Question 1

Find the limit.

Accepted Answer

The answer of Find the limit. ...

Question 2

Find the limit.

Accepted Answer

The answer of Find the limit. 
...

Question 3

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Accepted Answer

The region is bounded by the curves $y=x^2$ and $y=4$ and is rotated about the $y$-axis.

First, we need to find the points of intersection between the two curves:

$$x^2=4\quad\Rightarrow\quad x=\pm 2.$$

Thus, the region we are rotating is as follows:

[asy]
size(150);
import graph;
real f(real x)
{
	return x^2;
}
fill((2,0)--(2,4)--(-2,4)--(-2,0)--cycle,lightcyan);
draw(graph(f,-2.2,2.2),linewidth(1bp));
draw((-2.2,0)--(2.2,0),Arrows(4));
draw((0,-0.2)--(0,4.2),Arrows(4));
label("$y=x^2$",(2.2,f(2)),E);
label("$y=4$",(2.2,4),E);
[/asy]

When we rotate this region about the $y$-axis, we get a solid that is a "shell" of thickness $dx$. The volume of each shell is the circumference of the shell times the height of the shell times the thickness:

$$dV=2\pi x\cdot(4-x^2)\cdot dx.$$Thus, the total volume is given by the integral:

$$\int_{-2}^2 2\pi x\cdot(4-x^2)\cdot dx.$$Evaluating this integral gives:

\begin{align*}
\int_{-2}^2 2\pi x\cdot(4-x^2)\cdot dx&=\left.\pi\left(4x^2-\frac{x^4}{4}ight)ight|_{-2}^2\
&=\pi(16-4-\left(-16+\frac{4}{4}ight))\
&=\boxed{24\pi}.
\end{align*}

Question 4

Find the volume of the solid obtained by rotating about the y axis the region bounded by the curves.

Accepted Answer

To find the volume of the solid, we need to use the formula for volume by rotation:

$V=\int_a^b\pi[f(y)]^2dy$

We can see from the graph that the two curves intersect at the point (1,1), so we will use this as our upper bound of integration. We also know that the two curves are symmetric, so we only need to find the volume of half the region, and then multiply by 2.

We will first find the equation of the curve labeled $y=x^3-3x+2$:

$x=y^{\frac13}-\frac{2}{y^{\frac13}}$

Then, we can set up our integral:

$V=2\int_0^1 \pi \left[\left(y^{\frac13}-\frac{2}{y^{\frac13}}\right)\right]^2 dy$

Simplifying, we get:

$V=2\int_0^1 \pi \left(y^{\frac{2}{3}}-\frac{4}{y^{\frac{1}{3}}}+4y^{\frac{2}{3}}\right) dy$

$V=2\pi \left[\frac{3}{5}y^{\frac{5}{3}}-\frac{8}{3}y^{\frac{2}{3}}+\frac{12}{5}y^{\frac{5}{3}}\right]_0^1$

$V=\frac{16}{5}\pi$

Therefore, the answer is B.

Question 5

Use the laws of logarithms to write the expression as the logarithm of a single quantity. 4 ln 3 -  ln (x + 3)

Accepted Answer

The answer of Use the laws of logarithms to write...

Question 6

Differentiate the function.

Accepted Answer

The answer of Differentiate the function. 
...

Question 7

Solve the equation for x.

Accepted Answer

The answer of Solve the equation for x. ...

Question 8

Differentiate the function.

Accepted Answer

The answer of Differentiate the function. 
...

Question 9

Find the values of  for which  satisfies the equation

Accepted Answer

The answer of Find the values of  for which...

Question 10

If a bacteria population starts with 50 bacteria and doubles every three hours, then the number of bacteria after t hours is  . When will the population reach  ?

Accepted Answer

The answer of If a bacteria population starts with 50...

Question 11

Find the solution of the equation correct to four decimal places.

Accepted Answer

The answer of Find the solution of the equation correct...

Question 12

Differentiate the function.

Accepted Answer

The answer of Differentiate the function. 
...

Question 13

Solve each equation for x.

Accepted Answer

The answer of Solve each equation for x. ...

Question 14

Use Newton's method to find the roots of the equation correct to five decimal places. x ln x - 1 = 0

Accepted Answer

Newton's method is an iterative numerical technique. Starting with an initial guess, we apply the formula x_(n+1) = x_n - f(x_n)/f'(x_n). For the equation x ln x - 1 = 0, the derivative is ln x + 1. Starting with an initial guess close to the expected root, such as x_0 = 1.5, and iterating, we find the root converges to approximately 1.76322.

Question 15

Find the solution of the equation correct to four decimal places.

Accepted Answer

The answer of Find the solution of the equation correct...

Question 16

Find an equation of the tangent line to the curve  at the point  .

Accepted Answer

To find the equation of the tangent line, we need to find the derivative of the function at the given point.

Taking the derivative of the function 11eaa3e5_5566_f33e_9f8f_5b141d6989d7_TB5971_11 with respect to x gives:

dy/dx = 3x^2 + 5x - 12

Evaluate this at the point x = 2:

dy/dx (at x=2) = 3(2)^2 + 5(2) - 12 = 10

This is the slope of the tangent line at the point (2,14).

Now we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where m is the slope and (x1,y1) is the point on the line.

Using the point (2,14) and the slope 10, we get:

y - 14 = 10(x - 2)

Simplifying this, we get:

y = 10x - 6

So the equation of the tangent line is 11eaa3e5_5567_4162_9f8f_e186008bc3e7_TB5971_11.

Question 17

Find the limit.

Accepted Answer

The answer of Find the limit. ...

Question 18

Find the exponential function  whose graph is given.

Accepted Answer

The answer of Find the exponential function  whose graph...

Question 19

Use the laws of logarithms to expand the expression. ln

Accepted Answer

ln 11eaa3e5_5567_ddab_9f8f_4350401edd3e_TB5971_11 can be written as ln (11) + ln (eaa3e5_5567_ddab_9f8f_4350401edd3e_TB5971_11).

Using the laws of logarithms, we can further simplify ln (eaa3e5_5567_ddab_9f8f_4350401edd3e_TB5971_11) as ln (eaa3e5_5568_2bd1_9f8f_5d2714902a65_TB5971_11) - ln ((x + 5)(x - 6))

Therefore, ln 11eaa3e5_5567_ddab_9f8f_4350401edd3e_TB5971_11 = ln (11) + ln (eaa3e5_5568_2bd1_9f8f_5d2714902a65_TB5971_11) - ln ((x + 5)(x - 6))

The only option that matches this is D.

Question 20

Find the inverse function.

Accepted Answer

The answer of Find the inverse function. ...

Quiz 6: Inverse Functions

Find the limit.

Find the limit.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Find the volume of the solid obtained by rotating about the y axis the region bounded by the curves.

Use the laws of logarithms to write the expression as the logarithm of a single quantity. 4 ln 3 - ln (x + 3)

Differentiate the function.

Solve the equation for x.

Differentiate the function.

Find the values of for which satisfies the equation

If a bacteria population starts with 50 bacteria and doubles every three hours, then the number of bacteria after t hours is . When will the population reach ?

Find the solution of the equation correct to four decimal places.

Differentiate the function.

Solve each equation for x.

Use Newton's method to find the roots of the equation correct to five decimal places. x ln x - 1 = 0

Find the solution of the equation correct to four decimal places.

Find an equation of the tangent line to the curve at the point .

Find the limit.

Find the exponential function whose graph is given.

Use the laws of logarithms to expand the expression. ln

Find the inverse function.