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
White raven [17]
4 years ago
12

Find the values of m and b that make the following function differentiable.

Mathematics
1 answer:
KengaRu [80]4 years ago
7 0

f(x)=\begin{cases}x^2&\text{for }x\le2\\mx+b&\text{for }x>2\end{cases}

In order to be differentiable everywhere, f must first be continuous everywhere, which means the limits from either side as x\to2 must be the same and equal to f(2). By definition, f(2)=2^2=4, and

\displaystyle\lim_{x\to2^-}f(x)=\lim_{x\to2}x^2=4

\displaystyle\lim_{x\to2^+}f(x)=\lim_{x\to2}(mx+b)=2m+b

so we need to have 4m+b=4.

For f to be differentiable at x=2, the derivative needs to be continuous at x=2, i.e.

\displaystyle\lim_{x\to2^-}f'(x)=\lim_{x\to2^+}f'(x)

We then need to have

\displaystyle\lim_{x\to2}2x=\lim_{x\to2}m\implies\boxed{m=4}

Then

2m+b=4\implies\boxed{b=-4}

You might be interested in
DVDs are on sale for $24 each Felipe write an expression to 4×24 to find the cost of buying the four DVDs how can you rewrite fu
professor190 [17]

Answer:

The cost of buying the four DVDs is $96.

Step-by-step explanation:

Given  that,

Price of DVD's = $24 each

The expression is

4\times24

We need to calculate the cost of buying the four DVDs

Using given data

Price of one DVD= $24

So, price\ of\ 4\ DVDs= 24\times4

price\ of\ 4\ DVDs=$96

Hence, The cost of buying the four DVDs is $96.

3 0
4 years ago
Create the f(x) function based on the table. Answer should be in function notation
irinina [24]

Answer:

It will be

f(x)=x+4

4 0
3 years ago
What does 4/7x35= plz tell me
horrorfan [7]
Basically divide by 7 times 4
35/7= 5
5 times 4= 20
5 0
3 years ago
Read 2 more answers
What is the equation of the line that passes through the point (4, -4) and has a<br> slope of –3/4?
Gelneren [198K]

Answer:

y=-3/4x-1

Step-by-step explanation:

y=mx+b; y is the y coordinate; m is the slope; x is the x coordinate; you have to find b. So you replace all the numbers out which is given to you;

-4=-3/4(4)+b (you multiply -3/4 by 4 and get -3)

-4=-3+b (you add 3 to both sides and get -1)

-1=b (you now found b so plug it into the equation)

y=-3/4x-1

5 0
3 years ago
If x/-2=18 then what is x equal to
Elodia [21]

Answer: X = -36

Step-by-step explanation:

X/-2 = 18 we need to get the variable alone, so we need to get rid of the /-2 to get x alone. And whatever we do to one side we have to do to the other so:

X/-2 = 18

X/-2 • (-2) = 18 • (-2)

The /-2 cancels out so we are left with

X = -36

7 0
3 years ago
Other questions:
  • Why does it matter if the exponent is in the parentheses?
    12·1 answer
  • What is the solution to the system of equations? -7x-2y+z=-2 <br> 6x-y-z=-18<br> 5x+4y-z=18
    13·1 answer
  • Write a real world problem involves classifying a quadrilateral then solve
    13·1 answer
  • Dina picked 1/2 of a pound of berries. She divided them evenly among 5 containers. How many pounds of berries did Dina put in ea
    14·2 answers
  • What is 50 + 9 - 6 equals
    13·2 answers
  • you are renting a moving truck for a day. There is a dailyfee of $20 and a charge of $0.75 per mile. Your budget allows a maximu
    12·1 answer
  • The first and second steps to solve the equation 3x/5+5=20 are
    10·2 answers
  • Jack rides his bicycle 4 miles in 30 minutes. If he maintains this rate, how far will he ride in 3 hours?
    11·1 answer
  • A rectangular book measures 4 x 7. What is the length of its
    11·1 answer
  • Compute the correlation coefficient for the data set.
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!