함수와 그래프 전체 문제

Rewrite the following expressions in terms of exponentials and simplify.

a. $2\cosh (\ln x)$ b. $\cosh 4x+\sinh 4x$ c. $\cosh 2x-\sinh 2x$ d. $\ln (\cosh x+\sinh x)+\ln (\cosh x-\sinh x)$

[T] The demand D (in millions of barrels) for oil in an oil-rich country is given by the function $D(p)=150\cdot {(2.7)}^{-0.25p},$ where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.

[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] 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] 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] 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.

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.

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

Simplify the following trigonometric expressions.

${\cos }^{2}x-{\sin }^{2}x$

Solve the following trigonometric equations on the interval $x=[-2\pi ,2\pi ]$ exactly.

$6{\cos }^{2}x-3=0$

Solve the following trigonometric equations on the interval $x=[-2\pi ,2\pi ]$ exactly.

${\sec }^{2}x-2\sec x+1=0$

Solve the following logarithmic equations.

${5}^{x}=16$