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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.

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The correct option A: y = -4 cos Ф/4, is the cosine function for the graph.

<h3>Define the term cosine function?</h3>

The ratio between the angle's adjacent leg and the hypotenuse when it is regarded as a leg of a right triangle is a trigonometric function for an acute angle.

  • One of the three fundamental trigonometric functions, cosine is the complement of sine (co+sine) and one of the three main trigonometric functions.
  1. Y=cos(x) has its greatest value when x = 2nπ, wherein n is an integer.
  2. Y=cos(x) has a lowest value for x=  π+2nπ , wherein n is an integer.

For the given graph,

cosine function:  y = -4 cos Ф/4.

In which, -4 is the amplitude (maximum displacement from the x axis).

Negative sign shows, the displacement is taken along negative y-axis.

And,  Ф/4 is the phase angle.

Thus, the cosine function for the graph is  y = -4 cos Ф/4.

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6 0
11 months ago
$5.97 / 2 <br><br>please show your work! I'll give brainliest to whoever shows work.
Sonbull [250]

Answer:

2.99

Step-by-step explanation:

1.) 2 goes into 5 2 times. So the 2 goes on top of the division sign

2.) 2x2=4

3.) Move 4 below 5

4.) Subtract 5-4=1

5.) Bring down 9

6.) 2 Goes into 19 9 times

7.) 2x9=18

8.) 19-18=1

9.) Bring down the 7

10.) 2 Goes into 17 8 times

11.) 2x8= 16

12.) 17-16=1

13.) Add a zero onto the end, bring down that zero

14.) 2 goes into 10 5 times.

15.) Round answer to 2.99 because with money you only have two decimals

7 0
3 years ago
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Divide a quantity into two parts in a given ratio.
True [87]
You would just divide that quantity by 2.
3 0
2 years ago
The gear is one of the oldest mechanical devices. It has been used since ancient times for its ability to increase or decrease r
eduard

The large gear attached to the pedals connected to the smaller gear

attached to the wheel allows the bicycle to travel further per rotation.

Correct responses:

Part A: Approximately 10.47 inches

Part B: 300°The d

Part C: Approximately 125.66 inches

Part D: The distance traveled increases by approximately 62.83 inches

<h3 /><h3>Methods used to calculate the above values</h3>

Given:

The radius of the large central gear, <em>r</em> = 4 inches

Radius of the small gear = 2 inches

Part A:

The distance travelled by the larger gear when the angle of rotation is 150° is given as follows;

\displaystyle Distance \ travelled \ by \ larger \ gear = \mathbf{\frac{150^{\circ}}{360^{\circ}} \times 2 \times \pi \times  4 \, inches} = \frac{10}{3} \cdot \pi \ inches

  • \displaystyle Distance \ traveled \ by \ the \ outer \ edge = \frac{10}{3} \cdot \pi \ inches  \approx \underline{ 10.47 \ inches}

Part B:

When the small gear travels the same linear distance as the large gear in part A, we have;

\displaystyle Degree \ of \ rotation = \frac{\frac{10}{3} \cdot \pi }{2 \times \pi \times  2 } \times 360^{\circ}= 300^{\circ}

  • The degree of rotation of the small gear = <u>300°</u>

Part C:

The distance travelled, <em>C</em>, by a single rotation of the large gear is given as follows;

C = 2 × π × 4 inches = 8·π inches

The degree of rotation of the smaller gear following one rotation of the large gear is therefore;

Degree  \ of \ rotation = \displaystyle \frac{8\cdot \pi \ inches}{4 \cdot \pi \ inches } \times 360^{\circ} = \mathbf{ 720^{\circ}}

720° = 2 × 360°

1 complete rotation is equivalent to 360°.

Therefore'

720° is equivalent to two complete rotation.

Therefore, the smaller gear and the wheel rotates twice for each rotation of the large gear

The distance the bicycle travels = 2 × The circumference of the wheel

Therefore;

Distance traveled by the bicycle = 2 × 2 × π × 10 inches = 40·π inches

  • Distance traveled by the bicycle = 40·π inches ≈ <u>125.66 inches</u>

Part D:

If the radius of the small gear is 1.5 inches and the radius of the large gear is 4.5 inches, we have;

Number of rotation of the small gear for each rotation of the large gear = 3 rotations

Therefore, number of rotation of the wheel = 3

Distance the bicycle travels = 3 × 2 × π × 10 inches = 60·π inches

The difference in distance traveled = 60·π inches - 40·π inches = 20·π inches

  • The distance traveled by the bicycle increases by 20·π inches ≈ <u>62.83 inches</u>

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8 0
2 years ago
I will like to know how to solve this
nikitadnepr [17]

1 + 4x = -5 + 7x

<u>   -4x  </u>    <u>    - 4x </u>

 1       = -5 + 3x

<u>+5     </u>    <u>+5        </u>

6        = 3x

\frac{6}{3} = \frac{3x}{3}

2 = x

Answer: x = 2

4 0
3 years ago
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