Differentiate the function/equation with respect to x and solve for the derivative, dy/dx. The value of dy/dx at the given point is the slope of the tangent line to the curve at that point. Then use the point-slope formula to get the equation of the tangent.
1.
[tex]y = \dfrac{x^2}{x+2} \implies \dfrac{dy}{dx} = \dfrac{2x\times(x+2) - x\times1}{(x+2)^2} = \dfrac{x(x+4)}{(x+2)^2}[/tex]
When x = 2, the derivative is
[tex]\dfrac{dy}{dx}\bigg|_{x=2} = \dfrac{2(2+4)}{(2+2)^2} = \dfrac34[/tex]
Then the equation of the tangent line at (2, 1) is
[tex]y - 1 = \dfrac34 (x - 2) \implies \boxed{y = \dfrac{3x}4 - \dfrac12}[/tex]
2.
[tex]x^3 + 2y^2 = 10y \implies 3x^2 + 4y \dfrac{dy}{dx} = 10 \dfrac{dy}{dx} \implies \dfrac{dy}{dx} = \dfrac{3x^2}{10-4y}[/tex]
When x = 2 and y = 1, the derivative is
[tex]\dfrac{dy}{dx}\bigg|_{(x,y)=(2,1)} = \dfrac{3\times2^2}{10-4\times1} = 2[/tex]
Then the tangent at (2, 1) has equation
[tex]y - 1 = 2 (x - 2) \implies \boxed{y = 2x - 3}[/tex]
PLEASE HELP!!!!!!!!!
Answer:
correct answer is B.
Step-by-step explanation:
for more information see the attachment.
Answer:
no 2/B is the correct answer
need help with this please
The values of x and y at t=3 are -4.95 and 0.423 respectively.
The value of dx/dt and dy/dt are 0.71 and -2.97 respectively.
The value of the tangent slope at t=3 is 4.21.
The speed at t=3 is 3.05 units/sec.
Given the equation of x as the function of t is
x= 5 cos t
similarly, the equation of y as the function of t is
y= 3 sin t
At t=3 the value of x will be
x (at t=3) = 5 cos 3= 5(-0.989)= -4.95
At t=3 the value of y will be
y (at t=3) = 3 sin 3= 3(0.141)= 0.423
The derivative of the function of x with respect to t will be
dx/dt= d(5 cos t)/dt= 5d(cos t)/dt= -5 sin t
at t=3 the value of dx/dt will be
dx/dt (at t=3) = -5 sin 3= -5(0.141)= 0.71
The derivative of the function of y with respect to t will be
dy/dt= d(3 sin t)/dt= 3d(sin t)/dt= 3 cos t
at t=3 the value of dy/dt will be
dy/dt (at t=3) = 3 cos t= 3(-0.989)= -2.97
The tangent slope is dy/dx which can be calculated by
dy/dx= (dy/dt)(dt/dx)= (dy/dt)/(dx/dt)= 3 cos t/ -5 sin t= (-3/5) cot t
at t=3 the value of tangent slope will be
dy/dx (at t=3) = (-3/5) cot 3= 4.21
The speed at t=3 will be
speed v= [tex]\sqrt{v_{x} ^{2} + v_{y} ^{2} }[/tex]
= √(dx/dt)²+(dy/dt)²
at t=3
= √(0.71)²+(-2.97)²
= √9.325
= 3.05 unit/sec
Therefore the values of x and y at t=3 are -4.95 and 0.423 respectively.
The value of dx/dt and dy/dt are 0.71 and -2.97 respectively.
The value of the tangent slope at t=3 is 4.21.
The speed at t=3 is 3.05 units/sec.
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Hamish and Harry work as plumbers. harry earns a dollar more than 5/4 the amount Hamish earns per hour. the amount Harry earns per hour is 2$ less than 7/5 the amount Hamish earns per hour. how much to both of them earn per hour?
Answer:
Hamish earns $20 per hour, and Harry earns $26 per hour.
Step-by-step explanation:
Harry earns
$1+(5/4)•x and also (7/5)•x-$2
$1+(5/4)•x = (7/5)•x-$2
$1+$2= (7/5)•x-(5/4)•x
3=(7•4•x-5•5•x)/20
3=3x/20
3•20=3x
X=20, Hamish earns $20
Substitute x in one of the expressions
1+(5/4)•20=1+25=26, Harry earns $26
Expert Mathematicians, please help!
Answer:
Two functions which satisfies the H(x) are f(x) = 2x - 4, (x) = x⁵ and
f(x) = 2x, g(x) = x⁵ - 2
Step-by-step explanation:
The given question is based on the decomposition of a given function into two functions
Decomposition of a function :- functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed from those parts by function composition
so the function given in the question is H(x) = 2x⁵ - 4
now by using the concept of decomposition of a function we can find two functions f and g
therefore,
f(x) = 2x - 4
and g(x) = x⁵
similarly we can find another pair which satisfies the function H(x)
f(x) = 2x
and g(x) = x⁵ - 2
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the answer to this question
From indices : when you have a fraction like 3⁶/5
we can also write it as (5√3)^6
and we were given (c√a )^b
meaning that anywhere the letters are, we put them as the numbers there.
therefore: a = 3 , b = 6 , c = 5
Hope this helps and you understood it.
you're super free to ask any more questions eh.
Good luck ✅.
Given a function described by the table below, what is y when x is 5?
XY
26
48
59
612
Answer:
y = x + 4
when x is 5, y is 9
Step-by-step explanation:
the pattern from the table is that the y value is 4 more than the x value, so the equation would be y = x + 4.
if you plug in x = 5, then this would be y = 5 + 4 = 9
This is from Khan academy I have to attach a PNG if you can help me solve it! Thank you!
Step-by-step explanation:
[tex] \frac{5 {}^{ - 4x + 7} }{125 {}^{x} } [/tex]
[tex] \frac{5 {}^{ - 4x + 7} }{5 {}^{3x} } [/tex]
[tex]5 {}^{ - 4x + 7 - 3x} [/tex]
[tex]5 {}^{ - 7x + 7} [/tex]
Raul's pet store has a play area that can fit up to 30 cats and dogs. His son is allergic to cats so the pet store never has more than 8 cats in the play areas.
How many of each type of pet can they have at the
pet store?
Write a system of inequalities.
Answer:
Step-by-step explanation:
Let :
C = number of cats
D = number of dogs
Raul's pet store has a play area that can fit up to 30 cats and dogs.
C + D = 30
The pet store never has more than 8 cats in the play areas.
C < 9(there can never be 9 or more cats)
As or the number of dogs :
C + D = 30
C = 30 - D
Since we know that C < 9. To get the number of dogs allowed, we just plug in 30 - D for C.
30 - D < 9
30 - 9 < D
or
D > 21
(dogs have to be 22 or more)
What is the average rate of change from n = 1 to n = 2 for the sequence an = 2(2)n − 1
Answer: 2
Step-by-step explanation:
[tex]a_{n}=2(2)^{n-1}\\\\a_{1}=2(2)^{1-1}=2\\\\a_{2}=2(2)^{2-1}=4[/tex]
Average rate of change:
[tex]\frac{4-2}{2-1}=\boxed{2}[/tex]
50 cm
1.5
Path
Vegetables
50 cm
50 cm
2
5.5m
This plan shows a vegetable garden.
The width of the path is 50 cm throughout.
4.5m
Step-by-step explanation:
b because you have to multiply
how would i solve this problem
Answer:
30
Step-by-step explanation:
1500/50=30
1.9^4 × 10 × 2.9 x 10^1
3779.309 is the answer....
Write the equation in standard form. Then factor the left side of the equation
3x2 + 19x = 14 .
A. (3x – 2)(x + 7) = 0
B. (3x + 2)(x – 7) = 0
C. (2x – 3)(x + 7) = 0
D. (2x + 3)(x + 7) = 0
Find the solution to the following system by substitution.
5x + y = 20
y = 5x
A. (2,10)
B. (4,0)
C. (0,20)
D. (4,20)
Answer:
A. (2,10)
Step-by-step explanation:
5x + y = 20
y = 5x
The second equation is already solved for y, so substitute 5x for y in the first equation to find x.
5x + 5x = 20
10x = 20
x = 2
Now substitute 2 for x in the second original equation to find y.
y = 5(2)
y = 10
Answer: (2, 10) which is A.
enita is factoring the expression 32 a b minus 8 b. She determines the GCF and writes the factored expression as 8 b (4 a minus 0). Which best describes Venita’s error?
a. She incorrectly determined the GCF.
b. She subtracted the GCF from the second term in the expression instead of dividing.
c. She divided the GCF into the first term in the expression instead of subtracting.
d. She divided each piece of the expression by different factors.
This shows that the greatest common factor is 8, Hence Venita's error is that she incorrectly determined the GCF.
Greatest common factorGiven the following expression
32ab - 8
Find the factors of each terms
32ab = 8 * 4. * a * b
8 = 8 * 1
Since 8 is common to both factors, hence
32ab - 8 = 8(4ab -1)
This shows that the greatest common factor is 8, Hence Venita's error is that she incorrectly determined the GCF.
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Mama bear wants to bake a cake with chocolate frosting for her twins birthday party. She uses a package containing 495 grams of coco powder for the frosting. What will be the weight of the chocolate frosting if it is 55% cocoa powder?
The weight of the chocolate frosting if it is 55% cocoa powder will be 272.25 grams.
What is the percentage?The quantity of anything is stated as though it were a fraction of a hundred.
Mama bear wants to bake a cake with chocolate frosting for her twin's birthday party.
She uses a package containing 495 grams of cocoa powder for the frosting.
Then the weight of the chocolate frosting if it is 55% cocoa powder will be
⇒ 0.55 × 495
⇒ 272.25 grams
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The function f(x) = –x2 – 4x + 5 is shown on the graph. On a coordinate plane, a parabola opens down. It goes through (negative 5, 0), has a vertex at (negative 2, 9), and goes through (1, 0). Which statement about the function is true? The domain of the function is all real numbers less than or equal to −2. The domain of the function is all real numbers less than or equal to 9. The range of the function is all real numbers less than or equal to −2. The range of the function is all real numbers less than or equal to 9.
The true statement is:
"The range of the function is all real numbers less than or equal to 9."
Which statements are true?
Here we have the quadratic function:
[tex]f(x) = -x^2 - 4x + 5[/tex]
And we want to see which of the given statements are true.
The first one is:
"The domain is al real numbers less than or equal to -2"
This is false, for all quadratic functions the domain is the set of all real numbers (unless the domain is defined).
The second statement is:
" The domain of the function is all real numbers less than or equal to 9."
Also false
Third one:
" The range of the function is all real numbers less than or equal to −2"
The range of a quadratic function with a negative leading coefficient will be the set of all the values smaller than the y-value of the vertex.
In this case, the quadratic function is:
[tex]f(x) = -x^2 - 4x + 5[/tex]
So the vertex is at:
[tex]x = 4/(2*-1) = -2\\[/tex]
Then the y-value of the vertex is:
[tex]f(-2) = -(-2)^2 - 4*(-2) + 5 = -4 + 8 + 5 = 9[/tex]
So the range is the set of all real numbers less than or equal to 9.
So the above statement is false, and the final one:
"The range of the function is all real numbers less than or equal to 9."
Is the true statement.
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work hours
frequency
0 - <10
45
10 - <20
33
20 - <30
20
30 - <40
7
40 - <50
8
Use your calculator to find the mean and standard deviation
The mean and the standard deviation are 16.15 and 12.03, respectively
How to determine the mean and the standard deviation?The table of values is given as:
Work hours Frequency
0 - <10 45
10 - <20 33
20 - <30 20
30 - <40 7
40 - <50 8
Start by rewriting the table using the midpoint of each class
x f
5 45
15 33
25 20
35 7
45 8
The mean is then calculated as:
[tex]\bar x = \frac{\sum fx}{\sum f}[/tex]
This gives
[tex]\bar x = \frac{5 * 45 + 15 * 33 + 25 * 20 35 * 7 + 45 * 8}{45 + 33 + 20 + 7 + 8}[/tex]
Evaluate the sum and the products
[tex]\bar x = \frac{1825}{113}[/tex]
Divide
[tex]\bar x = 16.15[/tex]
The standard deviation is:
[tex]\sigma =\sqrt{\frac{\sum f(x - \bar x)^2}{\sum f}}[/tex]
This gives
[tex]\sigma= \sqrt{\frac{45 * (5 - 16.15)^2 + 33 * (15 - 16.15)^2 + 20 * (25 - 16.15)^2 + 7 * (35 - 16.15)^2 + 8 * (45 - 16.15)^2 }{45 + 33 + 20 + 7 + 8}}[/tex]
Evaluate the sum and the products
[tex]\sigma= \sqrt{\frac{16350.4425}{113}}[/tex]
Divide
[tex]\sigma= \sqrt{144.694181416}[/tex]
Take the square root
[tex]\sigma= 12.03[/tex]
Hence, the mean and the standard deviation are 16.15 and 12.03, respectively
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Which graph shows y=1/2⌈x⌉?
The correct answer is option B which is the graph that represents the equation y = (1/2)x will be the second option in the figure. At every value of y, the value of x is double.
What is a graph?A graph is the representation of the data on the vertical and horizontal coordinates so we can see the trend of the data.
The graph of the equation y =1/2x is represented in the answer below we can see in the graph that at every value of y the value of x is double for example if we put y = 1 then the value of x will be 2.
y = 1/2 x
at x = 2
y = 1/2 x 2
x = 1
Therefore the correct answer is option B which is the graph that represents the equation y = (1/2)x will be the second option in the figure. At every value of y, the value of x is double.
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What is the length of leg s of the triangle below? 45 √32 45° A. √4 B. 1.5 C. 16
Answer:
s = 4 unitsStep-by-step explanation:
Given is isosceles right triangle, since its interior angles are 45°, 45° and 90°.
The missing length is therefore same as the length of the other leg:
s = 4Note: This is not included in the answer choices.
Answer:
4 units
Step-by-step explanation:
We can use the converse theorem of the isosceles triangle to find the missing leg.
Converse Isosceles triangle theorem: If two angles are congruent in a triangle, then sides opposite to those angles are also congruent.
Therefore,
As there are two 45° angles, the opposite sides to those are also congruent.
∴ s = 4
in need of assistance
Answer:
60
Step-by-step explanation:
_---&$'_-4$':/-&$$--(:'##4--&&4___&$$_666&4&&&&&
Need number 7 the answer please.
Answer:
h(x) = -2|x+2| + 2
Step-by-step explanation:
let me know if you want an explanation :))
Given (x – 7)2 = 36, select the values of x.
Answer:
x = 25
Step-by-step explanation:
Solve by isolating x:
(x-7)2 = 36
x-7 = 18
x = 25
what is ⅛ in a percentage form
Answer:
12.5%
Step-by-step explanation:
1/8 × 100% = 12.5%
So 9f you multiply an eighth by one hundred percent and simplify accurately you'll get 12.5%
Rewrite exponential expressions
This is from Khan academy I have to attach a PNG if you can help me solve it! Thank you!
Answer:
Step-by-step explanation:
The equation for the pH of a substance is pH = -log[H*], where H+ is the concentration of hydrogen ions. A basic
solution has a pH of 11.2. An acidic solution has a pH of 2.4. What is the approximate difference in the concentratic
of hydrogen ions between the two solutions?
1.6x10-9
O4.0x10-3
O 6.7x10-1
O 1.6x10¹11
The approximate difference in the concentration of hydrogen ions between the two solutions is 4.0×10⁻³
What is pH value?The pH of a solution is a measurement of the concentration of hydrogen ions in the solution, as well as its acidity or alkalinity. Normally, the pH scale ranges from 0 to 14.
pH value for basic solution = 11.2
pH value for acidic solution = 2.4
Concentration for a basic solution:
[tex]\rm = 10^{-11.2}[/tex]
[tex]= 6.309\times10^{-12}[/tex]
Concentration for an acidic solution:
[tex]\rm = 10^{-2.4}[/tex]
[tex]= 3.981\times10^{-3}[/tex]
Difference between the concentration:
[tex]= 3.981\times10^{-3}- 6.309\times10^{-12}[/tex]
[tex]= 3.98\times10^{-3}[/tex]
[tex]= 4\times10^{-3}[/tex]
Thus, the approximate difference in the concentration of hydrogen ions between the two solutions is 4.0×10⁻³
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Question 6 of 15
Gloria invested $1200 in an account that earns 2.5% interest, compounded
annually. The formula for compound interest is A(t) = P(1 + i)t. How
much did Gloria have in the account after 5 years?
O A. $3662.11
OB. $6150.00
OC. $1470.00
OD. $1357.69
Answer:
D
Step-by-step explanation:
[tex]1200( {1.025}^{5} ) = 1357.69[/tex]
What are the zeros of the quadratic function f(x)=6x^2+12x-7?
(Please pick from one of the options in the photo)
The zeroes of the quadratic function f(x) = 6x² + 12x – 7 will be given below. Then the correct option is A.
The missing options are given below.
What is the solution of the quadratic equation by formula method?The quadratic equation is given as ax² + bx + c = 0.
Then the solution is given as
[tex]\rm x = \dfrac{-b\pm \sqrt{b^2 - 4ac}}{2a}[/tex]
The quadratic function is given below.
f(x) = 6x² + 12x – 7
Then the zeroes of the quadratic function will be
[tex]\rm x = \dfrac{-12\pm \sqrt{12^2 - 4 * 6 * (-7)}}{2 * 6}\\\\\\\rm x = \dfrac{-12\pm \sqrt{312}}{12}\\\\\\\rm x = -1 \pm \sqrt{\dfrac{13}{6}}\\\\\\x = -1 - \sqrt{\dfrac{13}{6}} , -1 + \sqrt{\dfrac{13}{6}}[/tex]
Then the correct option is A.
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Write an algorithm to find Area of square?
Step-by-step explanation:
AREA OF SQUARE:
A square is simply a rectangle with equal sides. As a result, the length and breadth of a square are equal. As a result, the area of a square is the product of two square sides, or in simpler terms, squares of the side.
Area = Side^{2}Area=Side
2
We use the following formula to compute the area of a rectangle after being given the length and width of the rectangle as input.
ALGORITHM:
Step 1 : Start
Step 2: Input Side
Step 3: Area = Side \times× Side
Step 4: Print area
Step 5: Stop
Question 9 of 10
For what value of x is the rational expression below equal to zero?
3x + 15
6-x
OA. -5
OB. 5
O C. 6
OD. -6
UBMIT
Answer:
A. -5
Explanation:
For (3x + 15)/(6 - x) to be equal to zero. Set the equation to 0.
Doing so:
(3x + 15)/(6 - x) = 0
3x + 15 = 0(6 - x)
3x + 15 = 0
3x = -15
x = -15/3
x = -5
Therefore, the value of x is -5 to equal to zero.
