Differentiate both sides with respect to x and solve for the derivative dy/dx :
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[x^2y^2+xy\right] = \dfrac{\mathrm d}{\mathrm dx}[2] \\\\ \dfrac{\mathrm d}{\mathrm dx}\left[x^2\right]y^2 + x^2\dfrac{\mathrm d}{\mathrm dx}\left[y^2\right] + \dfrac{\mathrm d}{\mathrm dx}\left[x\right]y + x\dfrac{\mathrm dy}{\mathrm dx} = 0 \\\\ 2xy^2 + x^2(2y)\dfrac{\mathrm dy}{\mathrm dx} + y + x\dfrac{\mathrm dy}{\mathrm dx} = 0 \\\\ (2x^2y+x)\dfrac{\mathrm dy}{\mathrm dx} = -2xy^2-y \\\\ \dfrac{\mathrm dy}{\mathrm dx} = -\dfrac{2xy^2+y}{2x^2y+x}[/tex]
This gives the slope of the tangent to the curve at the point (x, y).
If the slope of some tangent line is -1, then
[tex]-\dfrac{2xy^2+y}{2x^2y+x} = -1 \\\\ \dfrac{2xy^2+y}{2x^2y+x} = 1 \\\\ 2xy^2+y = 2x^2y+x \\\\ 2xy^2-2x^2y + y - x = 0 \\\\ 2xy(y-x)+y-x = 0 \\\\ (2xy+1)(y-x) = 0[/tex]
Then either
[tex]2xy+1 = 0\text{ or }y-x=0 \\\\ \implies y=-\dfrac1{2x} \text{ or }y=x[/tex]
In the first case, we'd have
[tex]x^2\left(-\dfrac1{2x}\right)^2+x\left(-\dfrac1{2x}\right) = \dfrac14-\dfrac12 = -\dfrac14\neq2[/tex]
so this case is junk.
In the second case,
[tex]x^2\times x^2+x\times x=x^4+x^2=2 \\\\ x^4+x^2-2 = (x^2-1)(x^2+2)=0[/tex]
which means either
[tex]x^2-1 = 0 \text{ or }x^2+2 = 0 \\\\ x^2 = 1 \text{ or }x^2 = - 2[/tex]
The second case here leads to non-real solutions, so we ignore it. The other case leads to [tex]x=\pm1[/tex].
Find the y-coordinates of the points with x = ±1 :
[tex]x=1 \implies y^2+y=2 \implies y=-2 \text{ or }y=1 \\\\ x=-1\implies y^2-y=2\implies y=-1\text{ or }y=2[/tex]
so the points of interest are (1, -2), (1, 1), (-1, -1), and (-1, 2).
What are the coordinates of the image of polygon KLMN after a reflection across the y-axis?
The logarithmic function f(x) = ln(x) is the ___ logarithmic function and has a base ___
Answers:
naturaleThe logarithmic function f(x) = ln(x) is the natural logarithmic function and has base e
=======================================
Explanation:
The ln(x) or Ln(x) is the natural log function. The order of L and n is backwards compared to "natural log", but you can think of it "log that is natural" or something along those lines.
The Ln(x) function is of the form [tex]\log_b(x)[/tex] where the base b is the special constant e = 2.718... (this constant is similar to pi = 3.14...)
Find the length of AB
Answer:
52
Step-by-step explanation:
AB is the hypotenuse of the right triangle
sin34 = opp/hyp = 29/hyp
hyp = 29/sin34 = 51.860457849...
Answer:
[tex]let \: |ab| \: be \: x \\ \\ \frac{ \sin(90) }{x} = \frac{ \sin(34) }{29} \\ x \sin(34) = 29 \sin(90) \\ x = \frac{29 \sin(90) }{ \sin(34) } \\ x = 51.86(2.d.p) \\ |ab| = 51.86[/tex]
The tree diagram represents an
experiment consisting of two trials.
(PICTURE INCLUDED FOR BETTER CONTEXT) urgent pls help
Answer:
answer is in photo.
Green one is main answer and blue one is formula
In a GP the second and fourth terms are 0.04 and I respectively find the common ratio and the first term
Answer:
common ratio=0.5, a1= 0.08
Step-by-step explanation:
r=a3/a2
r=a4/a3
compare both we get:
a3/a2=a4/a3
subtitute a2=0.04 and a4=1
a3/0.04=1/a3
(a3)^2=0.04*1
(a3)^2=0.04
taking square root in both sides
a3=0.02
For r, r=a3/a2
subtitute a3 and a2 above
r=0.02/0.04
r=0.5 common ratio
For a1
r=a2/a1
0.5=0.04/a1
a1=0.04/0.5
a1=0.08
A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.
The first term of the geometric sequence is 0.08.
The common ratio of the geometric sequence is 1/0.02.
What is a geometric sequence?A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.
We have,
Geometric sequence:
The second term = 0.04
The fourth term = 1
In a geometric sequence the nth term is given as:
[tex]a_{n}[/tex] = [tex]a_1 r^{n-1}[/tex]
Second term = 0.04
[tex]a_2[/tex] = 0.04
[tex]a_1 r^1 = 0.04[/tex]
[tex]a_1r[/tex] = 0.04 _____(1)
Fourth term = 1
[tex]a_4 = 1[/tex]
[tex]a_1 r^3[/tex] = 1 _____(2)
From (1) and (2) we get,
0.04/r = 1/r³
r³ = r / 0.04
r² = 1/0.04
r = 1/0.02 ____(3)
Putting (3) in (1) we get
[tex]a_1[/tex] = 0.04 x 0.02 = 0.08
Thus,
The first term of the geometric sequence is 0.08.
The common ratio of the geometric sequence is 1/0.02.
Learn more about geometric sequence here:
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B) Lim[2f(x)-3g(x)]
Please do all if possible:) thank you!
Recall some limit properties…
• limits distribute over sums:
[tex]\displaystyle \lim_{x\to c}(f(x)+g(x)) = \lim_{x\to c}f(x)+\lim_{x\to c}g(x)[/tex]
• limits distribute over products:
[tex]\displaystyle \lim_{x\to c}(f(x)\times g(x)) = \lim_{x\to c}f(x)\times\lim_{x\to c}g(x)[/tex]
• limits distribute over quotients, provided that the denominator doesn't approach 0 :
[tex]\displaystyle\lim_{x\to c}\frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x\to c}f(x)}{\displaystyle\lim_{x\to c}g(x)}[/tex]
• if f(x) is continuous at x = c, then the limit "passes through" a composition:
[tex]\displaystyle \lim_{x\to c}f(g(x)) = f\left(\lim_{x\to c}g(x)\right)[/tex]
# # #
(a) This limit is 2. At x = 2, we have f (2) = 1, but from either side of x = 2, we see f(x) approaching the point (2, 2). So
[tex]\displaystyle \lim_{x\to 2}(f(x)+g(x)) = \lim_{x\to2}f(x)+\lim_{x\to2}g(x) = 2+0 = 2[/tex]
(b) This limit does not exist. We would have
[tex]\displaystyle \lim_{x\to1}(2f(x)-3g(x)) = 2\lim_{x\to1}f(x)-3\lim_{x\to1}g(x)[/tex]
but g(x) approaches 2 from the left of x = 1, and g(x) approaches 1 from the right of x = 1. The one-sided limits don't match, so the two-sided limit doesn't exist.
(c) This limit is 0. It looks like f(x) passes through the origin, while g(x) ≈ 3/2 at x = 0. So
[tex]\displaystyle\lim_{x\to0}f(x)g(x) = \lim_{x\to0}f(x)\times\lim_{x\to0}g(x)=0\times\frac32 = 0[/tex]
(d) This limit does not exist since
[tex]\displaystyle \lim_{x\to-1}g(x)=0[/tex]
(e) This limit is 16. Nothing tricky here, just use the same property as in (c).
[tex]\displaystyle\lim_{x\to2}x^3f(x) = \lim_{x\to2}x^3\times\lim_{x\to2}f(x) = 8\times2=16[/tex]
(f) This limit is 1. f(x) is continuous at x = 1, while g(x) approaches 2 from the left.
[tex]\displaystyle\lim_{x\to1^-}f(g(x)) = f\left(\lim_{x\to1^-}g(x)\right) = f(2) = 1[/tex]
Find the length of the third side. If necessary, round to the nearest tenth.
Answer:
a = 6
Step-by-step explanation:
Since this is a right triangle, we can use the Pythagorean theorem
a^2+b^2 = c^2 where a and b are the legs and c is the hypotenuse
a^2 + 8^2 = 10 ^2
a^2 +64 = 100
a^2 = 100 -64
a^2 = 36
Take the square root of each side
sqrt(a^2) = sqrt(36)
a = 6
(1-\sqrt{x})/1+\sqrt{x}
Answer:
[tex] \frac{1 - 2 \sqrt{x} + x }{1 - x} [/tex]
Step-by-step explanation:
[tex] \frac{1 - \sqrt{x} }{1 + \sqrt{x} } [/tex]
Rationalize the denominator by multiplying by it conjugate.
[tex] \frac{1 - \sqrt{x} }{1 + \sqrt{x} } \times \frac{1 - \sqrt{x} }{1 - \sqrt{x} } = \frac{1 - 2 \sqrt{x} + x}{1 - x} [/tex]
why do you add them instead of multiplying them
Answer:
huh
Step-by-step explanation:
what is other way to write 3x(4+7)
Answer:
[tex]3 \times (4 + 7) =( 3 \times 4) + (3 \times 7) \\ thank \: you[/tex]
what is the approxiamate solution to this equation 4lnx-8=12
Answer:
143.5
Step-by-step explanation:
4lnx=20
lnx=5
lnx= logeX
logeX=5
x=e^5
that is approximately equal to 2.7^5= 143.5
Answer:
x=148.4
Step-by-step explanation:
Choose the most convenient method to graph the line y = –3x + 4.
Answer:
Step-by-step explanation:
Domain and Range. PLEASE HELP!!!!! Numbers 1 and 2 please.
Answer:
1) D: -8 < x ≤ 2, R: -4 ≤ y ≤ 11.
2) D: -8 ≤ x ≤ -3 U 2 ≤ x < ♾️,
R: -6 ≤ y ≤ -5 U -3 ≤ y < ♾️.
the ratio of two number is 4:7 and their hcf is 6 .find their lcm
168
this is the full process
There are 8 Pizza Huts in the
city, Altogether they sold 5,376
pizzas yesterday, If each Pizza
Hut sold the same number of
pizzas, how many pizzas did
each one sell?
Answer:
672
Step-by-step explanation:
Given: 8 pizza hut's sold a total of 5,376 pizzas
Each pizza hut sold the same amount of pizzas
So number of pizzas sold from 1 pizza hut = 5376/8 = 672
Anyone who knows this mathematic question.to help please. Am giving the brainliest.
Show two ways fivepeople cans hare a 3-segmant chewy fruit worm
Answer:
ethethe
Step-by-step explanation:
thethetht
Which of the following is an example of a complex number that is not in the set of real numbers?
A) -7
B)
[tex]2 + \sqrt{3} [/tex]
C) 4+9i
D)
[tex]\pi[/tex]
Answer:
C
Step-by-step explanation:
By convention, it is C.
B is irrational but still real
D is irrational but still real
A is an integer which is real.
Answer: 4 + 9i
On edge2020.
Simplify. √ −18
A -3i√2
B 3i√2
C 9i√2
D -3√2
10. Simplify: 6/√3
please help, show work
Answer: [tex]2\sqrt{3}[/tex]
Step-by-step explanation:
First, start by expanding the expression to this
[tex]\frac{6}{\sqrt{3}} * \frac{\sqrt{3} }{\sqrt{3} }[/tex]
Then multiply the fractions by numerators and denominators (separately)
[tex]\frac{6*\sqrt{3} }{\sqrt{3}*\sqrt{3} }[/tex]
It results in this, the radical 3 * radical 3 get rid of the square root and result in 3
[tex]\frac{6\sqrt{3} }{3}[/tex]
Reduce the fraction by dividing the 6 by 3 so your solution is
[tex]2\sqrt{3}[/tex]
14. The cost of 10 dozen of caps is Rs.800. How many caps can bebought for Rs.200.
Answer:
30 caps can be bought
Step-by-step explanation:
[tex]{ \sf{Rs.800 = (10 \times 12) \: caps}} \\ { \sf{Rs.200 = ( \frac{200 \times 10 \times 12}{800}) \: caps}} \\ \\ { \sf{ = 30 \: caps}}[/tex]
can someone help me with the question below. just number #1 please :)
Answer:
[tex]{ \bf{ \bar {MN}} \: means \: length \: between \: M \: and \: N}[/tex]
MN means line joining point M to point N.
(2x³)⁴= ? What is the right answer, with full explanation please:
A. 16x⁷
B. 16x¹²
C. 8x⁷
D. 8x¹²
The required simplified value of the given expression is 16x¹². Option B is correct.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
=(2x³)⁴
As we know that,
[xᵃ]ᵇ = xᵃˣᵇ
So,
(2x³)⁴ = 2⁴x³ˣ⁴
= 16x¹²
Thus, the required simplified value of the given expression is 16x¹².
Learn more about simplification here:
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Is this a function? Input: students first name, Output: students favorite color
Are the following sets of points coplanar? 8. E, B, and F 9. DB and FC 10. AC and ED 11. AE and Dc 12. F, A, B, and C 13. F, A, B, and D 15. FB and BD 14. plane Q and EC
Answer: 8.EB and F: yes
9.DB and FC:no
10.AC and ED:yes
11.AE and DC:yes
12.F,A,B and C:yes
13.F,A,B and D:no
14.plane Q and EC:yes
15.FB and BD:yes
Step-by-step explanation:
hope this helps have a good day!
Write 4 2/5 as an improper faction
Answer:
22/5
Step-by-step explanation:
4 [tex]\frac{2}{5}[/tex]
new numerator for improper fraction = (4 x 5) + 2 = 20 + 2 = 22
=> 22/5
√5•√10
please show a step by step explanation
please solve the equation for y
Answer:
Step-by-step explanation:
A = [tex]\frac{x+y}{2}[/tex]
2A = x + y
y = 2A - x
Please Help Me Do This I cant understand it no matter how hard I try. (Problem in the image)
Answer:
7.3
Step-by-step explanation:
The length of AB is found using the distance formula
We need to know the coordinates of A and B
A = (-5,-4) and B = (-3,3)
The distance is
d = sqrt ( (y2-y1)^2 + (x2-x1)^2)
= sqrt( (3 - -4)^2 +(-3 - -5)^2)
= sqrt( (3+4)^2 +(-3+5)^2)
= sqrt(7^2+2^2)
= sqrt(49+4)
= sqrt(53)
=7.280109889
To the nearest hundredth
=7.3
When the product of x5 × x3 × x6 is expressed as a monomial, we get
Answer:
[tex]x^{14}[/tex]
Step-by-step explanation:
[tex]x^{5}[/tex] × [tex]x^{3}[/tex] × [tex]x^{6}[/tex]
[tex]x^{5+3+6}[/tex]
[tex]x^{14}[/tex]