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
Hey, how are you?
I am trying to do 100 points for you guys just saying how you are daily!
Answers
Answer:
My day has been pretty okay!!
Step-by-step explanation:
Thank you for asking!!
Answer:
My day is going alright. Though, I am a bit stressed out since I am currently studying for an upcoming test.
Which one is the answer!
Answers
PLEASE HELP PLEAAE HURRY ANS NO LINKS PLEASE I REALLY NEED THE ANSWER
Answers
Answer:
3(x+15) is the answer
Step-by-step explanation:
The equation x+15 repeats 3 times so you would have to create an equation to multiply x+15 3 times.
3x+15 is wrong because its only multiplying x by 3 and ignoring 15
x+45 is wrong because it is only multiplying 15 by 3 and ignoring x
x+15 is wrong because it is not multiplied by 3
3(x+15) is correct because it is the only answer that multiples 3 with both terms
Hope this helps
Answer:
Lets say that x is the number 5,
Step-by-step explanation:
5 divided by 15 = 0.33333333333
5 divided by 0.33333333333 = 15
5 + 15 + 5 + 15 + 5 + 15=60
(I know im all over the place with this, but this is just what i do)
60 - 15 = 45
15 + 45 = 60
I know this is random but maybe you may figure it out or get a random hint??, 3(x+15) is 3x + 45
PRETTY SURE ITS 3(x+15) Please give me brainliest i sat here for like 24 mins trying to figure this out :(
a. After Pam bought the party supplies, what is the balance of her bank account compared with her
starting balance? Show your work.
Answers
Answer:
What is the original Balance, and what is the price of the party supplies?
Write in words. Use commas and hyphens.
38,492
Answers
Thirty-eight thousand, four hundred and ninety-two
Simply write out the number as you would say it out loud!
DONT CLICK ON THIS ok lol
Answers
She can only buy 2 whole packs.
Acording to the graph, 2 whole pakcs will cost $22, and 3 whole packs will cost $33. She can't afford 3 whole packs as we can see as she only has $30.50.
Let me know if you have any questions, thanks!
Answer:
u told me not to so i did it (its might be B)
Step-by-step explanation:
so i do the did the and the make than the marh and rhe barf and the ract and the rekt
What is the value of g(2)
Answers
Answer:
Its value is 9.8 m/s2
Answer:
9.8 m/s2
Step-by-step explanation:
The height of a model rocket, H(t), is a function of the time since it was
launched, t.
Time (seconds)
What is the domain of H(t)?
A. t <100
B. T>0
C. 0
D. 0
Answers
The domain of function H(t) is 0 ≤ t ≤ 20.
What is a function?A function has an input and an output.
A function can be one-to-one or onto one.
It simply indicated the relationships between the input and the output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
From the graph,
We see that the value of t is from 0 to 20 seconds.
Now,
The value of t can be written as an inequality:
0 ≤ t ≤ 20
Thus,
The domain of function H(t) is 0 ≤ t ≤ 20.
Learn more about functions here:
https://brainly.com/question/28533782
#SPJ7
A case of 24 pairs of the same kind of sport shoes cost a little more than $800 explain whether $28 per pair with tax included is a good estimate of the price
Answers
Write a equation in slope intercept form that is equivalent to 10x-2y=16
Answers
Answer:
y = 5x - 8
Step-by-step explanation:
10x -16 = 2y
y = 10/2x - 16/2
y = 5x - 8
Multiply 2 digit numbers with area models. 26x74
Answers
Answer:
26 x 74 is 1924
Step-by-step explanation:
Is just lining the numbers up.
One floor of a dollhouse is 1.6 feet tall. The dollhouse contains two levels of living areas and an attic for a total of 2.5 floors. Jaime estimates the height of the dollhouse by computing 2 times 3. How does her estimate compare to the actual height of the dollhouse?
Answer Options:
-Her estimate is greater than the actual height because both factors were rounded up.
-Her estimate is less than the actual height because both factors were rounded up.
-Her estimate is greater than the actual height because both factors were rounded down.
-Her estimate is less than the actual height because both factors were rounded down.
Answers
Answer:
Her estimate is greater than the actual height because both factors were rounded up.
Step-by-step explanation:
Hope this helps :)
Answer:
A. Her estimate is greater than the actual height because both factors were rounded up.
Step-by-step explanation:
Hope this helped! :)
Solve for w.
–1.01 = 1.4(w + 2) − 2.41
Answers
w=0
Step-by-step explanation:
-1.01=1.4(w+2)-2.41
-1.01=1.4w+2.8-2.41
-1.01=1.4w+0.39
-1.01-0.39=1.4w
+1.4=1.4w
1.4/1.4=w
0=w
So,the answer is w=O
I hope it is very helpful
1. What is 85% of 620?
1147
527
729
465
Answers
Answer:
465 I think is the answer
% / 100 = Part / Whole
replace the given values:
85 / 100 = Part / 620
Cross multiply:
= 85 x 620 = 100 x Part
= 52700 = 100 x Part
Now, divide by 100 and get the answer:
Part = 52700 / 100 = 527
35 more than twice the number is 65. Find the number.
Answers
Answer: 15
Step-by-step explanation:
2x + 35 = 65
2x = 30
x = 15
What is the value of n in the proportion below?
n/28 = 4/7
•1
•12
•14
•16
Answers
Answer:
n = 16
Step-by-step explanation:
Given the proportion, n/28 = 4/7:
Cross multiply the two fractions:
7n = 28 × 4
7n = 112
Divide both sides by 7 to solve for n:
7n/7 = 112/7
n = 16
Therefore, the value of n = 16.
Please mark my answers as the Brainliest, if you find this helpful :)
Explanation: 7 goes into 28 four times which would set the denominators as equal. But to finish converting the fraction, you would have to multiply 4 by four just like you did the denominator which gives you 16.
If a car travels 200km in 2.5 hours, how far would it travel in 6 hours at the same rate of speed?
Answers
Answer: 500km I believe
Step-by-step explanation:
hope this helps :)
Answer:
I think that the answer is 850
Step-by-step explanation:
find out how many km traveled 0.5 hours, then multiply by 12
-5/2x-19/10
Plss Answer this in fraction form, and simplified, I can't figure it out!
Answers
Answer:
= -25x-19/10
Step-by-step explanation:
Solve:-4(-6x - 3) =12
Answers
Answer:
Step-by-step explanat ion:
<=>2 4x+12 = 12
<=>24 x = 0
<=>x =0
What number would you get if you multiply the's numbers
1,2,3,4,5,6,7,8,9,10 ?
Answers
what percent of 235 is 148.05
Answers
Answer:
63% is the correct answer hope this helps
A large national park has an area of 91,417 acres. What is its area in hectares
Answers
Answer:
36995.1474
Step-by-step explanation:
multiply the area value by 2.471
What is the difference of 18 - 10 2/8 ?
Answers
Answer:
7 3/4
Step-by-step explanation:
[tex]18 - 10\frac{2}{4}[/tex] [tex]18 - 10\frac{1}{4}[/tex] [tex]18 - \frac{41}{4}[/tex] [tex]\frac{72}{4} - \frac{41}{4}[/tex] [tex]\frac{31}{4}[/tex] [tex]7\frac{3}{4}[/tex]Find the slope of the line through each pair of points
(10, -14), (11,11)
Answers
Hello there! I can help you with joy! :)
If we have 2 points, then we can find the slope using this formula:
m=[tex]\frac{y2-y1}{x2-x1}[/tex]
y2=11
y1=-14
x2=11
x1=10
[tex]\frac{11-(-14)}{11-10}[/tex]
[tex]\frac{11+14}{1}[/tex]
25
Hope this helps you!
#HaveAnAmazingDay
[tex]GraceRosalia[/tex]
21. You place a mat around a 25 in by 21 inch painting as shown. The mat
is twice as wide at the left and right of the painting as it is at the top an
bottom of the painting. The area of the mat is 714 square iinches (note:
the area of the border). How wide is the mat at the left and right of the
painting? At the top and bottom of the painting? (P.264 #66)
Answers
Using a quadratic equation according to the area of the rectangular mat, it is found that:
At the top and bottom of the painting, the width is 0.654 inches.At the left and right of the painting, the width is of 1.308 inches.The area of a rectangle of length l and width w is given by:
[tex]A = lw[/tex]
In this problem:
The length of the figure is of [tex]l = 25 + 4x[/tex].The width of the figure is of [tex]w = 21 + 2x[/tex].Area of 714 square inches, thus:[tex]lw = 714[/tex]
[tex](25 + 4x)(21 + 2x) = 714[/tex]
[tex]8x^2 + 134x + 525 = 714[/tex]
[tex]8x^2 + 134x - 189 = 0[/tex]
Which is a quadratic equation with [tex]a = 8, b = 134, c = -189[/tex].
The, to find the widths of the mat, we have to first find the value of x, solving the quadratic equation.
[tex]\Delta = 134^2 - 4(8)(-189) = 24004 [/tex]
[tex]x_{1} = \frac{-134 + \sqrt{24004}}{2(8)} = 0.654[/tex]
[tex]x_{2} = \frac{-134 - \sqrt{24004}}{2(8)} = -9.03[/tex]
We are interest in the positive value of x, so x = 0.654.
At the top and bottom of the painting, the width is of 0.654 inches.[tex]2(0.654) = 1.308[/tex]
At the left and right of the painting, the width is of 1.308 inches.A similar problem is given at https://brainly.com/question/14099239
Polynomials
Multiply (x - 5)(3x – 2)
Answers
9514 1404 393
Answer:
3x² -17x +10
Step-by-step explanation:
Use the distributive property and collect terms.
(x -5)(3x -2) = x(3x -2) -5(3x -2)
= (3x² -2x) +(-15x +10)
= 3x² -17x +10
2/5x11 in simplest form
Answers
Answer:
25 is already in the simplest form. It can be written as 0.4 in decimal form (rounded to 6 decimal places).
so its in its simplest form
Answer:
4 2/5 or 4.4
Step-by-step explanation:
2/5 x 11
22/5
Make sure to turn it into a mixed number.
show your solution need an answer, please need an answer, please.
Answers
Answer:
As follows,
Step-by-step explanation:
You can solve these either by factoring or using the quadratic formula or other ways,
1. x^2+2x+1=0
Factoring,
(x+1)^2=0
x+1=0
x=-1
2.y^2-y-2=0
Using quadratic formula,
(-b±√(b^2-4ac))/2a
(--1±√(-1^2-4*1*-2))/2*1
(1±√(1+8))/2
(1±√(9))/2
(1±3)/2
y=(1±3)/2
y=(1+3)/2 y=(1-3)/2
y=4/2 y=-2/2
y=2 y=-1
3.y^2+y+1=0
Using quadratic formula,
(-b±√(b^2-4ac))/2a
(-1±√(1^2-4*1*1))/2*1
(-1±√(1-4))/2
(-1±√(3))/2
No real roots
4. x^2+9x+18=0
Using quadratic formula,
(-9±√(9^2-4*1*18))/2*1
(-9±√(81-72))/2
(-9±√(9))/2
(-9±3)/2
x=(-9+3)/2 x= (-9-3)/2
x=-6/2 x=-12/2
x=-3 x=-6
5.x^2+5x+6=0
Using quadratic formula,
(-5±√(b^2-4ac))/2a
(-5±√(5^2-4*1*6))/2*1
(-5±√(25-24)/2
(-5±1)/2
x=(-5+1)/2 x=(-5-1)/2
x=-4/2 x=-6/2
x=-2 x=-3
6. 3n^2-13n-10=0
Using quadratic formula,
(-b±√(b^2-4ac))/2a
(-(-13)±√(-13^2-4*3*(-10)))/2*3
(13±√(169-(-120))/6
(13±√289)/6
The discriminant b^2−4ac>0
So there are 2 real roots.
x=(13±17)/6
x=30/6 or x=-4/6
x=5 or x=-2/3
Evaluate. {1−[−3−(5+4)]}⋅(−6)
Answers
Answer:
-78
Step-by-step explanation:
So we start with the innermost parentheses
(5+4) = 9
{1−[−3−(9)]} x -6
-3 - 9 = -12
{1 - [-12]}
Whenever you subtract a negative, the subtraction and negative cancel each other out, so it turns into addition
1 + 12 = 13
13 x -6 = -78
12 + 4[5% - 4(x + 5)] = 7 - 5x
Answers
Answer:
https://brainly.com/question/24983242 help
Step-by-step explanation:
12 + 4[5% - 4(x + 5)] = 7 - 5x
12 + 4•(5% - 4•(x + 5)) = 7 - 5•x
11x = -55
x = -5
Answer:
[tex]x = - \frac{34}{5} [/tex]
Step-by-step explanation:
12 + 4 (5% - 4(x + 5)) = 7 - 5x
12 + 4 (5/100 - 4x - 20) = 7 - 5x
12 + 4 (-399/20 - 4x) = 7 - 5x
12 - 399/5 - 16x = 7 - 5x
-399/5 - 16x = 7 - 5x
-16x + 5x = 7 + 339/5
-11x = 374/5
x = -34/5