Chapter 1

Graph the function. What is the amplitude and period? $$y=3 \sin x$$

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 \(p\) $$y=\frac{x}{5}$$

In Problems \(1-3,\) find the following: (a) \(\quad f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=5 x-1 \text { and } g(x)=3 x+2$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$10=2^{t}$$

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)?

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

Graph the function. What is the amplitude and period? $$y=4 \cos 2 x$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$5^{t}=7$$

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 \(p\) $$y=5 \sqrt{x}$$

In Problems \(1-3,\) find the following: (a) \(\quad f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=x-2 \text { and } g(x)=x^{2}+8$$

The exponential function \(y(x)=C e^{\alpha x}\) satisfies the conditions \(y(0)=2\) and \(y(1)=1 .\) Find the constants \(C\) and a. What is \(y(2) ?\)

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) \(\quad A=5.3(1.054)^{t}\) (c) \(\quad A=3500(0.93)^{t}\) (d) \(\quad A=12(0.88)^{t}\)

Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing. The height of a sand dune is a function of time, and the wind erodes away the sand dune over time.

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

Graph the function. What is the amplitude and period? $$y=-3 \sin 2 \theta$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$2=(1.02)^{t}$$

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 \(p\) $$y=\frac{8}{x}$$

In Problems \(1-3,\) find the following: (a) \(\quad f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=3 x \text { and } g(x)=e^{2 x}$$

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?

Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing. The amount of carbon dioxide in the atmosphere is a function of time, and is going up over time.

Find an equation for the line that passes through the given points. $$(-2,1) and (2,3)$$

Graph the function. What is the amplitude and period? $$y=3 \sin 2 x$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$130=10^{t}$$

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 \(p\) $$y=\frac{3}{x^{2}}$$

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

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?

Values of a linear cost function are in Table \(1.27 .\) What are the fixed costs and the marginal cost? Find a formula for the cost function. $$\begin{array}{c|c|c|c|c|c}\hline q & 0 & 5 & 10 & 15 & 20 \\ \hline C(q) & 5000 & 5020 & 5040 & 5060 & 5080 \\\\\hline\end{array}$$

Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing. The number of air conditioning units sold is a function of temperature, and goes up as the temperature goes up.

Find an equation for the line that passes through the given points. $$(4,5) and (2,-1)$4

Graph the function. What is the amplitude and period? $$y=5-\sin 2 t$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$10=e^{t}$$

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 \(p\) $$y=2^{x}$$

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

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.

The cost \(C,\) in millions of dollars, of producing \(q\) items is given by \(C=5.7+0.002 q .\) Interpret the 5.7 and the 0.002 in terms of production. Give units.

Table 1.12 gives values of a function \(w=f(t) .\) Is this function increasing or decreasing? Is the graph of this function concave up or concave down? $$\begin{array}{c|c|c|c|c|c|c|c} \hline t & 0 & 4 & 8 & 12 & 16 & 20 & 24 \\ \hline w & 100 & 58 & 32 & 24 & 20 & 18 & 17 \\ \hline \end{array}$$

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

Graph the function. What is the amplitude and period? $$y=4 \cos \left(\frac{1}{2} t\right)$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$100=25(1.5)^{t}$$

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 \(p\) $$y=\frac{3}{8 x}$$

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

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

If 12,000 is deposited in an account paying \(8 \%\) interest per year, compounded continuously, how long will it take for the balance to reach \(\$ 20,000 ?\)

(a) Give an example of a possible company where the fixed costs are zero (or very small). (b) Give an example of a possible company where the marginal cost is zero (or very small).

Graph a function \(f(x)\) which is increasing everywhere and concave up for negative \(x\) and concave down for positive \(x\)

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

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$50=10 \cdot 3^{t}$$

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 \(p\) $$y=\left(3 x^{5}\right)^{2}$$

For the functions \(f\) and \(g\) in Problems \(7-10,\) find (a) \(\quad f(g(1))\) (b) \(g(f(1))\) (c) \(\quad f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=x^{2}, g(x)=x+1$$

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.