함수와 그래프 전체 문제

[T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by $A(t)=100,000\cdot {e}^{0.055t}.$ Find the amount A accumulated in 5 years.

[T] An investment is compounded monthly, quarterly, or yearly and is given by the function $A=P{\left(1+\frac{j}{n}\right)}^{nt},$ where $A$ is the value of the investment at time $t,P$ is the initial principle that was invested, $j$ is the annual interest rate, and $n$ is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount $A$ accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.

[T] The concentration of hydrogen ions in a substance is denoted by $[{\text{H}}^{+}],$ measured in moles per liter. The pH of a substance is defined by the logarithmic function $\text{pH}=-\log [{\text{H}}^{+}].$ This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.

a. Find the pH of the following substances. Round answers to one digit. b. Determine whether the substance is an acid or a base. i. Eggs: $[{\text{H}}^{+}]=1.6\,\times \,{10}^{-8}$ mol/L ii. Beer: $[{\text{H}}^{+}]=3.16\,\times \,{10}^{-3}$ mol/L iii. Tomato Juice: $[{\text{H}}^{+}]=7.94\,\times \,{10}^{-5}$ mol/L

[T] Iodine-131 is a radioactive substance that decays according to the function $Q(t)={Q}_{0}\cdot {e}^{-0.08664t},$ where ${Q}_{0}$ is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.

[T] According to the World Bank, at the end of 2013 ($t=0$ ) the U.S. population was 316 million and was increasing according to the following model:

$P(t)=316{e}^{0.0074t},$ where P is measured in millions of people and t is measured in years after 2013.

a. Based on this model, what will be the population of the United States in 2020? b. Determine when the U.S. population will be twice what it is in 2013.

[T] The amount A accumulated after 1000 dollars is invested for t years at an interest rate of 4% is modeled by the function $A(t)=1000{(1.04)}^{t}.$

a. Find the amount accumulated after 5 years and 10 years. b. Determine how long it takes for the original investment to triple.

[T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.

a. Use the exponential function $Q={Q}_{0}{e}^{kt}$ to determine the value $k,$ which is the growth rate of the bacteria. Round to four decimal places. b. Determine approximately how long it takes for 200,000 bacteria to grow.

[T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.

a. Use the exponential function $P={P}_{0}{a}^{t}$ to determine the growth rate constant $a.$ Round to four decimal places. b. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.

[T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more intense was the San Francisco earthquake than the Japanese earthquake? See the definition of Richter Scale in (see text) in this section.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

A function is always one-to-one.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

$f\circ g=g\circ f,$ assuming f and g are functions.

A relation that passes the horizontal and vertical line tests is a one-to-one function.

A relation passing the horizontal line test is a function.

For the following problems, state the domain and range of the given functions:

$f={x}^{2}+2x-3,\quad g=\ln (x-5),\quad h=\frac{1}{x+4}$

h

For the following problems, state the domain and range of the given functions:

$f={x}^{2}+2x-3,\quad g=\ln (x-5),\quad h=\frac{1}{x+4}$

g

For the following problems, state the domain and range of the given functions:

$f={x}^{2}+2x-3,\quad g=\ln (x-5),\quad h=\frac{1}{x+4}$

$h\circ f$

For the following problems, state the domain and range of the given functions:

$f={x}^{2}+2x-3,\quad g=\ln (x-5),\quad h=\frac{1}{x+4}$

$g\circ f$

Find the degree, y-intercept, and zeros for the following polynomial functions.

$f(x)=2{x}^{2}+9x-5$

Find the degree, y-intercept, and zeros for the following polynomial functions.

$f(x)={x}^{3}+2{x}^{2}-2x$

Simplify the following trigonometric expressions.

$\frac{{\tan }^{2}x}{{\sec }^{2}x}+{\cos }^{2}x$