?Compare the functions \(f(x)=x^{5}\) and \(g(x)=5^{x}\) by graphing both functions in

Use the Laws of Exponents to rewrite and simplify each expression. (a) \(\frac{-2^{6}}{4^{3}}\) (b) \(\frac{(-3)^{6}}{9^{6}}\) (c) \(\frac{1}{\sqrt[4]{x^{5}}}\) (d) \(\frac{x^{3} \cdot x^{n}}{x^{n+1}}\) (e) \(b^{3}\left(3 b^{-1}\right)^{-2}\) (f) \(\frac{2 x^{2} y}{\left(3 x^{-2} y\right)^{2}}\) Equation Transcription: Text Transcription: -2^6 / 4^3 (-3)^6 / 9^6 1 / 4th root x^5 x^3 cdot x^n / x^n+1 b^3 (3^b-1)^-2 2x^2 y / (3x^-2 y)^2

Use the Laws of Exponents to rewrite and simplify each expression. (a) \(\frac{\sqrt[3]{4}}{\sqrt[3]{108}}\) (b) \(27^{2 / 3}\) (c) \(2 x^{2}\left(3 x^{5}\right)^{2}\) (d) \(\left(2 x^{-2}\right)^{-3} x^{-3}\) (e) \(\frac{3 a^{3 / 2} \cdot a^{1 / 2}}{a^{-1}}\) (f) \(\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}}\) Equation Transcription: Text Transcription: 3 sqrt 4/ /3 sqrt 108 27^2/3 2x^2 (3x^5)^2 (2x^-2)^-3 x^-3 3a^3/2 . a^½ / a^-1 Sqrt a sqrt b/ 3 sqrt ab

(a) Write an equation that defines the exponential function with base \(b>0 \). (b) What is the domain of this function? (c) If \(???? ? 1\), what is the range of this function? (d) Sketch the general shape of the graph of the exponential function for each of the following cases. (i) \(b >1 \) (ii) \(b=1 \) (iii) \(0< b<1\) Equation Transcription: Text Transcription: b > 0 b not = 1 b > 1 b =1 0 < b<1

(a) How is the number \(e\) defined? (b) What is an approximate value for \(e\)? (c) What is the natural exponential function? Equation Transcription: Text Transcription: e

Graph the given functions on a common screen. How are these graphs related? \(y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}\) Equation Transcription: Text Transcription: y ? 2^x , y ? e^x , y ? 5^x , y ? 20^x

Graph the given functions on a common screen. How are these graphs related? \(y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}\) Equation Transcription: Text Transcription: y ? e^x , y ? e^-x , y ? 8^x , y ? 8^-x

Graph the given functions on a common screen. How are these graphs related? \(y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}\) Equation Transcription: Text Transcription: y = 3^x , y=10^x , y = ( 1/ 3)^ x , y= ( 1/ 10 )^ x

Graph the given functions on a common screen. How are these graphs related? \(y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}\) Equation Transcription: Text Transcription: y = 0.9^x, y = 0.6^x, y = 0.3^x, y = 0.1^x

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3. \(g(x)=3^{x}+1\) Equation Transcription: Text Transcription: g(x)=3^x+1

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3. \(h(x)=2\left(\frac{1}{2}\right)^{x}-3\) Equation Transcription: Text Transcription: h(x) = 2 (½)^x - 3

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3. \(y=-e^{-x}\) Equation Transcription: Text Transcription: y = -e^-x

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3. \(y=4^{x+2}\) Equation Transcription: Text Transcription: y = 4^x+2

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3. \(y=1-\frac{1}{2} e^{-x}\) Equation Transcription: Text Transcription: y = 1 - 1/2e^-x

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3. \(y=e^{|x|}\) Equation Transcription: Text Transcription: y = e^|x|

Starting with the graph of \(y=e^{x}\), write the equation of the graph that results from (a) shifting 2 units downward. (b) shifting 2 units to the right. (c) reflecting about the \(x\)-axis. (d) reflecting about the \(y\)-axis. (e) reflecting about the \(x\)-axis and then about the \(y\)-axis. Equation Transcription: Text Transcription: ???? ? ????^???? x y

Starting with the graph of \(y=e^{x}\), find the equation of the graph that results from (a) reflecting about the line y = 4. (b) reflecting about the line x = 2.

Find the domain of each function. (a) \(f(x)=\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}\) (b) \(f(x)=\frac{1+x}{e^{\cos \ x}}\) Equation Transcription: Text Transcription: f(x) = 1 - e^x^2/1 - e^1-x^2 f(x) = 1 + x/e^cos x

Find the domain of each function. (a) \(g(t)=\sqrt{10^{t}-100}\) (b) \(g(t)=\sin \left(e^{t}-1\right)\)

Find the exponential function \(f(x)=C b^{x}\) whose graph is given. Equation Transcription: Text Transcription: f(x)=Cb^x

Find the exponential function \(f(x)=C b^{x}\) whose graph is given.

If \(f(x)=5^{x}\), show that \(\frac{f(x+h)-f(x)}{h}=5^{x}\left(\frac{5^{h}-1}{h}\right)\) Equation Transcription: Text Transcription: f(x)=5^x f(x + h) - f(x)/h=5^x (5^h - 1/h)

Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? I. One million dollars at the end of the month. II. One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, \(2^{n-1}\) cents on the \(n\)th day. Equation Transcription: Text Transcription: 2^n-1 n

Suppose the graphs of \(f(x)=x^{2}\) and \(g(x)=2^{x}\) are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that at a distance 2 ft to the right of the origin, the height of the graph of f is 48 ft but the height of the graph of t is about 265 mi.

Compare the functions \(f(x)=x^{5}\) and \(g(x)=5^{x}\) by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when \(x\) is large? Equation Transcription: Text Transcription: f(x)=x^5 g(x)=5^x x

Compare the functions \(f(x)=x^{10}\) and \(g(x)=e^{x}\) by graphing both functions in several viewing rectangles. When does the graph of \(g\) finally surpass the graph of \(f\) ? Equation Transcription: Text Transcription: f(x)=x^10 g(x)=e^x g f

Use a graph to estimate the values of \(x\) such that \(e^{x}>1,000,000,000\). Equation Transcription: Text Transcription: x e^x>1,000,000,000

A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. (a) Make a scatter plot of the data. (b) Use a calculator or computer to find an exponential curve \(f(t)=a \cdot b^{t}\) ? but that models the bacteria population \(t\) hours later. (c) Graph the model from part (b) together with the scatter plot in part (a). Use the graph to estimate how long it takes for the bacteria count to double. Equation Transcription: Text Transcription: f(????) ? a . b^t t

The table gives the population of the United States, in millions, for the years 1900 –2010. Use a graphing calculator (or computer) with exponential regression capability to model the US population since 1900. Use the model to estimate the population in 1925 and to predict the population in the year 2020.

A bacteria culture starts with 500 bacteria and doubles in size every half hour. (a) How many bacteria are there after 3 hours? (b) How many bacteria are there after \(t\) hours? (c) How many bacteria are there after 40 minutes? (d) Graph the population function and estimate the time for the population to reach 100,000. Equation Transcription: Text Transcription: t

A gray squirrel population was introduced in a certain region 18 years ago. Biologists observe that the population doubles every six years, and now the population is 600. (a) What was the initial squirrel population? (b) What is the expected squirrel population t years after introduction? (c) Estimate the expected squirrel population 10 years from now.

In Example 4 , the patient's viral load V was 76.0 RNA copies per mL after one day of treatment. Use the graph of V in Figure 11 to estimate the additional time required for the viral load to decrease to half that amount.

After alcohol is fully absorbed into the body, it is metabolized. Suppose that after consuming several alcoholic drinks earlier in the evening, your blood alcohol concentration (BAC) at midnight is 0.14 g/dL. After 1.5 hours your BAC is half this amount. (a) Find an exponential model for your BAC t hours after midnight. (b) Graph your BAC and use the graph to determine when your BAC reaches the legal limit of 0.08 g/dL.

If you graph the function \(f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}\) you'll see that \(f\) appears to be an odd function. Prove it. Equation Transcription: Text Transcription: f(x) = 1 - e^1/x/1 + e^1/x t

Graph several members of the family of functions \(f(x)=\frac{1}{1+a e^{b x}}\) where \(a>0\). How does the graph change when b changes? How does it change when a changes? Equation Transcription: Text Transcription: f(x) = 1/1 + ae^bx a >0

Graph several members of the family of functions \(f(x)=\frac{a}{2}\left(e^{x / a}+e^{-x / a}\right)\) where \(a>0\). How does the graph change as a increases? Equation Transcription: Text Transcription: f(x) = a/2(e^x/a + e^-x/a) a > 0