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Neko [114]
3 years ago
7

Help asap!!!!!!!!!!!!!!!!!!!!!!! 20 points!1

Mathematics
1 answer:
Darya [45]3 years ago
8 0

Answer:

24

Step-by-step explanation:

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Use differentiation rules to find the values of a and b that make the function f(x) = ( x 2 if x ≤ 2, ax3 + bx if x > 2 diffe
Maurinko [17]

Answer:

The values of a and b are  a=\frac{1}{4} and b = 1

Step-by-step explanation:

* <em>Lets explain how to solve the equation</em>

  f(x) =  {x²                x  ≤ 2

            {ax³ + bx     x > 2  

* We need to find the values of a , b that make the function

 differentiable at x = 2

- <em>At first for f(x) to be continuous at x = 2, substitute x by two in the</em>

<em>  the two expressions and equate them</em>

∵ f(x) = x² at x ≤ 2 and f(x) = ax³ + bx at x > 2

∴ f(2) = (2)² = 4 ⇒ (1)

∴ f(2) = a(2)³ + b(2)

∴ f(2) = 8a + 2b ⇒ (2)

- Equate (1) and (2)

∴ 8a + 2b = 4 ⇒ (3)

* <em>For f(x) to be differentiable when x = 2, the function must be </em>

<em>  continuous when x = 2 and the one-sided derivatives must be </em>

<em>  equal when x = 2</em>

# <u>Remember</u>: If f(x)=ax^{b} , then f'(x)=abx^{b-1}

 If f(x)=ax , then f'(x)=a

 If f(x)=a , then f'(x)=0

∵ f(x) = x²

∴ f'(x) = 2x

- Substitute x by 2

∴ f'(2) = 2(2) = 4

∴ f'(2) = 4 ⇒ (4)

∵ f(x) = ax³ + bx

∴ f'(x) = 3ax² + b

- substitute x by 2

∴ f'(2) = 3a(2)² + b

∴ f'(2) = 12a + b ⇒ (5)

- Equate (4) and (5)

∴ 12a + b = 4 ⇒ (6)

* Now we have system of equations

 8a + 2b = 4 ⇒ (3)

 12a + b = 4 ⇒ (6)

- Multiply equation (6) by -2 to eliminate b

∴ -24 a - 2b = -8 ⇒ (7)

- Add equations (3) and (7)

∴ -16a = -4

- Divide both sides by -16

∴ a = \frac{1}{4}

- substitute the value of a in equation (6)

∴ 12(\frac{1}{4})+b=4

∴ 3 + b = 4

- Subtract 3 from both sides

∴ b = 1

* The values of a and b are  a=\frac{1}{4} and b = 1

5 0
3 years ago
How do I find the function rule???<br> x: 0, 1, 2, 3<br> y: 5, 4, 3, 2
Makovka662 [10]

Answer: y=-1x+5

Step-by-step explanation:

1. Use slope-intercept formula (y = mx + b) to find the slope which is represented by the variable 'b'.

m = ∆y / ∆x

m = -1 / 1

m = -1

y= -1x + b

2. Use any cooresponding x and y values to substitute into the formula. Ex: (x,y) → (1,4)

y = mx + b

4 = -1(1) + b

4 = -1 + b

5 = b

3. Plug in the values found into the formula.

y = -1x + 5

8 0
2 years ago
Give an example of a rational function that has a horizontal asymptote of y = 2/9..
Ann [662]
The horizontal asymptote of a rational function tells us the limiting value of that function as it approaches infinity.

For a rational function to have a horizontal asymptote of
\frac{2}{9}
then,the highest degree of x in the numerator must be equal to the highest degree of x in the denominator.

The second condition is that,
the coefficient of the highest degree in the numerator and the coefficient of the highest degree in the denominator should be in the ratio 2:9.

Example are given in the graph above.

Here are some other examples,

y = \frac{2x - 1}{9x + 2}


y = \frac{1 - 6 {x}^{3} }{5x - 27 {x}^{3} }

7 0
3 years ago
Read 2 more answers
Which of the choices best describes this arithmetic sequence?<br> -3, 1, 5, 9, 13,...
zloy xaker [14]

Answer:

Add by 4

Step-by-step explanation:

-3+4=1

1+4=5

5+4=9

9+4=13

3 0
2 years ago
Which equation defines the graph of y=x^2 after it is shifted vertically 3 units down and horizontally 5 units left? A) y=(x-5)^
jeka94
The answers is B because you set x+5=0 and x=-5 so that means that the graph needs to be shifted 5 units to the left because 5 is negative and to move 3 units down you just put a -3 at the end of the equation like this y=(x+5)-3
3 0
3 years ago
Read 2 more answers
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