Answers
(1)
(a) Use the fact that [tex]\sqrt{x^2} = |x|[/tex] for all [tex]x[/tex]. Since [tex]x\to+\infty[/tex], we have [tex]x>0[/tex] and [tex]|x| = x[/tex].
[tex]\displaystyle \lim_{x\to\infty} \frac{\sqrt{9x^2 - 2}}{x + 4} = \lim_{x\to\infty} \frac{\sqrt{x^2} \sqrt{9 - \frac2{x^2}}}{x + 4} \\\\= \lim_{x\to\infty} \frac{ x \sqrt{9 - \frac2{x^2}}}{x+4} \\\\= \lim_{x\to\infty} \frac{\sqrt{9 - \frac2{x^2}}}{1 + \frac4x} \\\\= \frac{\sqrt9}1 = \boxed{3}[/tex]
(b) This time [tex]x\to-\infty[/tex], so [tex]x < 0[/tex] and [tex]|x| = -x[/tex].
[tex]\displaystyle \lim_{x\to\infty} \frac{\sqrt{9x^2 - 2}}{x + 4} = \lim_{x\to\infty} \frac{\sqrt{x^2} \sqrt{9 - \frac2{x^2}}}{x + 4} \\\\= \lim_{x\to\infty} \frac{ \boxed{-x} \sqrt{9 - \frac2{x^2}}}{x+4} \\\\= (-1) \times \lim_{x\to\infty} \frac{\sqrt{9 - \frac2{x^2}}}{1 + \frac4x} \\\\= -\frac{\sqrt9}1 = \boxed{-3}[/tex]
(c) We immediately have
[tex]\displaystyle \lim_{x\to\infty} (x - \sqrt x) = \boxed{\infty}[/tex]
since [tex]x > \sqrt x[/tex] for all [tex]x > 1[/tex].
(d) Introduce a difference of squares by factoring in the limand's conjugate. The rest mirrors what we did in (a)/(b).
[tex]\displaystyle \lim_{x\to\infty} \left(\sqrt{x^2 + 12x} - x\right) = \lim_{x\to\infty} \frac{\left(\sqrt{x^2+12x} - x\right) \left(\sqrt{x^2+12x} + x\right)}{\sqrt{x^2 + 12x} + x} \\\\ = \lim_{x\to\infty} \frac{\left(\sqrt{x^2+12x}\right)^2 - x^2}{\sqrt{x^2+12x}+x} \\\\ = \lim_{x\to\infty} \frac{12x}{\sqrt{x^2+12x}+x} \\\\ = \lim_{x\to\infty} \frac{12}{\sqrt{1 + \frac{12}x} + 1} = \frac{12}{\sqrt1 + 1} = \boxed{6}[/tex]
(e) Divide through by the highest-degree exponential term.
[tex]\displaystyle \lim_{x\to\infty} \frac{12e^{2x} - 3e^{3x}}{2e^{2x} + 4e^{3x}} = \lim_{x\to\infty} \frac{12e^{-x} - 3}{2e^{-x} + 4} = \frac{0 - 3}{0 + 4} = \boxed{-\frac34}[/tex]
(2) By definition of the derivative, we have
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{f(x+h) - f(x)}h[/tex]
For [tex]f(x) = \sqrt{x^2+1}[/tex], the limit becomes
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\sqrt{(x+h)^2+1} - \sqrt{x^2+1}}h[/tex]
Factor in the conjugate.
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\left(\sqrt{(x+h)^2+1} - \sqrt{x^2+1}\right) \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\left(\sqrt{(x+h)^2+1}\right)^2 - \left(\sqrt{x^2+1}\right)^2}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\bigg((x+h)^2+1\bigg) - (x^2+1)}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{2xh + h^2}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{2x + h}{\sqrt{(x+h)^2+1} + \sqrt{x^2+1}}[/tex]
[tex]\implies \boxed{f'(x) = \displaystyle \lim_{h\to0} \frac{x}{\sqrt{x^2+1}}}[/tex]
(3) The tangent line to
[tex]y = \frac1{x^2+1}[/tex]
at the point (2, 1/5) has slope equal to the derivative [tex]\frac{dy}{dx}[/tex] when [tex]x = 2[/tex]. Compute the derivative; since [tex]y = \frac1{f(x)^2}[/tex] where [tex]f(x)[/tex] is the function from the previous problem, using the chain rule gives
[tex]y = \dfrac1{f(x)^2} \implies \dfrac{dy}{dx} = -\dfrac{2f'(x)}{f(x)^3} = -\dfrac{2 \times \frac{x}{\sqrt{x^2+1}}}{\left(\sqrt{x^2+1}\right)^3} \\\\ \implies \dfrac{dy}{dx} = -\dfrac{2x}{(x^2+1)^2}[/tex]
The tangent line at (2, 1/5) then has slope
[tex]\dfrac{dy}{dx}\bigg|_{x=2} = -\dfrac{2\times2}{(2^2+1)^2} = -\dfrac4{25}[/tex]
Using the point-slope formula, the equation of the tangent line is
[tex]y - \dfrac15 = -\dfrac4{25} (x - 2) \implies \boxed{y = -\dfrac{4x - 13}{25}}[/tex]
Related Questions
Order the numbers from least to greatest 0.607, 607, 6.07, 60.7
Answers
Answer:
0.607, 6.07, 60.7, 607
Step-by-step explanation:
The value of an asset, currently priced at $100 000, is expected to increase by 20% a
year.
(a) Find its value in 10 years’ time.
(b) After how many years will it be worth $1 million?
Answers
Answer:
A) it will become 300,000 in the next 10 years.
5 by 8 of the children in the field are girls. There are 45 boys. How many girls are there?
Answers
Answer:
5+8 = 13
5:8
Girls
8÷5 × 45
== 28
hope it helped
Answer:
75 girls
Step-by-step explanation:
[tex]\dfrac{5}{8} \ are\ girls[/tex]
[tex]\dfrac{8-5}{8}=\dfrac{3}{8} \ are \ boys[/tex]
Let the total children = x
[tex]\dfrac{3}{8} \ of \ x = 45\\\\\dfrac{3}{8}*x = 45\\\\x =45*\dfrac{8}{3}= 15 * 8\\[/tex]
x = 120 children
Number of girls = 120 - 45 = 75
-1+3[(10-12)/(-16) ]
Answers
-1 + 3(-2/-16)
-1 + 3(1/8)
-1 + 3/8
-8/8 + 3/8
-5/8
How to find t= 2,500•(1.1)^100 round to the nearest whole number
Answers
Answer:
2600
Step-by-step explanation:
you added 2;500 to the 100
Find the area of the figure. (Sides meet at right angles.)
15 yd
9 yd
9yd
5 yd
5 yd
5 yd
4 yd
4 yd
Answers
Answer:
115 square yards
Step-by-step explanation:
yeah-ya....... right?
The area of the figure is 115 yd².
What is a rectangle?A rectangle is a two-dimensional shape where the length and width are different.
The area of a rectangle is given as:
Area = Length x width
We have,
A figure:
This can be assumed to have two rectangles and one square.
Two rectangles are equal with dimensions:
Length = 9 yd
Width = 5 yd
The square has side = 5 yd.
Now,
Area of the figure.
= 2 x area of rectangle + area of the square
= 2 x (length x width) + side²
= 2 x (9 x 5) + 5²
= 2 x 45 + 25
= 90 + 25
= 115 yd²
Thus,
The area is 115 yd².
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(-11[(3-25 divided by |-2|+5]
Answers
Answer:
(-11[2-25/ l-2l + 5]
(11[2-25/2+5]
(11[23/7]
11(3.28571428571)
36.1428571429
Help help pelesse please
Answers
Answer:
450
Step-by-step explanation:
9×50=450
have a good day
70 percent of 38.20 is what
Answers
Answer:
26.74
Step-by-step explanation:
38.20 x 0.70
= 26.74
Are 6/9 and2/3 proportion
Answers
Answer:
yes, they are proportional
Step-by-step explanation:
6/9 is just three time as big as 2/3 but they are basically the same fraction
Answer:
Step-by-step explanation:
yes, it is proportional
6/9 = 2/3
18 = 18
Write the equation of a line that goes through the point (2, 4) that is parallel to the
line y = 3x - 6.
Answers
Answer:
y = 3x - 2
Step-by-step explanation:
Parallel lines have the same gradient in this case 3 is the gradient.
We will use the point (2,4) to form our own equation.
4 will be y, 3 will be m, 2 will be x
All we need is to find the C in the equation y=mx + c
4 = 3(2) + c
4 = 6 + c
C = -2
Therefore;
y = 3x - 2
answer please. bots will obvious be deleted
Answers
Answer:
A right triangle needs to have all the sides of the same length. Since this obviously does not, it is not a right triangle. Infact, this is a scalene triangle because all of its sides are unequal.
You and your mom enter a drawing with 3 different prizes. A total of 10
people entered the drawing, and prizes are awarded randomly.
There are 720 ways to award the prizes.
What is the probability that you win
prize?
first prize and your mom wins second
Answers
Answer:
Step-by-step explanation:
Number of people who entered the drawing = 10
Number of prizes = 3
Number of ways to award the prize = 720 ways
The total number of ways that 3 prizes can be awarded to the 10 people can be determined as:
10*9*8 = 720
The total number of ways in which I win the first prize and my mom wins the second prize would be 1*1*8 = 8
Thus the probability that I win the first prize and my mom wins the second prize would be 8/720
Which of the following is true?
A. |−8| < 6
B. |−6| < |−8|
C. |8| < |−6|
D. |−6| < −8
Answers
Answer:
B. |-6| < |-8|
Step-by-step explanation:
I used a calculator, pls let me know if im incorrect
Prove the polynomial identity.
(4x^2−2)^2−(4x^2)^2=4(1−2x)(1+2x)
Drag and drop the lines of the proof in the correct order to complete the proof.
Answers
Answer:
(4x^2 - 2)^2 - (4x^2)^2 = 4(1 - 2x)(1 + 2x)
16x^4 - 16x^2 + 4 - 16x^4 = 4(1 - 2x)(1 + 2x)
4 - 16x^2 = 4(1 - 2x)(1 + 2x)
4(1 - 4x^2) = 4(1 - 2x)(1 + 2x)
4(1 - 2x^2)(1 + 2x) = 4(1 - 2x)(1 + 2x)
1. Jean bought the new Iphone for $1080 (Tax included) on an installment plan. The installment
agreement included a 10% down payment and 24 monthly payments of $50 each.
a. How much is the down payment?
b.what’s the total amount of the monthly payments?
c.how much did Jean pay for the phone on the installment plan?
d.what is the finance charge (interest)?
Answers
Answer:
jndfjhasbfjashdfbasjdfbjashbdcjashdbjcbjhasdcbjasdcbjashdcbjashcdbjadhcbasdhcb
Step-by-step explanation:
Danielle and 8.25% commission on everything she sells at the electronics store where she works. She also earns a base salary of $650 per week. How much did she earn last week if she sold $4,900 in electronics merchandise? Round your intermediate calculations and answer to the nearest cent.
Answers
Step-by-step explanation:
She earns $1046 that week
PLEASE HELP ITS MATH THANK YOUUUU
Answers
Explanation:
24 * 5 = 120
show the steps and name the properties you used to solve the equation below
log3=log2-log(3x+7)
Ps. if you just try to get points I will have your answer removed which takes away the points
Answers
Answer:
0.477121=−log(3x+7)+0.30103
Step 1: Flip the equation.
−log(3x+7)+0.30103=0.477121
Step 2: Add -0.30103 to both sides.
−log(3x+7)+0.30103+−0.30103=0.477121+−0.30103
−log(3x+7)=0.176091
Step 3: Divide both sides by -1.
log(3x+7)=−0.176091
Step 4: Solve Logarithm.
log(3x+7)=−0.176091
10log(3x+7)=10−0.176091
3x+7=0.666667
3x+7−7=0.666667−7(Subtract 7 from both sides)
3x=−6.333333
3x
3
=
−6.333333
3
(Divide both sides by 3)
x=−2.111111
Step-by-step explanation:hope this helps
4. 3/16. × 7 12/32.
Answers
Answer:
1 49/128
Step-by-step explanation:
3/16. × 7 12/32
3/16 * 236/32
177/128
1 49/128
13. There are 6 times as many dogs as cats. If the total number of dogs
and cats is 21, how many dogs are there?
Answers
Answer: The answer you would be looking for is 126.
Step-by-step explanation:
If theres 21 you are multiplying by 6 and that is 126
b) In 3x?, the 3 is called the____
please help
Answers
Answer:
3 is the coefficient
Step-by-step explanation:
The number in front of the variable is called the coefficient
3x
3 is the coefficient
x is the variable
he product of two whole numbers is 864 and their sum is 60 . What are the two numbers?
Answers
Answer:
Step-by-step explanation:
xy = 864
x + y = 60 Solve for y in the second equation
y = 60 - x
x * (60 - x) = 864 Remove the brackets.
60x - x^2 = 864 Put in order of powers and subtract 864
-x^2 + 60x - 864 =0 Multiply through by -1
x^2 - 60x + 864 = 0
This factors
(x - 36)(x - 24) = 0
Either
x - 36 = 0 in which case
x = 36
or
x - 24 = 0
x = 24
The two numbers are x = 36 and y = 24
of x = 24 and y = 36
You are rolling artwork to ship so it fits perfectly in a cylindrical shipping tube. The volume of the shipping tube is
169.56 cubic inches. If the radius of the shipping tube is 1.5 inches, what is the maximum height of the artwork you
can ship and what formula can you use to calculate this height?
O 6 inches; height =
volume
3.14 x (radius)?
O 18 inches; height:
volume
3.14 x radius
O 24 inches, height
volume
3.14 x (radius)2
36 inches; height
volume
3.14 x radius
O 54 inches; height
volume
3.14
Answers
The volume of a cylinder is found by using the formula V = pi x r^2 x h where r is the radius and h is the height.
Using the given volume and radius calculate the height:
169.56 = pi x 1.5^2 x h
169.56 = pi x 2.25 x h
169.56 = 7.065 x h
H = 169.56/7.065
H = 24 inches
The answer is the height is 24 inches
Answer:
b
Step-by-step explanation:
HELPPPPP PICTURE BELOW! MATHHHH
Answers
Answer:
8
Step-by-step explanation:
if it's saying that 32/52 is equal to x/13, then we know that 32/52 is divisible by the mystery fraction.
since both fractions have a denominator, divide them.
52÷13=4
then, divide the answer, 4, by 32.
32÷4=8.
therefore, your answer is 8.
hope this helps!
Answer:
8
Step-by-step explanation:
A trick with equal fractions will be that the cross multiply will create equal answers
13(32) = 52(?)
To find the missing number, just divide the known product by the missing number's multiplicand.
? = 13(32)/52 = 8
To verify
13(32) = 416
52(8) = 416
Yup!
What is the difference of 9.2 and 0.03
Answers
Answer:
9.17
Step-by-step explanation:
9.20-0.03=9.17
Answer:
9.17
Step-by-step explanation:
9.20
-0.03
_____
9. 17
What kinds of education did Martin Luther king JR have in high school and college?
Answer ASAP
Answers
Step-by-step explanation:
graduated from Booker T. Washington High School and was admitted to Morehouse College at age 15. Graduates from Morehouse college and enters Crozer Theological Seminary.
How to convert the mixed number to a fraction
6 and 1 over 7?
Answers
Answer:
43/7
Step-by-step explanation:
6 1/7
First, you need to multiply 6 by 7, then add 1
43/7
Answer:
[tex]\frac{43}{7}[/tex]
Step-by-step explanation:
6 x 7 + 1
there is 6 whole fractions so it would be [tex]\frac{7}{7}[/tex] 6 times then add the [tex]\frac{1}{7}[/tex]
which would be [tex]\frac{43}{7}[/tex]
hopefully that made sense!
What is the Lcm of 22 and 33 using prime factorization
Answers
My answer would be 66
The LCM of 22 and 33, using prime factorization, is 66.
What is Number system?A number system is defined as a system of writing to express numbers.
We have to find LCM of 22 and 33 using prime factorization
Factor each term into prime factors
22 = 2 x 11
33 = 3 x 11
Identify all of the unique prime factors of these two numbers.
Finally, we can find the LCM by multiplying each of these prime factors raised to their highest power that appears in either of the two factorizations.
LCM(22, 33) = 2x 3 x 11
LCM(22, 33) = 66
Therefore, the LCM of 22 and 33, using prime factorization, is 66.
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Rewrite 4^-9 using a positive exponent.
Answers
Step-by-step explanation:
using a positive exponent
4^-9 = 1/ 4^9
__________
[tex] \: [/tex]
[tex] {a}^{ - m} = \frac{1}{ {a}^{m} } [/tex]
Step by step :[tex] {4}^{ - 9} = ...[/tex]
[tex] \longmapsto \frac{1}{ {4}^{9} } [/tex]
[tex] \longmapsto \frac{1}{4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4} [/tex]
[tex] \longmapsto \blue{ \frac{1}{262.144}} [/tex]
