Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\geq{0}$ is $d\left(t\right)=\left(5-0.22t\right)^2$.

b. At what time is the tank empty?

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\geq{0}$ is $d\left(t\right)=\left(5-0.22t\right)^2$.

c. What is an appropriate domain for $d$?

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3). Simplify or evaluate the following expressions.

g(1/z)

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(y⁴)

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(g(y))

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(F(x))

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

a. $\text{ƒ}(x+1)$

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

$f\left(2\right(x-1\left)\right)$

Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

Solve the following equations.

$\sin3x=\frac{\sqrt{2}}{2},0\leq{x}\lt{2\pi}$

Solve the following equations.

$\cos3x=\sin3x,0\leq{x}\lt2\pi$

Solve the following equations.

$\tan^22\theta=1,0\lt{\theta}\lt{\pi}$

Solve the following equations.

{Use of Tech} $\sin2\theta=\frac15,0\lt{\theta}\lt{\frac{\pi}{2}}$

Slope functions Determine the slope function S (x) for the following functions

$\text{ƒ}\left(x\right)=\mid x\mid$

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

ƒ o g

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

ƒ o G

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

a. Find A(2) .

ƒ(t) =6 <IMAGE>

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

b. Find A(6).

ƒ(t) =6 <IMAGE>

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

c. Find a formula for A(x)

ƒ(t) =6 <IMAGE>

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o g o ƒ

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

a. Find A(2) .

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

c. Find a formula for A(x)

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Shifting a graph Use a shift to explain how the graph of $\text{ƒ}\left(x\right)=\sqrt{-x^2+8x+9}$ is obtained from the graph of $g\left(x\right)=\sqrt{25-x^2}$ . Sketch a graph of ƒ.

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=x^2$ into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=\text{ƒ}(2x-4)$

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=x^2$ into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=6\text{ƒ}\left(\frac{x-2}{3}\right)+1$

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=\sqrt{x}$  into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=2\text{ƒ}(2x-1)$

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=\sqrt{x}$  into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=3\sqrt{x-1}-5$

Missing piece Let g(x) = x² + 3 Find a function ƒ  that produces the given composition.

(ƒ o g) (x) = x⁴ + 6x² + 9

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

$g\left(x\right)=-3x^2$

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.

(g o ƒ ) (x) = x²⸍³ + 3