Answers
(i) The graph of f is symmetric about the y-axis - this is another way of saying f is an even function, which means
[tex]f(x) = f(-x) \iff \dfrac{ax+b}{x^2-c} = \dfrac{-ax+b}{x^2-c}[/tex]
The denominator remains unchanged, so
ax + b = -ax + b
Solve for a :
ax = -ax
2ax = 0
a = 0
(The alternative is that x = 0, but if that were the case then there would be infinitely many choices for a.)
So
[tex]f(x) = \dfrac{b}{x^2-c}[/tex]
(ii) Since
[tex]\displaystyle \lim_{x\to2^+}f(x) = +\infty[/tex]
it follows that x - 2 must be a factor of the denominator. If that's true, then
x ² - c = (x - 2) (x - r )
for some other root r of x ² - c. Expanding the right side gives
x ² - c = x ² - (2 + r )x + 2r
so that
-(2 + r ) = 0 ===> r = -2
2r = -c ===> -c = -4 ===> c = 4
and so
[tex]f(x) = \dfrac{b}{x^2-4}[/tex]
(iii) Finally, we're given that f ' (1) = -2. Differentiating f gives
[tex]f'(x) = -\dfrac{2bx}{(x^2-4)^2}[/tex]
so that
[tex]f'(1) = -\dfrac{2b}{(1^2-4)^2} = -\dfrac{2b}{9}[/tex]
Then
-2b/9 = -2 ===> b = 9
and we find that
[tex]f(x) = \dfrac{9}{x^2-4}[/tex]
Related Questions
PLEASE help!! BRAINLIEST to correct answer!!
Answers
The general equation for a circle is:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Where the center is (h,k) and the radius is r.
The equation we are given is:
[tex](x-2)^2+(y+3)^2=80[/tex]
From this, the center is (2,-3), and the radius is [tex]\sqrt{80}[/tex], or [tex]4\sqrt{5}[/tex].
Let me know if you have any questions, thanks!
Find the mean rounded to the nearest tenth median mode and range 59,50,54,50,50
Answers
Answer:
Mean = 52.6
Median = 50
Mode = 50
Range = 9
Step-by-step explanation:
How to find the mean
The mean of a set of numbers is the sum divided by the number of terms.You add 59,50,54,50,50 then divide it by 5 because there are 5 numbersHow to find the median
Arrange the data in an ascending order and the median is the middle value. If the number of values is an even number, the median will be the average of the two middle numbers.How to find the mode
The mode is the element that occurs most in the data set. In this case, 50 occurs 3 times.How to find the range
Subtract the minimum data value from the maximum data value to find the data range. In this case, the data range is 59 − 50 = 9Hope this helps
The number of times the people in a group surveyed on the street have gone to the cinema this month are:
5, 5, 0, 8, 4, 5, 6, 4, 3, 5, 3, 5, 3, 7, 4, 3, 6, 7, 6, 2, 1, 4, 2, 0, 2
Fill in a frequency table and calculate the mean of the distribution.
Answers
Ok so we draw a frequency table by like say if we have 3 2’s then we add a column of like 0-5 and say like 3 frequencies,
I know it’s confusing at first but just keep trying
And to find average we add all the numbers and we divide by how many numbers there are.
Hope it helps!
If it doesn’t I will explain again.
Can I get brainliest answer?
PLEAZE HELP SOON FASTES ANSWER GETS BRAINLIEST
Answers
Answer:
36.8 divided by 16=2.3
2.60 x 2.3= 5.98
I think this is right..
Step-by-step explanation:
If there is 16 ounces is 1 pound then divide 36.8 by 16
then the answer should be multiplied by the amount the peach cost per pound.
For BRAINILY please help I need to pass this class
Answers
Answer:
70
Step-by-step explanation:because they are all the same angles
5r-8=9r-12 answer this plsss
Answers
Answer:
R=1
Step-by-step explanation:
5
−
8
+
8
=
9
−
1
2
+
8
ask you teacher how to solve linear equation he or she could help with that
What is the end behavior of the graph of the polynomial function y = 10x9 – 4x?
o As X-0, y = 0 and as X+0,7-00,
o As X→-0, y = 0 and as X>0, y →--.
As X →-00, y →- and as X>0, y →00
X
As X→-00, y →-00 and as X0, y →-00
y
Answers
Answer:
[tex]x \rightarrow + \infty \: \Rightarrow \: y \rightarrow + \infty \\ x \rightarrow - \infty \: \Rightarrow \: y \rightarrow - \infty[/tex]
x → ∞ y → ∞ and as x →- ∞ y → -∞ is the end behavior of the graph of the polynomial function y = 10x⁹ – 4x
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The meaning of end behavior is how the function ‘y’ (in this case) moves when x tends towards both ∞ and - ∞.
As x → ∞ y → ∞.
This is apparent from the upward movement of the graph of the function y=10x⁹ –4x
As x →- ∞ y → -∞
This inference is also drawn from the above graph i.e. as x values become more negative the graph of y also travels towards infinity i.e. southward on the LHS of the above graph.
Even though the graph of the function remains flat in between x = -4 to x = 4.
After these points, the graph moves towards infinity exponentially.
Hence, x → ∞ y → ∞ and as x →- ∞ y → -∞ is the end behavior of the graph of the polynomial function y = 10x⁹ – 4x
To learn more on Graph click:
https://brainly.com/question/17267403
#SPJ7
please help me solve this problem and figure out how to solve it
Answers
directly find the square root of 36
Answers
Answer:
The square root of 36 is 6.
Answer:
square root of 36 is 6 your welcome
Find the degree of the monomial.
7a765
Answers
Answer: 1^ degree
Step-by-step explanation: because A is raised to 1 and being the only literal part the degree is one
What is the valeu of x?
1/2x -2 =4
Answers
Answer:
12
Step-by-step explanation:
1/2x = 6
x = 12
Answer:
x=12
Step-by-step explanation:
half of 12 is 6. 6-2=4
Can someone send me proxys that aren't blocked by iboss
and how do I removed chrome extensions that were added by a chrome administrator
I'm tryin to play fort night on geforce now browser but its blocked
Answers
Answer:
unblock miniproxy.com
Step-by-step explanation:
Can some please help me work this problem. TIA
Answers
Answer:
the question is what problem
Step-by-step explanation:
I NEED HELP IM STUCK ITS CONFUSING,
A board is 88cm in length and must be cut so that one piece is 24cm longer than the other piece. Find the length of each piece. Round your answers to the nearest centimeter, if necessary
Answers
Answer:
64
Step-by-step explanation:
88-24=64 which is the right answer
I need help fast please!
Answers
Answer:
y=x+9
Step-by-step explanation:
x to the second power plus 24 to the second power equals 30 to the second power .
Answers
Answer:
x = 18 or -18
Step-by-step explanation:
x^2 + 24^2 = 30^2
simplify
x^2 + 576 = 900
subtract 576 from both sides
x^2 = 324
Take square root of 324
+/- √324
x = 18 x = -18
18 or -18 is the answer
what is 7019 - 1135 plezzzzzz
Answers
Answer:
The answer to 7019 - 1135 is 5884
is 16/6 a rational numer
Answers
Please Help!! I really need it!
Answers
Answer:
1. 2:1
2. 1:2
3. 2:1:3
4. 11:4
5. 2:1
Answer:
Step-by-step explanation:
1) Apples : Mangoes = 20 : 10 = 2 :1
2) Ponkans : Guavas = 15 : 30 = 1 : 2
3) Apples : Mangoes : Guavas = 20 : 10 : 30: = 2 : 1 : 3
4) All fruits = apples + mangoes + ponkans + guavas = 20 + 10 + 15 + 30 = 75
All fruits : Apples = 75 : 20 = 15 : 4
4) apples + guavas = 20 + 30 = 40
Mangoes + Ponkans = 10 + 15 = 25
apples + guavas : mangoes + Ponkans = 40 : 25 = 8 : 5
Hey!!:D
this is pretty urgent (only have to the end of the period to finish) hurry pleaseeee lol
Answers
Answer:
A,C, and D.
Step-by-step explanation: I promise
2 and 4, Also could do brainliest i'm trying to level up.
Work out the area of this trapezium.
15cm
6cm
9cm
3cm
Answers
Answer:
33 cm
Step-by-step explanation:
my bad if im wrong havent done this in a while
Answer:
Step-by-step explanation:
area of trapeziumm=1/2(15+3)×6=9×6=54 cm²
Find the new amount after increasing $30 by 15%. I need help it’s due soon! Thanks!
Answers
Answer:
$34.5
Step-by-step explanation:
15% of $30 is 4.5
So 30+4.5=34.5
15 / 100 × 30 = 4.5
30 + 4.5 = 34.5
The answer is 34.5
How to put y=14x-62 in standard form, pls explain.
Answers
Answer:
14x - y = 62
Step-by-step explanation:
Hi there!
We are given the equation y=14x-62 and we want to put it into standard form.
Standard form is ax + by = c, where a, b, and c are free integer coefficients, but a and b cannot be equal to zero, and a cannot be negative
The equation is currently in slope-intercept form, which is y=mx+b, where m is the slope and b is the y intercept
Since standard form has the term with x in it on the left side, we can move 14x to the left side (subtract 14x from both sides)
y = 14x-62
-14x -14x
_________________
-14x + y = -62
Remember that a (coefficient of x) cannot be negative. So multiply everything by -1 to change the sign of all the terms
-1(-14x + y) = -1(-62)
14x - y = 62
Hope this helps!
Sandeep put some business cards into a basket. Then, he drew 2 business cards out of the
basket.
Is this sample of the business cards in the basket likely to be biased?
Yes or No
Answers
Jack had to get three more pails of water than Jill did. The product of the number of
pails they had to get was 40. How many pails of water did Jill have to get?
Answers
Answer:
Impossible?
Step-by-step explanation:
Answer:
5 pails
because jack has 5 pail and he must get 3 more pail to times 5 and get 40
A 20 foot ladder is set against the side of a house so that it reaches up 16 feet. If Christian grabs the ladder at its base and pulls it 2 feet farther from the house, how far up the side of the house will the ladder reach now? (The answer is not 14 ft.) Round to the nearest tenth of a foot.
Answers
Answer:
14.2
Step-by-step explanation:
The ladder forms a right angles triangle,
We have,
h=20
p=16
b=x
Using pythogorean theorem,
b=√(20^2-16^2)
√144
b=12
Now,
Since, she pulls it 2 cm,
b=x+2
b=14
p=?
Using pythogorean theorem,
p=√(20^2-14^2)
√204
p=14.2828568571
Rounding to the nearest tenth,
p=14.2
The side of the ladder will reach up is 14.3 feet.
What is trigonometric Ratios?"Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. It is with respect to any of its acute angles are known as the trigonometric ratios of that particular angle".
For the given situation,
Hypotenuse, h = 20 feet
Perpendicular, p = 16 feet
Base = b
By Pythagoras theorem,
[tex]h^{2} = p^{2} + b^{2}[/tex]
⇒[tex]20^{2} = 16^{2}+b^{2}[/tex]
⇒[tex]b^{2} = 400-256[/tex]
⇒[tex]b^{2}=144[/tex]
⇒[tex]b=\sqrt{144}[/tex]
⇒[tex]b=12[/tex]
The ladder at the house is pulled 2 feet farther.
So, base = [tex]12+2[/tex]
⇒[tex]b=14[/tex]
Now, the side of the ladder will reach up
[tex]p^{2} = h^{2}-b^{2} \\[/tex]
⇒[tex]p^{2}=20^{2}-14^{2} \\[/tex]
⇒[tex]p^{2}=400-196[/tex]
⇒[tex]p^{2}=204[/tex]
⇒[tex]p=\sqrt{204}[/tex]
⇒[tex]p=14.28[/tex]
⇒[tex]p=14.3[/tex] (rounded off to nearest tenth)
Hence we can conclude that the side of the ladder will reach up is 14.3 feet.
Learn more about trigonometric ratios here
brainly.com/question/9085166
#SPJ2
find the value of x.
Answers
Answer:
Step-by-step explanation:
RQ=QS=SR
5x =3x + 10 =6x-5
5x =3x +10
5x-3x =10
2x =10
X=5
A total points scored in a basketball game was 21. How many were 2 points, and how many were 3 point shots. using a linear equation to figure
Answers
Answer:
im not sure
Step-by-step explanation:
is this sss or sas or asa or aas
Answers
Answer:um
Step-by-step explanation:
please help i do not get it
Answers
so, 29-5x would be the equation.
answer: 29-40 = -11. :)
explanation: multiply 5 times 8, because 9am to 4pm is 8 hours. this is 40. take your starting temp, 29 minus 40.
hope this helps