Answer:
A)
[tex]f'(x) = \left\{ \begin{array}{lIl} \frac{1}{2} & \quad x >2 \\ 0& \quad x =2\\1&\quad x<2 \end{array} \right.[/tex]
B) Continuous but not differentiable.
Step-by-step explanation:
So we have the piecewise function:
[tex]f(x) = \left\{ \begin{array}{lIl} \frac{1}{2}x+5 & \quad x >2 \\ 6& \quad x =2\\x +4&\quad x<2 \end{array} \right.[/tex]
A)
To write the differentiated piecewise function, let's differentiate each equation separately. Thus:
1)
[tex]\frac{d}{dx}[\frac{1}{2}x+5}][/tex]
Expand:
[tex]\frac{d}{dx}[\frac{1}{2}x]+\frac{d}{dx}[5][/tex]
The derivative of a linear equation is just the slope. The derivative of a constant is 0. Thus:
[tex]\frac{d}{dx}[\frac{1}{2}x+5}]=\frac{1}{2}[/tex]
2)
[tex]\frac{d}{dx}[6][/tex]
Again, the derivative of a constant is 0. Thus:
[tex]\frac{d}{dx}[6]=0[/tex]
3)
We have:
[tex]\frac{d}{dx}[x+4][/tex]
Expand:
[tex]\frac{d}{dx}[x]+\frac{d}{dx}[4][/tex]
Simplify:
[tex]=1[/tex]
Now, let's substitute our original equations for the differentiated equations. The inequalities will stay the same. Therefore:
[tex]f'(x) = \left\{ \begin{array}{lIl} \frac{1}{2} & \quad x >2 \\ 0& \quad x =2\\1&\quad x<2 \end{array} \right.[/tex]
B)
For a function to be differentiable at a point, the function must be a) continuous at that point, and b) the left and right hand derivatives must be equivalent.
Let's first determine if the function is continuous at the point. Remember that a function is continuous at a point if and only if:
[tex]\lim_{x \to n^-} f(n)= \lim_{x \to n^+}f(n)=f(n)[/tex]
Let's find the left hand limit of f(x) at it approaches 2.
[tex]\lim_{x \to 2^-}f(x)[/tex]
Since it's coming from the left, let's use the third equation:
[tex]\lim_{x \to 2^-}f(x)\\=\lim_{x \to 2^-}(x+4)[/tex]
Direct substitution:
[tex]=(2+4)=6[/tex]
So:
[tex]\lim_{x \to 2^-}f(x)=6[/tex]
Now, let's find the right-hand limit:
[tex]\lim_{x \to 2^+}f(x)[/tex]
Since we're coming from the right, let's use the first equation:
[tex]\lim_{x \to 2^+}(\frac{1}{2}x+5)[/tex]
Direct substitution:
[tex](\frac{1}{2}(2)+5)[/tex]
Multiply and add:
[tex]=6[/tex]
So, both the left and right hand limits are equivalent. Now, find the limit at x=2.
From the piecewise function, we can see that the value of f(2) is 6.
Therefore, the function is continuous at x=2.
Now, let's determine differentiability at x=2.
For a function to be differentiable at a point, both the right hand and left hand derivatives must be equivalent.
So, let's find the derivative of the function as x approahces 2 from the left and from the right.
From the differentiated piecewise function, we can see that as x approaches 2 from the left, the derivative is 1.
As x approaches 2 from the right, the derivative is 1/2.
Therefore, the right and left hand derivatives are not the same.
Thus, the function is continuous but not differentiable.
x+2/2×>1/2 in rational in equalities
Hi there!
~
[tex]x + \frac{2}{2} (\frac{1}{2} )[/tex]
= [tex]x + \frac{1}{2}[/tex]
Hope this helped you!
\dfrac{4}{9},-\dfrac{4}{3},4, 9 4 ,− 3 4 ,4,
Answer:
[tex]-\frac{8}{9}[/tex]
Step-by-step explanation:
[tex]\dfrac{4}{9}-\dfrac{4}{3}\\\\\mathrm{Least\:Common\:Multiplier\:of\:}9,\:3:\quad 9\\\\\mathrm{Adjust\:Fractions\:based\:on\:the\:LCM}\\\\\frac{4}{3}=\frac{4\times\:3}{3\times\:3}=\frac{12}{9}\\\\=\frac{4}{9}-\frac{12}{9}\\\\\mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}:\\\\\quad \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}\\\\=\frac{4-12}{9}\\\\=\frac{-8}{9}\\\\=-\frac{8}{9}[/tex]
Answer:
what?
Step-by-step explanation:
calculate the length of the base of a parallelogram if the area is 135cm and it's height is 9cm
Answer:
15cm
Step-by-step explanation:
The formula to calculate the area of a parallelogram is base x height.
Since we know the height and area, we can write the equation:
135 = base x 9
Now we can just divide 9 from both sides to solve for base:
135/9=basex9/9
So we get:
Base = 135/9=15
The length of the base is 15cm.
Hope this helped!
Answer:
lol
Step-by-step explanation:
2. haha
3. haha.5.jsj
please me asap :)) will give 40 points
i only need 4-101
x = -3
y = 5
x = -2
y = 0
x = -1
y = -3
x = 0
y = -4
x = 1
y = -4
x = 2
y = 0
x = 3
y = 5
I need help please no one helps me
Answer:
the answer is A
bro please help mee
Step-by-step explanation:
solve for x: -2(x+3)= -2 - 6
0
3
all real numbers
no solution
Answer:
0
Step-by-step explanation:
-2 is multiply by x and 3 then from left the number are replace in right then like term are simplify.
malik's solution to the equation [tex]\frac{2}{5}[/tex]×-4y=10
Answer:
10
Step-by-step explanation:
you put the answer on the board
Identify the sequence below as arithmetic, geometric, or neither. Then find the next two terms a5 and a6
10, 30, 90, 270,
Answer:
Geometric sequence of common ratio = 3
a5 would be: 810,
and a6 = 2430
Step-by-step explanation:
The sequence is a geometric sequence because each term is produced by multiplying the previous one by the factor 3 (called the common ratio)
then terms a5 and a6 would be: 810, and 2430 respectively
write a sentence as an equation
b is 380 more than the quantity y time 268
Answer:
b is → b =
380 more → + 380
y time 268 → 268y
combine
b = 268y + 380
solve for n
8+8+n-4=3n-2n
Answer:
No solution
Step-by-step explanation:
8+8-4 = 12
12+ n = 3n-2n
3n-2n
12+n3n-2n
subtract n on both sides
12=0n
This makes it no solution
The Cobb family’s natural gas bill was 85.50 for 95 cubic feet of gas, What is the unit rate? (Cost per cubic foot of gas?)
Answer:
I will not tell you
Step-by-step explanation:
Not telling
What part of 20 is 25?
Answer:
80%
Step-by-step explanation:
just look it up on google thyats what i did
Wyatt owes $472.22 on his credit card. He returns two items, one for $49.63 and the other for $47.58.
Then he makes purchases of $39.57 and $32.21. What is the new balance on the card?
Answer:
Holy bloop you gotta learn
Step-by-step explanation:
you dumb
go to school
and learn
What is the value of x & y?
Answer:
x = 16, y = 9
Step-by-step explanation:
The 2 triangles are similar by the AA postulate, thus the ratios of corresponding sides are equal, that is
[tex]\frac{x}{24}[/tex] = [tex]\frac{12}{18}[/tex] = [tex]\frac{2}{3}[/tex] ( cross- multiply )
3x = 48 ( divide both sides by 3 )
x = 16
----------------------------------------------------
[tex]\frac{y}{6}[/tex] = [tex]\frac{18}{12}[/tex] = [tex]\frac{3}{2}[/tex] ( cross- multiply )
2y = 18 ( divide both sides by 2 )
y = 9
Are these the correct answers? Why or why not?
Answer:
Yes, they are correct
Step-by-step explanation:
The width of the interval is 9 − 5 = 4. So the minimum possible value of the area is 4m, and the maximum possible value of the area is 4M.
How many years old are you have lived 1 billion second
One year Hank had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched) of any male pitcher at his school, with an ERA of
3.05. Also, Sally had the lowest ERA of any female pitcher at the school with an ERA of 2.96. For the males, the mean ERA was 3.848 and the standard deviation was
0.742. For the females, the mean ERA was 5.007 and the standard deviation was 0.722. Find their respective Z-scores. Which player had the better year relative to
their peers, Hank or Sally? (Note: In general, the lower the ERA, the better the pitcher.)
Tests
Hank had an ERA with a z-score of
Sally had an ERA with a z-score of
Answer:
Hank had an ERA with a z-score of 25.554
Step-by-step explanation:
Which term has a degree of 8? a.) 9xy^8 b.)-2x^3y^5 c.) 3x^2y^4 d.) 8xy
Answer:
A
Step-by-step explanation:
I honestly guessed,but i just took the test and got it right.
Answer:
-2x^3y^5
Step-by-step explanation:
Mr.rodriguez invests half money in land a tenth in stocks and a twentieth in bonds he puts the remaining 35,000 in his savings account what is the total amount of money that Mr Rodriguez saves and invests
Answer:
Step-by-step explanation:
21. Chandra is retiling her bathroom floor. Each tile is in the shape of a
square, and is 6.5 inches long. How many square inches does one tile
cover? *
WILL MARK AS BRAINLIEST
Answer:
42.25 square inches
Step-by-step explanation:
the formula for area of a square is length x width
a square has four equal sides
so the length and width of the square are 6.5
so you would multiply 6.5 by itself: 6.5x6.5=42.25
so the tile overs 42.25 sq inch
I need help with this ASAP
Answer:
x=5
Step-by-step explanation
Move all terms containing x to the left side of the equation.
4x-8=12
Move all terms not containing x to the right side of the equation.
4x=20
Divide each term by 4 and simplify.
x=5
Nidhi is creating a rectangular garden in her back yard. The length of the garden is 10 feet. The perimeter of the garden must be at least 40 feet and no more than 76 feet. Write and solve a compound inequality to find the range of values for the width, w, of the garden.
Answer:
10<=w<=33
Step-by-step explanation:
length is 10 so width can only add up to 20 less than those
40-20=20
20/2=10
76-20=66
66/2=33
p.s. <= is greater than or equal to oops
if a gymnasium is 80 feet long and 60 feet wide, what will the distance be between opposite corners
Answer:
140
Step-by-step explanation:
Lena has $12 she bought a pack of craft paper for six dollars and a stapler for two dollars which expression correctly shows the money Lena has left
Answer:
6+2=8 12-8=4 so she has $4 left
Step-by-step explanation:
Answer:
12-6-2=4
or
6+2=8, 12-8=4
70% of 120 = 120x70/100
How do you figure this out
find the mode in the set 3,5,7,13,7,9,3
Answer:
3 and 7
Step-by-step explanation:
The mode of a set is the number that appears the most frequently. There can be multiple modes in a set. Looking at the given set, we see that the numbers that show up the most are 3 and 7 so that is our answer.
Answer:
3 and 7
Step-by-step explanation:
The mode means the number that appears the most.
hope this helps you :)
Prove that root 7 is an irrational number.
Answer: Root 7 would equal 2.64575131106 and would go on forever.
Step-by-step explanation: It isn't one of the perfect squares. For example, the first 10 perfect square are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Answer:
it is an irrational number and here is why
Step-by-step explanation:
So think of square root as it's answer is something that you you multiply its self by its self and you get that answer. Ex: 8 x 8 = 64 square root of 64 is 8. So in that case there is nothing you can exactly multiply by itself to get 7 so in that case it's a decimal or an irrational number.
HOPE THIS HELPS!
Sam can prepare 30 wall paintings in four weeks.How much time will he need to do 90 wall painting s
(-2.8)^6 (Round to three decimal places as needed)
Answer:
481.890
Step-by-step explanation:
481.890304 rounded to the nearest hundredth will be 481.890
Hello what is 50+3 ? c h a t w i t h p e p s
Answer:
50+3=53
Step-by-step explanation:
mark as Brainliest
Answer:
53
Step-by-step explanation:
5+5+5+5+5+5+5+5+5+5+1+1+1