Answers
(a) f(x) is continuous at x = 1 if the limits of f(x) from either side of x = 1 both exist and are equal:
[tex]\displaystyle \lim_{x\to1^-}f(x) = \lim_{x\to1} (2x-x^2) = 1[/tex]
[tex]\displaystyle \lim_{x\to1^+}f(x) = \lim_{x\to1} (x^2+kx+p) = 1 + k + p[/tex]
So we must have 1 + k + p = 1, or k + p = 0.
f(x) is differentiable at x = 1 if the derivative at x = 1 exists; in order for the derivative to exist, the following one-sided limits must also exist and be equal:
[tex]\displaystyle \lim_{x\to1^-}f'(x) = \lim_{x\to1^+}f'(x)[/tex]
Note that the derivative of each piece computed here only exists on the given open-ended domain - we don't know for sure that the derivative *does* exist at x = 1 just yet:
[tex]f(x) = \begin{cases}2x-x^2 & \text{for }x\le1 \\ x^2+kx+p & \text{for }x>1\end{cases} \implies f'(x) = \begin{cases}2 - 2x & \text{for }x < 1 \\ ? & \text{for }x = 1 \\ 2x + k & \text{for }x > 1 \end{cases}[/tex]
Compute the one-sided limits of f '(x) :
[tex]\displaystyle \lim_{x\to1^-}f'(x) = \lim_{x\to1} (2 - 2x) = 0[/tex]
[tex]\displaystyle \lim_{x\to1^+}f'(x) = \lim_{x\to1} (2x+k) = 2 + k[/tex]
So if f '(1) exists, we must have 2 + k = 0, or k = -2, which in turn means p = 2, and these values tell us that we have f '(1) = 0.
(b) Find the critical points of f(x), where its derivative vanishes. We know that f '(1) = 0. To assess whether this is a turning point of f(x), we check the sign of f '(x) to the left and right of x = 1.
• When e.g. x = 0, we have f '(0) = 2 - 2•0 = 2 > 0
• When e.g. x = 2, we have f '(2) = 2•2 - 2 = 2 > 0
The sign of f '(x) doesn't change as we pass over x = 1, so this critical point is not a turning point. However, since f '(x) is positive to the left and right of x = 1, this means that f(x) is increasing on (-∞, 1) and (1, ∞).
(c) The graph of f(x) has possible inflection points wherever f ''(x) = 0 or is non-existent. Differentiating f '(x), we get
[tex]f'(x) = \begin{cases}2-2x & \text{for }x<1 \\ 0 & \text{for }x=1 \\ 2x+k & \text{for }x>1\end{cases} \implies f''(x) = \begin{cases}- 2 & \text{for }x < 1 \\ ? & \text{for }x = 1 \\ 2 & \text{for }x > 1 \end{cases}[/tex]
Clearly f ''(x) ≠ 0 if x < 1 or if x > 1.
It is also impossible to choose a value of f ''(1) that makes f ''(x) continuous, or equivalently that makes f(x) twice-differentiable. In short, f ''(1) does not exist, so we have a single potential inflection point at x = 1.
From the above, we know that f ''(x) < 0 for x < 1, and f ''(x) > 0 for x > 1. This indicates a change in the concavity of f(x), which means x = 1 is the only inflection point.
Related Questions
You and eight friends plan a surprise party. Each of you contributes the same amount of
money m for food. Their mom also contributes $20.
a. Write an algebraic expression for the total amount of money contributed for food.
b. Evaluate your expression if each person contributed $10.50
Answers
Answer:
a. 9m + 8×$20 = $340
b. 9×$10.50 + 8×$10.50 = $178.50
Step-by-step explanation:
Its self-explanatory
in the sequence 5,11,19,29,41
find the 25th term
Answers
Answer:
25th term = 701
Step-by-step explanation:
see attached
have a great day :)
Help to solve for x by completing the square
Answers
here u go, if u need it clearer I will re write just tell me
Answer:10.4 or -0.4
Step-by-step explanation:
The explanation is in the attached file below
A bakery give me three options 12 cupcakes for $29 24 cupcakes for $56 50 cupcakes for $129 find each unit price to the nearest cent
Answers
a = 29 / 12
b = 56 / 24
c = 129 / 50
a = 2.42
b = 2.33
c = 2.58
b is cheapest, 24 for $56
Lesson 13 Exit Ticket 5.4
Name
Date
1. Solve. Draw a rectangular fraction model, and write a number sentence to show your thinking.
I do not know how to do this
Answers
A rectangular fraction model can be drawn based on this idea: Annie had a total of cookies 20 but she gave 9 cookies to her friends, how many cookies does she have?
What is a rectangular fraction model?A rectangular fraction model refers to using a rectangle and dividing it to represent and understand fractions. This might be easier than representing fractions by only using numbers.
How to draw a rectangular fraction model?The first step is to think about a possible situation or word problem. After this, you can draw the fraction model. For example:
Annie had a total of 20 cookies but she gave 9 cookies to her friends, how many cookies does she have?For this, divide a rectangle in the total of parts (20) and color the total of parts that Annie has (11). This means the total of cookies she has is 11/20.
Learn more about rectangles in: https://brainly.com/question/10046743
y=x+2
y+1= – 4x
Is this ordered pair a solution of the system of equations? Help:(
Answers
Put value of y in Equation 1 in Equation 2.
[tex] \looparrowright \sf y + 1 = - 4x[/tex]
[tex] \\ \\ [/tex]
[tex] \looparrowright \sf (x + 2) + 1 = - 4x[/tex]
[tex] \\ \\ [/tex]
[tex] \looparrowright \sf x + 2 + 1 = - 4x[/tex]
[tex] \\ \\ [/tex]
[tex] \looparrowright \sf x +3= - 4x[/tex]
[tex] \\ \\ [/tex]
[tex] \looparrowright \sf x + 4x= - 3[/tex]
[tex] \\ \\ [/tex]
[tex]\looparrowright \sf 5x= - 3[/tex]
[tex] \\ \\ [/tex]
[tex]\looparrowright \sf x= \dfrac{ - 3}{5} [/tex]
[tex] \\ \\ [/tex]
Let's find value of y :
[tex] \\ \\ [/tex]
[tex]\looparrowright \sf y=x+2[/tex]
[tex] \\ \\ [/tex]
[tex]\looparrowright \sf y= \dfrac{ - 3}{5} +2[/tex]
[tex] \\ \\ [/tex]
[tex]\looparrowright \sf y= \dfrac{ - 3 + 10}{5} [/tex]
[tex] \\ \\ [/tex]
[tex]\looparrowright \sf y= \dfrac{7}{5} [/tex]
Solution:
[tex]\looparrowright \sf \bigg( \dfrac{ - 3}{5} , \dfrac{7}{5} \bigg)[/tex]
What is the solution to the equation below?
4 + square root 6x + 4 = 8
Answers
[tex]4+ \sqrt{6x + 4 = 8} [/tex]
[tex] \sqrt{6x + 4 = 4}[/tex]
[tex]6x + 4 = 16[/tex]
[tex]6x = 12[/tex]
[tex]x = \frac{12}{6} = 2[/tex]
Option C is correct, x=2 is the solution to the equation 4+√(6x+4)=8
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is 4+√(6x+4)=8
We have to solve for x
Subtract 4 from both sides
√(6x+4)=4
Square on both sides
6x+4=16
Subtract 4 from both sides
6x=12
Divide both sides by 6
x=2
Hence, Option C is correct, x=2 is the solution to the equation 4+√(6x+4)=8
To learn more on Equation:
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#SPJ2
The population of a Midwest industrial town decreased from 213,000 to 209,000 in just 2years. Assuming that this trend continues, what will the population be after an additional 3 years?
Answers
Answer:
203,000
Step-by-step explanation:
You need to find the pattern first, so you need to subtract 213,000 from 209,000 to get 4,000. Now that is the pattern for 2 years we need it for 1 year. So divide 4,000÷2=2,000 for 1 year. Next, you need to multiply 2,000×3=6,000. That is the decrease of the population for 3 years. Finally, subtract 6,000 from 209,000.
Hope you get an A! <3
PLS ANSWER AS SOON AS POSSIBLE
Answers
Step-by-step explanation:
[tex]36 - {b}^{2} - {c}^{2} + 2bc[/tex]
Factor out -1
[tex] - ( - 36 + {b}^{2} + {c}^{2} - 2bc[/tex]
[tex] -( (b + c) { }^{2} - 36[/tex]
[tex] - (b + c + 6)( b + c - 6)[/tex]
Help help ASAP please
Answers
Answer:
y = -4x + 15
Step-by-step explanation:
slope of y = x/4 - 2 is 1/4
slope of line perpendicular to above line is -4
so y = -4x + b
use (4, -1)
-1 = -4(4) + b
b = 16 - 1 = 15
so
y = -4x + 15
90 kids in the band 20% of the kids own their own instruments how many kids own their own instruments
Answers
Answer:
18
Step-by-step explanation:
20% = .20
90 x .20=18
Hence, 18 kids own their own instruments
[RevyBreeze]
The equation of line p is y=−4x+1. Line q is perpendicular to line p and passes through (−4, 1). What is the equation of line q?
Write the equation in slope-intercept form with no spaces. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.
Answers
Answer:
[tex]\displaystyle y = \frac{1}{4}\, x + 2[/tex].
Step-by-step explanation:
The question is asking for the slope-intercept equation of a line. The slope-intercept form equation of a line is in the form [tex]y = m\, x + b[/tex], where:
[tex]m[/tex] (a constant) is the slope of the line, and[tex]b[/tex] (also a constant) is the [tex]y[/tex]-intercept of that line.For example, the equation of the first line, [tex]y = -4\, x + 1[/tex], is given in slope-intercept form. The slope of that line is [tex]m_{1} = (-4)[/tex] while the [tex]y[/tex]-intercept of that line is [tex]b_{1} = 1[/tex].
Both [tex]m[/tex] and [tex]b[/tex] need to be found for the second line.
Start by finding the slope [tex]m_{2}[/tex] of the second line.
Make use of the fact that if two lines in a plane are perpendicular to one another, the product of their slopes would be [tex](-1)[/tex]. In other words, if a line with slope [tex]m_{1}[/tex] is perpendicular to a line with slope [tex]m_{2}[/tex], then [tex]m_{1}\, m_{2} = (-1)[/tex]. Rearrange to obtain [tex]m_{2} = (-1) / m_{1}[/tex].
In this question, since the slope of the first line is [tex]m_{1} = (-4)[/tex], the slope of the second line (perpendicular to the first line) would be [tex]m_{2} = (-1) / (-4) = (1/4)[/tex].
Thus, the equation of the second line would be of the form [tex]\displaystyle y = (1/4)\, x + b_{2}[/tex] for some constant [tex]b_{2}[/tex] that needs to be found.
Make use of the fact that the second line goes through the point [tex](-4,\, 1)[/tex] to find the value of [tex]b_{2}[/tex]. Since the point [tex](-4,\, 1)\![/tex] is on the second line, the equation of the second line should hold after substituting in [tex]x = (-4)[/tex] and [tex]y = 1[/tex]. That is:
[tex]1 = (1/4)\times (-4) + b_{2}[/tex].
Solve this equation for [tex]b_{2}[/tex]:
[tex]b_{2} = 2[/tex].
Therefore, the equation of the second line would be [tex]y = (1/4)\, x + 2[/tex].
You roll a six sided die. Find the probability of 1) event A rolling a 3.
2) event B rolling a 7 3) event C rolling a number less than 5.
Answers
Answer:
total sum (s)= 6
probability of getting a 3 = 3÷6
= 1÷2
Step-by-step explanation:
probability of getting a 7 =0÷7
=0
probability of getting less than 5 =4÷6
2÷3
Find the slope of each line points (3,4) and (5,4)
Answers
Step-by-step explanation:
Slope formula:
[tex] \frac{y2}{x2} - \frac{y1}{x1} = m[/tex]
m = slope
Substitute the points into the formula format:
[tex] \frac{4}{5} - \frac{4}{3} = m[/tex]
Solve:
[tex] \frac{0}{2} = m[/tex]
The slope is 0.
The line doesn't increase or decrease.
Two pools are being filled with water. To start, the first pool contains 2164 liters of water and the second pool contains 2500 liters of water. Water is being added to the first pool at a rate of 31.75 liters per minute. Water is being added to the second pool at a rate of 21.25 liters per minute.
After how many minutes will the two pools have the same amount of water?
How much water will be in each pool when they have the same amount?
Answers
The amount of water in each pool at a particular time are illustrations of linear functions
Both pools will have the same amount of water after 32 minutesBoth pools will have 3180 liters at 32 minutesHow to determine the time they have the same amountFor the first pool, we have the following parameters
Initial amount = 2164 litersRate = 31.75 liters per minuteSo, the equation of the amount of water in the pool is:
y = 2164 + 31.75x
For the second pool, we have the following parameters
Initial amount = 2500 litersRate = 21.25 liters per minuteSo, the equation of the amount of water in the pool is:
y = 2500 + 21.25x
When both pools have the same amount of water, we have:
2164 + 31.75x = 2500 + 21.25x
Collect like terms
31.75x - 21.25x = 2500 - 2164
Evaluate the like terms
10.5x = 336
Solve for x
x = 32
This means that they will have the same amount of water after 32 minutes
Substitute 32 for x in y = 2164 + 31.75x
This gives
y = 2164 + 31.75 * 32
y = 3180
Both pools will have 3180 liters at 32 minutes
Read more about linear functions at:
https://brainly.com/question/15602982
x/5+7=3
tellme fastplease
Answers
Answer:
-20
Step-by-step explanation:
x/5=3-7
x/5=-4
x=-4×5
x=-20
What is 3/4 - 2/6
Subtracting Proper Fractions
Answers
Determine the cost of flooring a home office with carpet tiles, ceramic tiles, and laminate tiles. (Just flooring-not labor,tools,equipment,etc). Show all calculations. Include units with final and answers in space provided. Round to the nearest tenth if needed.
a) Carpet price $46.44 per box of 16 carpet tiles. Each tile 18"x 18"
Sold by the box only
b) Ceramic tile $2.98 for each individual tile. Each tile 12"x"12
Sold by the individual tile
c) Laminate cost $34.88 for a box that covers 17.5 square feet.
Sold by the box only
Question- 1.) How much flooring will be needed for the new office?
2.) Calculate the cost of flooring the new office with carpet tiles.
3.)Calculate the cost of flooring the new office with ceramic tiles
4.) Calculate the cost of flooring the new office with laminate flooring
Show all Work
(Please Help Thank you so much)
Answers
The radius of a semicircle is 1 kilometer. What is the semicircle's area?
Answers
Answer:
1.6
Step-by-step explanation:
area of a semicircle is A = pi * r^2/2
So pi*1^2/2
3.14/2
1.57079632679
Rounded to the nearest tenth
1.6
pls can u gus tell me the ans and explanation
Answers
Step-by-step explanation:
the volume of the tank is
80×50×45 = 180,000 cm³
3/4 of that volume is
180000 × 3/4 = 135,000 cm³
so, we want to know how long it takes to fill 135,000 cm³.
the water intake flow is 9 l / minute.
now we need to know that a cube of 10 cm side length contains exactly 1 liter of water.
such a cube has the volume
10×10×10 = 10³ = 1000 cm³
so, 1000 cm³ = 1 liter.
and we need to fill 135,000 cm³.
how many times fit 1000 cm³ into 135,000 cm³ ?
135000/1000 = 135 times.
and that means it would take 135 minutes to fill it, if the flow was 1 l / minute.
but it is faster (9 times faster) : 9 l / minute
so, how many minutes ?
135 / 9 = 15 minutes
it takes 15 minutes to fill 3/4 of the tank.
Answer:
15 minutes
Step-by-step explanation:
The volume (V) of the tank is calculated as
V = lbh (l is the length, b the breadth and h the height (
here l = 80, b = 50, h = 45 , then
V = 80 × 50 × 45 = 180000 cm³
1 litre = 1000 cm³ , then
9 litres = 9 × 1000 = 9000 cm³
Divide the volume of the tank by 9000 for time in minutes
180000 ÷ 9000 = 20 minutes
It takes [tex]\frac{3}{4}[/tex] × 20 = 15 minutes to fill [tex]\frac{3}{4}[/tex] of the tank
12. Find one solution for each equation. Assume all angles are acute angles
a. cos(0 + 4°) = sin(30 +2°)
Answers
Answer:
21 degrees
Step-by-step explanation:
I'm assuming that the 0 is supposed to be theta, which I'm replacing with x to make it easier for me to type.
Because these angles are acute, we can use the fact that if two angles A and B are complementary, then sinA = cosB.
So we can get that x+4+3x+2=90
4x+6=90
4x=84
x=21
A hotel room cost $79 per night. Meg's family is getting 2 rooms for 5 nights for their trip.
What is the total hotel bill, without tax and fees?
A. 790
B. 880
C. 1056
D. 1120
Answers
Shaunda said that buying 4 towels for $17 was a better buy than buying 2 towels for $9. She found her answer by doubling the terms in the ratio 9/2 and comparing the first terms in the ratio. Is she correct? Use unit prices to support your answer
Answers
Answer:
The cost for a towel at 4 for $17 is:
17/4 = 4.25 $/towel
The cost for a towel at 2 for $9 is:
9/2 = 4.5 $/towel
So Shaunda is correct, as the cost per towel for 4 at $17 is lower.
Use the number lines below to fill in an equivalent fraction.
1/4
Answers
Explanation:
Math…
1/4 is also equal to:
1/4 = 2/8
1/4 = 4/16
1/4 = 8/32
1/4 = 16/64
1/4 = 32/128
1/4 = 64/256
1/4 = 128/512
1/4 = 256/1024
Hope this helps!
Find the value of xif 3(x + 5 ) =-6
Answers
Answer:
-7
Step-by-step explanation:
A equation is given to us and we need to find out the value of x . The given equation is ,
[tex]\longrightarrow 3( x +5)=-6 [/tex]
Simplify the parenthesis ,
[tex]\longrightarrow 3x + 3(5)=-6[/tex]
Simplify,
[tex]\longrightarrow 3x +15 = -6[/tex]
Subtracting 15 to both sides ,
[tex]\longrightarrow 3x =-6-15[/tex]
Simplify,
[tex]\longrightarrow 3x = -21 [/tex]
Divide both sides by 3,
[tex]\longrightarrow x = -7 [/tex]
This is the required answer!
If ∠JAM and ∠JAK are supplementary angles, then what is the measure of ∠KAL?
A.
60°
B.
33°
C.
55°
D.
57°
Answers
Answer:
D) 57
Step-by-step explanation:
180 - 123
AABC and ADEF are similar. Find the missing side length.
Answers
Answer:
/\ABC------/\DEF
the ratio of the the corresponding sites must be same
AB/AC=DE/DF
AB/30=4/5
AB=(4/5)30
AB=24
HOPE IT HELPED.
PLEASE MARK BRAINLIEST
answer this please? :)
Answers
Answer:
yes
Step-by-step explanation:
see the attachement
Astronomers measure large distances in light-years. One light-year is the distance that light
can travel in one year, or approximately 5.88x10^12 miles. Suppose a star is 2.23x10^2 light-
years from Earth. In scientific notation, approximately how many miles is it?
A. 2.23x10^14
B. 1.31x10^14
C. 1.31X10^15
D. 5.88x10^12
Answers
Answer:
it's C
Step-by-step explanation:
Astronomers measure large distances in light-years. One light-year is the distance that light can travel in one year, or approximately 5.88\times 10^{12}5.88×1012 miles. Suppose a star is 2.23\times 10^22.23×102 light years from Earth.
To find : In scientific notation, how many miles away a star is from earth?
Solution :
One light-year is the distance that light can travel in one year is 5.88\times 10^{12}5.88×1012 miles.
In 2.23\times 10^22.23×102 lights year the distance is
=5.88\times 10^{12}\times 2.23\times 10^2=5.88×1012×2.23×102
=1.31124\times10^{15}=1.31124×1015
Therefore, Option C is correct.
Solve for x.
−4/5x+3/10<8/10
Answers
Answer:
x > -5/8
Step-by-step explanation:
Simplify by combining x and 4/5 and then moving 4 to the left side of x
-x * 4/5 + 3/10 < 8/10
-4x/5 + 3/10 < 8/10
Now we cancel the common factor of 8 and 10.
Factor 2 out of 8
-4x/5 + 3/10 < 2(4)/10
2 from 10
-4x/5 + 3/10 < 4/5
Now move all the terms not containing x to the right side
Lets subtract 3/10 from both sides
-4x/5 < 4/5 - 3/10
Now we multiply by 2/2 to write 4/5 with a common denomi.
-4x/5 < 4/5 * 2/2 - 3/10
Now write with a common denom of 10 and multiply by 1
-4x/5 < 4*2/5 * 2 - 3/10
5 by 2
-4x/5 < 4 * 2/10 - 3/10
Combine
-4x/5 < 4 * 2 - 3/10
Simplify the numerator by multiplying then subtracting
-4x/5 < 8 - 3/10
-4x/5 < 5/10
Cancel the common factor of 5 and 10...
-4x/5 < 5(1)/10
-4x/5 5* 1/5 * 2
-4x/5 < 1/2
Now we divide by -1
-4x/5)/-1 > 1/2)/-1
4x/5 > 1/2)-1
4x/5 > -1/2
Multiply both sides by 5 and cancel common factors. (5)
4x * 5 > -1/2 * 5
4x > -1/2 * 5
4x > -5/2
Now divide by 4 in each term
4x/4 > -5/2)/4
x > -5/2)/4
Multiply the numer by the reciprocal of the denom
x > -5/1 * 1/4
x > -5/4 * 2
x > -5/8