1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
cricket20 [7]
3 years ago
14

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 (−3 3 , 1)

(astroid)
Mathematics
1 answer:
vovikov84 [41]3 years ago
8 0

Answer with Step-by-step explanation:

We are given that an equation of curve

x^{\frac{2}{3}}+y^{\frac{2}{3}}=4

We have to find the equation of tangent line to the given curve at point (-3\sqrt3,1)

By using implicit differentiation, differentiate w.r.t x

\frac{2}{3}x^{-\frac{1}{3}}+\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}=0

Using formula :\frac{dx^n}{dx}=nx^{n-1}

\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}=-\frac{2}{3}x^{-\frac{1}{3}}

\frac{dy}{dx}=\frac{-\frac{2}{3}x^{-\frac{1}{3}}}{\frac{2}{3}y^{-\frac{1}{3}}}

\frac{dy}{dx}=-\frac{x^{-\frac{1}{3}}}{y^{-\frac{1}{3}}}

Substitute the value x=-3\sqrt3,y=1

Then, we get

\frac{dy}{dx}=-\frac{(-3\sqrt3)^{-\frac{1}{3}}}{1}

\frac{dy}{dx}=-(-3^{\frac{3}{2}})^{-\frac{1}{3}}=-\frac{1}{-(3)^{\frac{3}{2}\times \frac{1}{3}}}=\frac{1}{\sqrt3}

Slope of tangent=m=\frac{1}{\sqrt3}

Equation of tangent line with slope m and passing through the point (x_1,y_1) is given by

y-y_1=m(x-x_1)

Substitute the values then we get

The equation of tangent line is given by

y-1=\frac{1}{\sqrt3}(x+3\sqrt3)

y-1=\frac{x}{\sqrt3}+3

y=\frac{x}{\sqrt3}+3+1

y=\frac{x}{\sqrt3}+4

This is required equation of tangent line to the given curve at given point.

You might be interested in
Determine the area of this figure. Round your answer to the nearest tenth place.
denis-greek [22]
36 feet okay it is 36 feet of the area
5 0
3 years ago
FYI, I don't have many points.
zimovet [89]

Answer:

ok i see that question it would be 3

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
What is 1/125 as a decimal?
svlad2 [7]
1/125 in a decimal is 0.008.
5 0
3 years ago
Read 2 more answers
A bond quoted at 93 (1/8) is equal to ___ per 1,000.00 of the face amount.
Ksju [112]
The number a bond is the percentage of the value of the bond that the bond is worth. Because the bond is quoted at 93, the bond is worth $930 per $1,000.
5 0
3 years ago
Read 2 more answers
What is the greatest common factor that could be used to reduce 36/90
TEA [102]
I got 18 for this one............
3 0
3 years ago
Read 2 more answers
Other questions:
  • Find the 7th term of the geometric progression which begins<br> 2, -6, 18,...
    5·1 answer
  • Please i need an urgent help doing this question help
    9·1 answer
  • To​ ​ get​ ​ an​ ​ accurate​ ​ result,​ ​ a​ ​ sample​ ​ size​ ​ must​ ​ be
    8·1 answer
  • Which is the equation of the graphed line written in standard form? y = x x – 1/2y = 0 x – y = 0 y =1/2x
    15·1 answer
  • suppose a culture of bacteria starts with 9000 bacteria. after one you,r the count is 10000. find the equation that models the n
    10·1 answer
  • When 3x+2≤5(x-4) is solved for x, the solution is?
    11·1 answer
  • Please help!! I'm super confused:(
    15·1 answer
  • What's the 8th term of the sequence defined by the given rule: F(1)=3, F (n-1 )+10
    14·2 answers
  • Now consider the family of lines ax + by = c such that a, b and c are consecutive multiples of each other. what would that mean?
    5·1 answer
  • N is an integer.<br> Write the values of n such that -15 &lt; 3 ≤ 6<br><br> Help please ❤️❤️
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!