Use implicit differentiation to find the slope of the tangent line at the given point:
Answers
Answer:
[tex]\displaystyle \frac{dy}{dx}\Big|_{(1, 1)}=0[/tex]
Step-by-step explanation:
We are given the equation:
[tex](x^2+y^2)^2=4x^2y[/tex]
And we want to find the slope of the tangent line at the point (1,1).
Recall that the slope of the tangent line to a point of a function is given by the function's derivative.
Thus, find the derivative of the equation. Take the derivative of both sides with respect to x:
[tex]\displaystyle \frac{d}{dx}\left[(x^2+y^2)^2\right]=\frac{d}{dx}\left[4x^2y\right][/tex]
Let's do each side individually.
Left:
We can use the chain rule:
[tex](u(v(x))'=u'(v(x))\cdot v'(x)[/tex]
Let v(x) be x² + y² and u(x) is x². Thus, u'(x) is 2x. Therefore:
[tex]\displaystyle \frac{d}{dx}\left[(x^2+y^2)^2\right]=2\left(x^2+y^2\right)\left(\frac{d}{dx}\left[x^2+y^2\right]\right)[/tex]
Differentiate:
[tex]\displaystyle \frac{d}{dx}[(x^2+y^2)^2]=2(x^2+y^2)\left(2x+2y\frac{dy}{dx}\right)[/tex]
Factor. Therefore, our left side is:
[tex]\displaystyle 4(x^2+y^2)\left(x+y\frac{dy}{dx}\right)[/tex]
Right:
We have:
[tex]\displaystyle \frac{d}{dx}\left[4x^2y\right][/tex]
We can move the constant multiple outside:
[tex]\displaystyle =4\frac{d}{dx}\left[x^2y\right][/tex]
We will use the Product Rule:
[tex]\displaystyle =4\left(\frac{d}{dx}\left[x^2\right]y+x^2\frac{d}{dx}\left[y\right]\right)[/tex]
Differentiate:
[tex]\displaystyle =4\left(2xy+x^2\frac{dy}{dx}\right)[/tex]
Therefore, our entire equation is:
[tex]\displaystyle 4(x^2+y^2)\left(x+y\frac{dy}{dx}\right)=4\left(2xy+x^2\frac{dy}{dx}\right)[/tex]
To find the derivative at (1,1), substitute and evaluate dy/dx when x = 1 and y = 1. Hence:
[tex]\displaystyle 4((1)^2+(1)^2)\left((1)+(1)\frac{dy}{dx}\right)=4\left(2(1)(1)+(1)^2\frac{dy}{dx}\right)[/tex]
Evaluate:
[tex]\displaystyle 4((1)+(1))\left(1+\frac{dy}{dx}\right)=4\left(2+\frac{dy}{dx}\right)[/tex]
Further simplify:
[tex]\displaystyle 8\left(1+\frac{dy}{dx}\right)=8+4\frac{dy}{dx}[/tex]
Distribute:
[tex]\displaystyle 8+8\frac{dy}{dx}=8+4\frac{dy}{dx}[/tex]
Solve for dy/dx:
[tex]\displaystyle 4\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=0[/tex]
Therefore, the slope of the tangent line at the point (1, 1) is 0.
Related Questions
A baker needs 2 pounds of flour and 3 sticks of butter to
make 64 cupcakes. At these rates, how many pounds of
flour and how many sticks of butter would the baker
need to make 352 cupcakes?
Answers
Answer:
11 pounds of flour and 16.5 sticks of butter
Step-by-step explanation:
2/64 = 0.03125
3/64 = 0.046875
0.03125 * 352 = 11
0.046875 * 352 = 16.5
Answer:
15 1/2 sticks of butter, and 11 pounds of flower, hoped this helped.
Step-by-step explanation:
1) The price of a dress is $89. Susan has a coupon for $50 off ger entire purchase and another coupon for 50% off one item. The store's policy does not allow for both coupons to be used at once. Which monetary incentive would give Susan a better buy. $50 off coupon is a better buy 50 of one item is a better buy.
Answers
First coupon:
89 - 50 = $39 (discount price)
Second coupon:
50% = 0.5
89 * 0.5 = 44.5
89 - 44.5 = $44.5 (discount price)
39 < 44.5
Therefore, the better option is using the $50 off one item.
Best of Luck!
Write an absolute value equation that has the solutions x= -6 and x = 10.
The equation is blank
Answers
HERES THE EXAMPLE: If | x | = 3, we know that x may be either +3 or - 3,
since both are 3 units from the origin.
Notice that the answer is "OR", since x cannot be both -3 and +3 at the same time.
ANSWER: so the x=-6 would be would just be six units -6 to 0 would be 6
the x=10 would be -10
Two lines and a ray intersect at a
point as shown in the diagram
Write and solve an equation to find the value of 2 Find the measure of
DAE
110°
A
D
(+4)
30
E
Answers
Answer:
m<DAE = 80°
Step-by-step explanation:
Given that m<CAB = 110°,
m<DAE = (x + 4)°,
m<EAF = 30°,
m<EAF + m<DAE = m<DAF (angle addition postulate)
30° + (x + 4)° = m<DAF (substitution)
(34 + x)° = m<DAF
<DAF and <CAB are vertically opposite angles. Vertically opposite angles are said to be congruent. Therefore:
m<DAF = m<CAB
(34 + x)° = 110° (substitution)
The equation above can be used to solve for x as follows:
34 + x = 110
x = 110 - 34 (Subtracting 34 from both sides)
x = 76
m<DAE = (x + 4)°
Plug in 76, to replace x
m<DAE = 76 + 4
m<DAE = 80°
Evaluate the following expression
Answers
Answer:
1/216
Step-by-step explanation:
6^-3= 1/6^3
=1/216
Answer;
6³-3³
6×6×6-3×3×3=
216-27=199
199-1=answer
my number eight dosent work please understand
1. Ilove ordering cookies from Tiff's Treats (not
sponsored, although I wouldn't object!). If Tiff's
Treats charges a delivery fee of $1 and costs $8.50
per half-dozen, which equation relates my cost, c,
and the number of half-dozens, h, I ordered?
a) C = 9.500h
b) C = 8.50h + 1
c) c = h + 8.50
d) h = 9.50c
e) None of these
Answers
c = cost
h = # of half-dozens ordered
c = $8.50h + $1 is the correct answer.
w + 1200+209 = 2000
This might seem easy but I seriously don’t know how to get to the answer
Answers
Answer:
591
Step-by-step explanation:
added the 2 in the equation and minused 2000 from that
Can someone please help? This is due today, and I have NO CLUE
Answers
Step-by-step explanation:
Multiplication......
What is the common difference between successive terms in the sequence?
2, 8, 14, 20, 26, ...
O-6
0-4
O4
O 6
Answers
The common difference between successive terms in this arithmetic sequence is 6.
Option D is the correct answer.
What is an arithmetic sequence?It is a sequence where the difference between each consecutive term is the same.
We have,
The common difference between successive terms in the sequence 2, 8, 14, 20, 26, ... is 6.
To see why, notice that each term in the sequence is obtained by adding 6 to the previous term.
For example, to get from 2 to 8, we add 6. To get from 8 to 14, we add another 6.
This pattern continues for each successive term in the sequence.
Thus,
The common difference between successive terms in this arithmetic sequence is 6.
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60 is 1/100 of _______
Answers
Answer:
that is 60 percent (calculated percentage %) of number 100? Answer: 60.
what is 60% of 86 ?
Answers
Answer:
51.6
Step-by-step explanation:
Answer:
51.6
Step-by-step explanation:
60% = 0.6
0.6x86=51.6
Hey, I know the correct answer is C. May someone please explain or give me an example why the answer is C.
Answers
Answer:
3ny
Step-by-step explanation:
For this problem, the price of a widget is represented by 'n' amount of dollars. Additionally, the store was stocked with 4y widgets.
To find the total cost of the widgets sold, we will use the information that at the end of the day, the store only had y widgets remaining.
From here we can say the following to represent the total cost of the widgets sold:
cost = n * (4y - y)
cost = n * (3y)
cost = n3y
You can rewrite this cost to be 3ny as in the answer choice.
Cheers.
Which graph shows a polynomial function of an odd degree?
Answers
Answer:
THE ANSWER IS C ON EDGE.
Step-by-step explanation:
If both sides of the graph are pointing in the same way, THE FUNCTION HAS AN EVEN DEGREE.
The degree of a polynomial is the highest power of the polynomial.
Graph 3 has an odd degree.
From the attachments, we have the following highlights
The first graph crosses the x-axis, 4 timesThe second graph crosses the x-axis, 6 timesThe third graph cross the x-axis, 3 timesThe fourth graph cross the x-axis, 2 timesThe above highlights mean that:
Graphs 1, 2 and 4 have even degrees, because they have an even number of x-interceptsGraphs 3 have an odd degree, because it has an odd number of x-interceptsHence, graph 3 has an odd degree.
Read more about degrees of polynomials at:
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Please help me I’ll give brainliest
Answers
Answer:
B, C, D
Step-by-step explanation:
what number cubed equals 2744
Answers
What is the answer and steps to the problem 11(8r+3)-2(-9+6r)
Answers
(88r+33) + (18-12r)
88r+33+ 18- 12r
76r + 51
Find the common factors of:
(i) 24 and 28 (ii) 12 and 18 (iii) 36 and 48
Answers
(i) 1, 2, 4
(ii) 1, 2, 3, 6
(iii) 1, 2, 3, 4, 6, 12
:)
A=1/2 BH where A: area, B: base, & H: height. If the Base of a triangle is 7 ft and the Height of the triangle is 4 ft, what is the Area of the triangle? Question 12 options: 56 ft 28 ft 14 ft 37 ft
Answers
Answer:
[tex]\boxed {\tt A= 14 \ ft^2}[/tex]
Step-by-step explanation:
The area of a triangle can be found using the following formula.
[tex]A=\frac{1}{2} bh[/tex]
where [tex]b[/tex] is the base and [tex]h[/tex] is the height.
In this triangle, the base is 7 feet and the height is 4 feet.
[tex]b= 7 \ ft\\h= 4 \ ft[/tex]
Substitute the values into the formula.
[tex]A=\frac{1}{2} bh[/tex]
[tex]A=\frac{1}{2} (7\ ft)(4 ft)[/tex]
Multiply 7 ft and 4 ft.
[tex]A=\frac{1}{2} (28 \ ft^2)[/tex]
Multiply 28 square feet by 1/2 or divide 28 square feet by 2.
[tex]A= 14 \ ft^2[/tex]
The area of the triangle is 14 square feet.
Which postulate does this diagram demonstrate?
Answers
Find the length of side CB in the following right triangle
Round your answer to the nearest tenth.
Answers
Answer:
6cm
Step-by-step explanation:
a2 + b2 = c
(1.4)2 + (2)2 = c
1.96 + 4 = 5.96
=5.96cm (round it)
final answer = 6cm
Answer:
CB = 1.4 cm
Step-by-step explanation:
Given is a right angled triangle.
[tex] {x}^{2} + {1.4}^{2} = {2}^{2} \\ {x}^{2} + 1.96= 4 \\ {x}^{2} = 4 - 1.96 \\ {x}^{2} = 2.04 \\ x = \sqrt{2.04} \\ x = 1.42828569 \\ x \approx 1.4[/tex]
CB = 1.4 cm
(-4.5)to the power of 0
Answers
Answer:
-1
Step-by-step explanation:
Please help due in 10 min
Answers
Answer: the first option
Joanie weighs 102 pounds. Her father weighs 2.2 times as much as Joanie. How much does her father weigh?
Answers
Answer:
224.4
Step-by-step explanation:
102x2.2=224.4
Could I please have Brainliest.
The speed of a car traveling on a straight road increases from 63 m/s to 75
m/s in 4.2 seconds. What was the car's acceleration? m/s2 = meters per
second squared
Answers
Initial velocity = 63m/s
Final velocity = 75m/s
Time taken = 4.2
We have to find acceleration of car[tex].[/tex]
_________________________________
◈ Acceleration is defined as the rate of change of speed.
a = (v - u) / tIt is a vector quantity having both magnitude as well as direction.
SI unit : m/s²
⇒ a = (v - u) / t
⇒ a = (75 - 63) / 4.2
⇒ a = 12/4.2
⇒ a = 2.85 m/s²The speed of a car traveling on a straight road increases from 63 m/s to 75 m/s in 4.2 seconds. What was the car's acceleration?
Answer:[tex] \rm\red {Given:} [/tex]
u ( initial velocity ) = 63m/sv ( final velocity ) = 75m/st ( time taken ) = 4.2s[tex] \rm\red {To \: find:} [/tex]
a ( acceleration ) =?[tex] \rm\red {Calculation:} [/tex]
We know that,
[tex] {\huge {\boxed {\tt { a = \dfrac{v-u} {t}}}}} [/tex]
Substituting values-
[tex] \sf { \qquad \leadsto a = \dfrac{75-63}{4.2}} [/tex]
[tex] \sf { \qquad \leadsto a = \dfrac{12}{4.2}} [/tex]
[tex] \sf { \qquad \leadsto a = \dfrac{\cancel {12} {} ^{\: \: 6} \times 10}{\cancel {42} {} ^{\: \: 21} }} [/tex]
[tex] \sf { \qquad \leadsto a = \dfrac{\cancel6 {} ^{\: \: \: 2}\times 10 }{\cancel {21} {} ^{\: \: 7}}} [/tex]
[tex] \sf { \qquad \leadsto a = \dfrac{2 \times 10}{7}} [/tex]
[tex] \sf { \qquad \leadsto a = \dfrac{20}{7} = 2.85} [/tex]
[tex] \therefore [/tex] Acceleration of the car was [tex] \sf { 2.8m/{s} ^{2}} [/tex]
_________________________________I don’t know how to do this
Answers
Answer:
60yd
Step-by-step explanation:
First you multiply 3yd, and 6yd to get the area of 18 yards.
Second, you multiply the length of the triangle, so 6*7 is 42.
Finally you add both of the areas together to get the area of the full shape.
and ixl is the worst lol
The frequency table shows the distribution of 1,200 students at a high school. To the nearest percent, what percentage of females are freshmen or sophomores? Frequency Table of Student at a High School Seniors Total Freshmen Sophomores Juniors 168 128 166 146 Males 608 Females 160 592 134 162 136 328 264 306 302 1.200 Total 27% 50% 53% 23% My Progress
Answers
Answer:
23 percent
Step-by-step explanation:
The percentage of females who are freshmen or sophomores will be 50%.
What is the percentage?The quantity of anything is stated as though it were a fraction of a hundred. A quarter of 100 can be used to express the ratio. Per 100 is what the term percent signifies. The symbol 'percent' is used to symbolize it.
The frequency table shows the distribution of 1,200 students at a high school.
The frequency table is given below.
Then the percentage of females who are freshmen or sophomores will be
P = (134 + 162) / 592 x 100
P = 296 / 592 x 100
P = 0.5 x 100
P = 50%
Thus, the correct option is B.
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3^-4=????? HURRRRYYYYYYYYYYYYYYYYY
Answers
Answer:
Here’s the answer :)
Step-by-step explanation:
Answer:
[tex] {81}^{ - 1} \: or \: \frac{1}{81} [/tex]
Step-by-step explanation:
[tex] {3}^{ - 4} [/tex]
Apply rule :
[tex] {a}^{ - 1} = \frac{1}{ {a}^{1} } [/tex]
[tex] \frac{1}{ {3}^{4} } = \frac{1}{81} \: or \: {81}^{ - 1} [/tex]
If a line has a slope 2/3 a line that is perpendicular has a slope of what
Answers
Answer:
-1.5
Step-by-step explanation:
The product of slopes of two perpendicular lines = -1
If line one has a slope of 2/3;
Let line two, perpendicular to line one, has a slope of a
2/3 × a = -1
a = -1 ÷ 2/3 = -3/2 = -1.5
A rectangular prism and its dimensions are shown in the diagram.
Dimensions:length 7.5 width 5 height 6.5
Answers
Answer:
Surface Area =237.5
Answer:237.5
Step-by-step explanation:
The table represents the equation y = 2 – 4x The x-values are the The y-values are the The missing value in the table for x = –1 is y =
Answers
Answer:
1.) inputs
2.) output
3.) 6
Step-by-step explanation:
Hopefully this helps you ;)
pls mark brainlest :)
The missing value for x=-1 is y=6
How will you solve the equation?By substituting the known values in the equation gives the unknown value.
Given equation is y=2-4x
To find y when x=-1
Substitute x=-1 in given equation,
we get y=2-4(-1)
y=6
Hence, the value of y when x=-1 is 6
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