Chapter 1

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=5 \sqrt{x}$$

For \(g(x)=x^{2}+2 x+3\), find and simplify: (a) \(g(2+h)\) (b) \(g(2)\) (c) \(g(2+h)-g(2)\)

World wind energy generating \(^{59}\) capacity, \(W\), was 18,000 megawatts in 2000 and has been increasing at a continuous rate of approximately \(27 \%\) per year. Assume this rate continues. (a) Give a formula for \(W\), in megawatts, as a function of time, \(t\), in years since 2000 . (b) When is wind capacity predicted to pass 250,000 megawatts?

Solve for \(t\) using natural logarithms. $$5^{t}=7$$

The following functions give the populations of four towns with time \(t\) in years. (i) \(\quad P=600(1.12)^{t}\) (ii) \(\quad P=1,000(1.03)^{t}\) (iii) \(\quad P=200(1.08)^{t}\) (iv) \(\quad P=900(0.90)^{t}\) (a) Which town has the largest percent growth rate? What is the percent growth rate? (b) Which town has the largest initial population? What is that initial population? (c) Are any of the towns decreasing in size? If so, which one(s)?

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$ 7 y+12 x-2=0 $$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{3}{x^{2}}$$

If \(f(x)=x^{2}+1\), find and simplify: (a) \(f(t+1)\) (b) \(f\left(t^{2}+1\right)\) (c) \(f(2)\) (d) \(2 f(t)\) (e) \([f(t)]^{2}+1\)

The half-life of nicotine in the blood is 2 hours. A person absorbs \(0.4 \mathrm{mg}\) of nicotine by smoking a cigarette. Fill in the following table with the amount of nicotine remaining in the blood after \(t\) hours. Estimate the length of time until the amount of nicotine is reduced to \(0.04 \mathrm{mg}\). $$ \begin{array}{c|c|c|c|c|c|c} \hline t \text { (hours) } & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \text { Nicotine (mg) } & 0.4 & & & & & \\ \hline \end{array} $$

Solve for \(t\) using natural logarithms. $$10=2^{t}$$

Each of the following functions gives the amount of a substance present at time \(t\). In each case, give the amount present initially (at \(t=0\) ), state whether the function represents exponential growth or decay, and give the percent growth or decay rate. (a) \(\quad A=100(1.07)^{t}\) (b) \(A=5.3(1.054)^{t}\) (c) \(A=3500(0.93)^{t}\) (d) \(\quad A=12(0.88)^{t}\)

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$ 3 x+2 y=8 $$

The population of a city, \(P\), in millions, is a function of \(t\), the number of years since 1970 , so \(P=f(t)\). Explain the meaning of the statement \(f(35)=12\) in terms of the population of this city.

Sketch a possible graph of sales of sunscreen in the northeastern US over a 3 -year period, as a function of months since January 1 of the first year. Explain why your graph should be periodic. What is the period?

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=2^{x}$$

If you deposit $$\$ 10,000$$ in an account earning interest at an \(8 \%\) annual rate compounded continuously, how much money is in the account after five years?

Solve for \(t\) using natural logarithms. $$2=(1.02)^{t}$$

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$ 12 x=6 y+4 $$

Let \(W=f(t)\) represent wheat production in Argentina, \({ }^{4}\) in millions of metric tons, where \(t\) is years since 1990 . Interpret the statement \(f(12)=9\) in terms of wheat production.

For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=x^{2}, g(x)=x+1$$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=3 \sin x$$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{3}{8 x}$$

If you need $$\$ 20,000$$ in your bank account in 6 years, how much must be deposited now? The interest rate is \(10 \%\), compounded continuously.

Solve for \(t\) using natural logarithms. $$130=10^{t}$$

A company has cost and revenue functions, in dollars, given by \(C(q)=6000+10 q\) and \(R(q)=12 q\). (a) Find the cost and revenue if the company produces 500 units. Does the company make a profit? What about 5000 units? (b) Find the break-even point and illustrate it graphically.

A town has a population of 1000 people at time \(t=0\). In each of the following cases, write a formula for the population, \(P\), of the town as a function of year \(t\). (a) The population increases by 50 people a year. (b) The population increases by \(5 \%\) a year.

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$ -4 y+2 x+8=0 $$

The concentration of carbon dioxide, \(C=f(t)\), in the atmosphere, in parts per million (ppm), is a function of years, \(t\), since 1960 . (a) Interpret \(f(40)=370\) in terms of carbon dioxide. \({ }^{5}\) (b) What is the meaning of \(f(50)\) ?

For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=\sqrt{x+4}, g(x)=x^{2}$$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=3 \sin 2 x$$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\left(3 x^{5}\right)^{2}$$

If a bank pays \(6 \%\) per year interest compounded continuously, how long does it take for the balance in an account to double?

Solve for \(t\) using natural logarithms. $$50=10 \cdot 3^{t}$$

Suppose that \(q=f(p)\) is the demand curve for a product, where \(p\) is the selling price in dollars and \(q\) is the quantity sold at that price. (a) What does the statement \(f(12)=60\) tell you about demand for this product? (b) Do you expect this function to be increasing or decreasing? Why?

The gross domestic product, \(G\), of Switzerland was 310 billion dollars in 2007 . Give a formula for \(G\) (in billions of dollars) \(t\) years after 2007 if \(G\) increases by (a) \(3 \%\) per year (b) 8 billion dollars per year

Find the relative, or percent, change. \(S\) changes from 400 to 450

Find an equation for the line that passes through the given points. $$ (0,2) \text { and }(2,3) $$

For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=e^{x}, g(x)=x^{2}$$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=-3 \sin 2 \theta$$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{5}{2 \sqrt{x}}$$

For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=1 / x, g(x)=3 x+4$$

Suppose $$\$ 1000$$ is invested in an account paying interest at a rate of \(5.5 \%\) per year. How much is in the account after 8 years if the interest is compounded (a) Annually? (b) Continuously?

Solve for \(t\) using natural logarithms. $$100=25(1.5)^{t}$$

The demand curve for a quantity \(q\) of a product is \(q=\) \(5500-100 p\) where \(p\) is price in dollars. Interpret the 5500 and the 100 in terms of demand. Give units.

A product costs $$\$ 80$$ today. How much will the product cost in \(t\) days if the price is reduced by (a) $$\$ 4$$ a day (b) \(5 \%\) a day

Find the relative, or percent, change. \(\underline{B}\) changes from 12,000 to 15,000

Find an equation for the line that passes through the given points. $$ (0,0) \text { and }(1,1) $$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=4 \cos 2 x$$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=3 \cdot 5^{x}$$

Let \(f(x)=x^{2}\) and \(g(x)=3 x-1\). Find the following: (a) \(f(2)+g(2)\) (b) \(f(2)=g(2)\) (c) \(f(g(2))\) (d) \(g(f(2))\)