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
viva [34]
3 years ago
14

The graphs of the quadratic functions f(x) = 6 – 10x2 and g(x) = 8 – (x – 2)2 are provided below. Observe there are TWO lines si

multaneously tangent to both graphs.(a) The line simultaneously tangent to both graphs having the LARGEST slope has equation: (Two decimal places of accuracy.)y = ___ x + ___(b) The other line simultaneously tangent to both graphs has equation:(Two decimal places of accuracy.)y = ___ x + ___
Mathematics
1 answer:
natta225 [31]3 years ago
6 0

Answer:

a) y = 7.74*x + 7.5

b)  y = 1.148*x + 6.036

Step-by-step explanation:

Given:

                                  f(x) = 6 - 10*x^2

                                  g(x) = 8 - (x-2)^2

Find:

(a) The line simultaneously tangent to both graphs having the LARGEST slope has equation

(b) The other line simultaneously tangent to both graphs has equation,

Solution:

- Find the derivatives of the two functions given:

                                f'(x) = -20*x

                                g'(x) = -2*(x-2)

- Since, the derivative of both function depends on the x coordinate. We will choose a point x_o which is common for both the functions f(x) and g(x). Point: ( x_o , g(x_o)) Hence,

                                g'(x_o) = -2*(x_o -2)

- Now compute the gradient of a line tangent to both graphs at point (x_o , g(x_o) ) on g(x) graph and point ( x , f(x) ) on function f(x):

                                m = (g(x_o) - f(x)) / (x_o - x)

                                m = (8 - (x_o-2)^2 - 6 + 10*x^2) / (x_o - x)

                                m = (8 - (x_o^2 - 4*x_o + 4) - 6 + 10*x^2)/(x_o - x)

                                m = ( 8 - x_o^2 + 4*x_o -4 -6 +10*x^2) /(x_o - x)

                                m = ( -2 - x_o^2 + 4*x_o + 10*x^2) /(x_o - x)

- Now the gradient of the line computed from a point on each graph m must be equal to the derivatives computed earlier for each function:

                                m = f'(x) = g'(x_o)

- We will develop the first expression:

                                m = f'(x)

                                ( -2 - x_o^2 + 4*x_o + 10*x^2) /(x_o - x) = -20*x

Eq 1.                          (-2 - x_o^2 + 4*x_o + 10*x^2) = -20*x*x_o + 20*x^2

And,

                              m = g'(x_o)

                              ( -2 - x_o^2 + 4*x_o + 10*x^2) /(x_o - x) = -20*x

                              -2 - x_o^2 + 4*x_o + 10*x^2 = -2(x_o - 2)(x_o - x)

Eq 2                       -2 - x_o^2 + 4*x_o+ 10*x^2 = -2(x_o^2 - x_o*(x + 2) + 2*x)

- Now subtract the two equations (Eq 1 - Eq 2):

                              -20*x*x_o + 20*x^2 + 2*x_o^2 - 2*x_o*(x + 2) + 4*x = 0

                              -22*x*x_o + 20*x^2 + 2*x_o^2 - 4*x_o + 4*x = 0

- Form factors:       20*x^2 - 20*x*x_o - 2*x*x_o + 2*x_o^2 - 4*x_o + 4*x = 0

                              20*x*(x - x_o) - 2*x_o*(x - x_o) + 4*(x - x_o) = 0

                               (x - x_o)(20*x - 2*x_o + 4) = 0  

                               x = x_o   ,     x_o = 10x + 2    

- For x_o = 10x + 2  ,

                               (g(10*x + 2) - f(x))/(10*x + 2 - x) = -20*x

                                (8 - 100*x^2 - 6 + 10*x^2)/(9*x + 2) = -20*x

                                (-90*x^2 + 2) = -180*x^2 - 40*x

                                90*x^2 + 40*x + 2 = 0  

- Solve the quadratic equation above:

                                 x = -0.0574, -0.387      

- Largest slope is at x = -0.387 where equation of line is:

                                  y - 4.502 = -20*(-0.387)*(x + 0.387)

                                  y = 7.74*x + 7.5          

- Other tangent line:

                                  y - 5.97 = 1.148*(x + 0.0574)

                                  y = 1.148*x + 6.036

You might be interested in
Fill in the bubble for the correct answer choice.
dalvyx [7]
She arrived to school at 8:08
8 0
3 years ago
PLZ HELP!!!! ISSA PROBLEM!!!!! WILL GIVE BRAINLIEST!!!!!
UNO [17]
The answer would be
130 degrees

4 0
3 years ago
Read 2 more answers
Find the greatest common factor of 30 and 75.
yarga [219]

Answer:

15

Step-by-step explanation:

prime factorization of 30 is 2*3*5

prime factorization of 75 is 3*5*5

3*5 is common

8 0
3 years ago
What is the square root of 3
grandymaker [24]
There isn't an exact value for the square root of 3. It is roughly 1.732
4 0
3 years ago
Hello can someone help
exis [7]

Answer:

$149.44

Step-by-step explanation:

If each orange cost $0.64, you first multiply 0.64 and 154 and you get $98.56. You then subtract that number from $248 and you get your answer.

7 0
2 years ago
Read 2 more answers
Other questions:
  • Writing work/road work
    11·1 answer
  • Help my sister step by step show work so she can understand because I forgot how to do this stuff
    10·1 answer
  • Please answer this correctly without making mistakes
    5·1 answer
  • Help me pls offering good points
    10·1 answer
  • 50% of Brady's Halloween candy has been eaten. He has 68 pieces left. How much did he have​
    7·2 answers
  • Jill has 64 ounces of lemonade. Each day, her brother drinks 16 ounces. Then at the end of each day Jill adds 25% of the remaini
    15·2 answers
  • What is the answer to 22 +10X=
    10·1 answer
  • What degree is an obtuse angle<br>​
    10·2 answers
  • At the beginning of the school year, there
    10·2 answers
  • 25 3/2 in radical form
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!