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
anzhelika [568]
3 years ago
14

Find an equation for the line tangent to the curve at the point defined by the given value of t.​ Also, find the value of d^2y /

dx^2 at this point.
x = 3t^2 + 7, y = t^6, t = -1
Mathematics
1 answer:
Ilya [14]3 years ago
5 0

The tangent line has slope \frac{\mathrm dy}{\mathrm dx}, which we can find using the chain rule:

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}

We have

x(t)=3t^2+7\implies\dfrac{\mathrm dx}{\mathrm dt}=6t

y(t)=t^6\implies\dfrac{\mathrm dy}{\mathrm dt}=6t^5

so that

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{6t^5}{6t}=t^4

When t=-1, the tangent line has slope 1.

To compute the second derivative \frac{\mathrm d^2y}{\mathrm dx^2}, first notice that the first derivative \frac{\mathrm dy}{\mathrm dx} turns out to be a function of t. Denote this first derivative by f(t). Then by the chain rule, we find

\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm df}{\mathrm dx}=\dfrac{\frac{\mathrm df}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}=\dfrac{4t^3}{6t}=\dfrac23t^2

and at t=1, we get a value of 2/3.

You might be interested in
PLEASE HELP ASAP!!! MARKING BRAINLIEST!!!
saul85 [17]

Answer:

got here just in time phew :3

Step-by-step explanation:

3 0
2 years ago
two right triangles are similar if an acute angle of one triangle is congruent to an acute angle of the other triangle. a. true
MArishka [77]
Answer:  True

One angle of a right triangle  = 90°
Let x =  the acute angle of one angle.
Let y =  the angle of the other angle.

Because the sum of angles in a triangle is 180°,
x + y + 90 = 180
x + y = 90
y = 90 - x

Therefore the angles for each of the two triangles are
x, 90° - x, 90°.

The two triangles are therefore similar because of AAA (Angle, Angle, Angle).
5 0
3 years ago
Read 2 more answers
Linda. owed 8 lawns this week and earned $56. how much would Linda have earned if she had mowed 10 lawns?
Drupady [299]
56 / 8 = 7
7 * 10 = 70

She would have earned $70

6 0
3 years ago
Your friend comes from a large family. This list shows the ages of all of the children in her family. What is the median age of
Lyrx [107]

Answer:

It is 11 and 4/7 I am pretty sure hope I helped!

5 0
2 years ago
21. Which number below is equivalent to 3.82 x 10-5?
IceJOKER [234]

Answer:

19.10

Step-by-step explanation:

If im correct,

You subtract 10 and 5. Wich is 5. Then multiply that by 3.82.

3 0
3 years ago
Other questions:
  • Find the value of y <br><br>thank you in advance for the help! ​
    13·1 answer
  • For the feasibility region shown below, find the maximum value of the function P=3x+2y.
    5·2 answers
  • I need help, please can someone help me ?
    15·1 answer
  • Help me pls now its late and i need to turn it in.
    15·2 answers
  • A specialty muffin company makes muffins for $0.12 per muffin. The starting cost of the company is $300. How many muffins must t
    7·2 answers
  • What is the perimeter of the shaded triangle?
    11·1 answer
  • Write the expression for when you translate the graph of y = 3x − 5 one unit to the right.
    12·1 answer
  • Find the slope of the line containing the points (7,5) and (9,5).
    12·2 answers
  • Vanessa has a mortgage of $92,000 at 5 1/2% for 20 years. Find the total interest
    9·2 answers
  • An object is traveling at a steady speed of 10 1/10 mi/h. How long will it take the object to travel 1 9/10 mi?
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!