Direct measurements of the atmospheric carbon dioxide $\left(\mathrm{CO}_2\right)$ concentration have been made since about $1959$, when the measured concentration was about $316$ parts per million. In $2017$, the concentration was about $407$ parts per million. Assume that this growth can be modeled with an exponential function of the form $Q=Q_0 \times(1+r)^t$. a. By experimenting with various values of the fractional growth rate $r$, find an exponential function that fits the given data for $1959$ and $2017$. b. Use this exponential model to predict when the $\mathrm{CO}_2$ concentration will be $560$ parts per million (twice its preindustrial level). c. Research recent trends in the carbon dioxide concentration. Does your model seem to fit, overestimate, or underestimate its recent rise? Explain.

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The maximum speed of a semitrailer truck up a steep hill varies with the weight of its cargo. With no cargo, it can maintain a maximum speed of $50$ miles per hour. With $20$ tons of cargo, its maximum speed drops to $40$ miles per hour. At what load does a linear model predict a maximum speed of $0$ miles per hour?

In $2000$, the median home price in New York City was about $\$ 300,000$, and from $2000$ to $2006$, home prices in New York City rose at an average rate of about $11 \%$ per year. If prices had continued to rise at that rate, what would the median home price have been in $2017$? Compare to the actual median price of about $\$ 400,000$ in $2017$.

Complete the following sentences.

The number of deaths due to poisoning in the United States in $2009(39,000)$ is $\underline{\qquad \qquad}$ percent greater than the number of deaths due to falls $(26,100)$ (most recent data).

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The cost of a particular private school includes a one-time initiation fee of $\$ 2000$, plus annual tuition of $\$ 10,000$. How much will it cost to attend this school for $6$ years?

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The amount of sugar in a fermenting batch of beer decreases with time at a rate of $0.1$ gram per day, starting from an initial amount of $5$ grams. When is the sugar gone?

A small city had a population of $110,000$ in $2010$. Concerned about rapid growth, the residents passed a growth control ordinance limiting population growth to $2 \%$ each year. If the population grows at this $2 \%$ annual rate, what will the population be in $2020?$ What is the maximum growth rate cap that will prevent the population from reaching $150,000$ in $2025?$

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The cost of publishing a poster is $\$ 2000$ for setting up the printing equipment, plus $\$ 3$ per poster printed. What is the total cost to produce $2000$ posters?

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The diameter of a tree increases by $0.2$ inch with each passing year. When you started observing the tree, its diameter was $4$ inches. Estimate the time at which the tree started growing.

Use the radiometric dating formula to answer the following question. The half-life of carbon- $14$ is about $5700$ years. a. You find a piece of cloth painted with organic dyes. By analyzing the dye in the cloth, you find that only $63 \%$ of the carbon-$14$ originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has $12.3 \%$ of the carbon-$14$ that it must have had when it was alive. Estimate when the wood was cut. c. Is carbon-$14$ useful for establishing the age of the Earth? Why or why not?

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. A group of climbers begin climbing at an elevation of $6500$ feet and ascend at a steady rate of $600$ vertical feet per hour. What is their elevation after $3.5$ hours?

Use the radiometric dating formula to answer the following question. Uranium- $238$ has a half-life of $4.5$ billion years. a. You find a rock containing a mixture of uranium-$238$ and lead. You determine that $60 \%$ of the original uranium- $238$ remains; the other $40 \%$ decayed into lead. How old is the rock? b. Analysis of another rock shows that it contains $55 \%$ of its original uranium-$238$; the other $45 \%$ decayed into lead. How old is the rock?

For the following function involving the variables listed, use your intuition or additional research, if necessary, to do the following. a. Describe an appropriate domain and range for the function. b. Make a rough sketch of a graph of the function. Explain the assumptions that go into your sketch. c. Briefly discuss the validity of your graph as a model of the true function. (weight of car, average gas mileage)

For the following functions find out the slope of the graph and the intercept. Then sketch the graph for values of $x$ between $-10$ and 10 .

$$y=2 x+4$$