stock. A possible formula for $v(t)$ is $v(t)=\ldots(\ldots)^{t}$ . Round the second answer to 3 decimal places.

Assume that all important features are shown in the following graph of $y=f(x)$ . What is $\lim _{x \rightarrow-\infty} f(x)$ ? For $\infty$ or $-\infty$ , enter "inf" or "-inf".

Let $P(t)=4,500 e^{0.043 t}$ give the size of a population of animals in year $t$ . What will the population be after 12 years? Round to the nearest whole number.

The population of a city is increasing exponentially. In 2000 , the city had a population of 40,000 . In 2005 , the population was 58,502. The formula for $P(t)$ , the population of the town $t$ years after 2000 , is given by $p(t)=\ldots(\ldots)^{t}$ .Round your second answer to 3 decimal places.

Let $\left(x_{0}, y_{0}\right)$ be the intersection of the graphs of the two exponential functions $y=a e^{b x}$ and $y=c e^{d x}$ , where $0<a<c$ . If $a$ is increased, does $x_{0}$ increase, decrease, or stay the same?

Kevin buys a new CD player for $\$ 300$ , and finds two years later when he wants to sell it that it is only worth $\$ 82$ . Assuming the value of the CD player decreases exponentially, the formula for $V(t)$ , the value of the CD player after $t$ years, is given by $V(t)=\ldots(\ldots)^{t}$ . Round your second answer to 2 decimal places.

Is the formula for a function representing a quantity which begins at an amount $35 \%$ larger than $N$ in year $t=0$ and grows at a continuous annual rate of $r \%$ given by $f(t)=1.35 N e^{r t /100}$ ?

What is $\lim _{x \rightarrow \infty}-5 e^{3 x}$ ? If necessary, enter "inf" for $\infty$ and "-inf" for $-\infty$ .

The populations of 4 species of animals are given by the following equations: $P_{1}=390(0.75)^{t} P_{2}=100(1.09)^{t} P_{3}=230(0.82)^{t} P_{4}=600(1.05)^{t}$ What is the largest initial population of the 4 species?

In the following figure, the functions $f, g, h$ , and $p$ can all be written in the form $y=a b^{t}$ . Which one has the largest value for $b$ ?

The following table gives values from an exponential or a linear function. Determine which, and find values for $a$ and $b$ so that $f(x)=a+b x$ if the function is linear, or $f(x)=a(b)^{x}$ if the function is exponential. a= ---------------,b= ------------

Is the function graphed exponential?

Find $\lim _{t \rightarrow \infty} 20,000(0.82)^{t}$ . For $\infty$ or $-\infty$ , enter "inf" or "-inf".

The graph below shows the quantity of a drug in a patient's bloodstream over a period of time $t$ , in minutes. Which of the following scenarios best describes the graph?

In 2006, the cost of a particular piece of computer equipment was $\$ 490$ and going down at a rate of $14 \%$ per year. Assuming this percentage remains constant, what is the formula for $C$ , the cost of this equipment in dollars, as a function of $t$ , the number of years since 2006 ?

The price of an item increases due to inflation. Let $p(t)=12.50(1.024)^{t}$ give the price of the item as a function of time in years, with $t=0$ in 2004 . Estimate $p^{-1}(75)$ to 2 decimal places.

Let $f(x)$ be given in the table below. Find the value of $k$ if $f(x)$ is exponential.

Write the formula for the price $p$ of a gallon of gas in $t$ days if the price is $\$ 3.85$ on day $t$ $=0$ and the price increases by $\$ 0.08$ per day.

A biologist measures the amount of contaminant in a lake 2 hours after a chemical spill and again 11 hours after the spill. She sets up two possible models to determine $Q$ , the amount of the chemical remaining in the lake as a function of $t$ , the time in hours since the spill. The first model assumes the contaminant is leaving the lake at a constant rate, which she determines to be 3 tons\hour. Using this model, she estimates that the lake will be free from the contaminant 20 hours after the spill. Thus, Q(2)= ---------- and Q(11)= ----------- The second model assumes that the amount of contaminant decreases exponentially. In this model, she finds that $Q(t)=\ldots(\ldots)^{t}$ . Round both answers to 3 decimal places.

The graph of $P(t)=1.7^{-t}+29$ has a horizontal asymptote at $P(t)=$ ---------. (If there is no horizontal asymptote, enter "DNE".)