Answers
(1) f(x) = (1 - x³) / (x - 1)
(a) The domain is the set of values that this function can take on. If x = 1, the denominator becomes 0 and the function is undefined. Any other value of x is okay, though, since for x ≠ 1, we have
f(x) = (1 - x³) / (x - 1) = - (1 - x³) / (1 - x) = -(x² + x + 1)
which is defined for all x. This also tells us that the plot of f(x) is a parabola with a hole at x = 1. So, the domain is the interval (-∞, 1) ∪ (1, ∞).
(b) The range is the set of values that the function actually does take on. Taking the simplified version of f(x), we can complete the square to write
-(x² + x + 1) = -(x² + x + 1/4 - 1/4) - 1 = -(x + 1/2)² - 3/4
which is represented by a parabola that opens downward, with a maximum value of -3/4. So the range is the interval (-∞, -3/4).
(c) Judging by the plot of f, the limits at both negative and positive infinity are -∞.
(d) Same answer as part (a).
(2) f(x) = x³ - x
(a) The derivative of f at x = 3, and hence the slope of the tangent line to this point, is
[tex]f'(3)=\displaystyle\lim_{h\to0}\frac{f(3+h)-f(3)}h[/tex]
[tex]f'(3)\displaystyle=\lim_{h\to0}\frac{((3+h)^3-(3+h))-24}h[/tex]
[tex]f'(3)=\displaystyle\lim_{h\to0}\frac{(27+27h+9h^2+h^3)-(3+h)-24}h[/tex]
[tex]f'(3)=\displaystyle\lim_{h\to0}\frac{26h+9h^2+h^3}h[/tex]
[tex]f'(3)=\displaystyle\lim_{h\to0}(26+9h+h^2)=\boxed{26}[/tex]
(b) The tangent line at x = 3 has equation
y - f (3) = f ' (3) (x - 3)
y - 24 = 26 (x - 3)
y = 26 x - 54
We also want to find any other tangent lines parallel to this one, which requires finding all x for which f '(x) = 26. We could use the same limit definition as in part (a), but to save time, we exploit the power rule to get
f '(x) = 3 x² - 1
Then solve for when this is equal to 26:
3 x² - 1 = 26 ==> x² = 9 ==> x = ±3
The other tangent line occurs at x = -3, for which we have f (-3) = -24, and so the equation for the tangent is
y - f (-3) = 26 (x - (-3))
y + 24 = 26 (x + 3)
y = 26 x + 54
Related Questions
Which proportion can be used to show that the slope of JL is equal to the slope of MP? (sorry for the horrible quality)
Answers
Answer:
The proportion can be used to show that the slope of JL is equal to the slope of MP is [tex]\frac{0-4}{-4-(-7)}[/tex] = [tex]\frac{-4-8}{-1-(-10)}[/tex] ⇒ G
Step-by-step explanation:
The rule of the slope of a line is [tex]m=\frac{y2-y1}{x2-x1}[/tex] , where (x1, y1) and (x2, y2) are two points on the line
∵ The coordinates of the point J are (-7, 4)
∵ The coordinates of the point L are (-4, 0)
∴ x1 = -7 and y1 = 4
∴ x2 = -4 and y2 = 0
→ Substitute them in the rule above to find the slope of LJ
∴ [tex]m_{JL}=\frac{0-4}{-4-(-7)}[/tex]
∵ The coordinates of the point M are (-10, 8)
∵ The coordinates of the point P are (-1, -4)
∴ x1 = -10 and y1 = 8
∴ x2 = -1 and y2 = -4
→ Substitute them in the rule above to find the slope of MP
∴ [tex]m_{MP}=\frac{-4-8}{-1-(-10)}[/tex]
∵ The slope of JL = the slope of MP
∴ [tex]\frac{0-4}{-4-(-7)}[/tex] = [tex]\frac{-4-8}{-1-(-10)}[/tex]
The proportion can be used to show that the slope of JL is equal to the slope of MP is [tex]\frac{0-4}{-4-(-7)}[/tex] = [tex]\frac{-4-8}{-1-(-10)}[/tex]
The list shows numbers in order from least to greatest. Negative 15, negative 3 and four-fifths, negative 1.5, blank, 2.3, 5 and two-thirds Which is an integer that can be inserted on the blank line in the list? A. –2 B. Negative 1 and one-third C. 0 D. 1.2
Answers
Answer:
The correct option is option C
Step-by-step explanation:
Here we want an integer that can be inserted into the blank.
We have:
-15 , -3 4/5 , -1.5 , blank , 2.3 , 5 2/3
The integer that must be inserted into the blank should have a value between -1.5 and 2.3
What this means is that it should be greater than -1.5 but less than 2.3
Option A is incorrect since -2 itself is less than -1.5
option B is incorrect as it not an integer ( integer are whole numbers)
option C is correct
D. option D is incorrect since it is not an integer
Answer:
C
Step-by-step explanation:
A.Name a constant
B.Name the coefficients
Answers
Answer:
7 & 13 are constants
8 & 3 are coefficients
Step-by-step explanation:
One package of blackberries costs $3. How many packages of blackberries can you buy for $15?
Answers
Answer:
5 packages
Step-by-step explanation:
One package of blackberries cost $3
Let x represent the number of packages that will cost $15
1 = $3
x= $15
3x= 15
Divide both sides by the coefficient of x which is 3
3x= 15/3
x= 5
Hence 5 packages of blackberries will cost $15
uestion 1:
Damon wants to sell his motorcycle that he paid $4,000 for 3 years ago. The motorcycle depreciated (decreased in value) at a constant rate each month over a 3-year period. If x represents the monthly depreciation amount, write an expression that shows how much Damon can sell his motorcycle for today.
Answers
Answer:
4,000 -x3
Step-by-step explanation:
x times 3 is equal to the decrease so therefore you take the decrease away form 4/oo
Answer:
3y-x=4,000
Step-by-step explanation:
please help
Which values of k would cause this system of equations to have no solution? Check all that apply.
6x + 4y = 14,
3x + 2y = k
-2
5
7
10
21
Answers
Answer:
a , b , d , e
Step-by-step explanation:
56x + 40 = 48x + 104
Answers
Answer:
56x-48x=104-40
8x=24
X=3
Step-by-step explanation:
the answer is 8
]explanation subtract 56-48 and 104-40 you should get 8x=64
divide 8/8 and 64/8 and the answer is 8
translation: 4 units left and 4 units up
J(−1, −2), A(−1, 0), N(3, −3)
Answers
Answer:
J(-5,2), A(-5,4), and N(-1,1)
Step-by-step explanation:
what is the sale tax on a $14,500 truck if the tax rate is 9%
Answers
Answer:
$15, 805.00
Step-by-step explanation:
Tax is based on the state you live in.
write the number whose prime factorization is give: 2×2×2×7
Answers
Answer:
56
Step-by-step explanation:
Multiply it out 2×2×2×7 = 56
which numerical pattern in nonlinear?
A. 3, 11, 19, 27,
B. 1, 3, 9, 27
C. 1, 4, 7, 10,
D. 2, 3, 4, 5
Answers
Answer:
I am going with B.1,3,9,27
Step-by-step explanation:
A,C and D the patterns are from addition ie. A +8, C+3 and D +1 but B it's ×3
In the figure, <6 and <3 are
Answers
Answer:B
Step-by-step explanation
State whether the given pair of sets are equal, equivalent, both, or neither. {0,9}; {8, 1)
Answers
Answer:
Equivalent
Step-by-step explanation:
A set is a well defined collection of objects.
Two sets are said to be equal if they have the same elements.
Two sets are said to be equivalent if they have the same number of elements.
Given sets are [tex]\{0,9\}\,,\,\{8,1\}[/tex]
These sets are not equal as both the sets have different elements.
Order of both the sets is 2 (number of elements in both sets is 2)
So,
these two sets are equivalent.
Each picture shows how a mapping, f, maps elements of a domain onto a range.
Which mapping, f, is NOT a function?
Domain
Range
Domain
Range
-6
2.
5
6 00
4
7
7
1
9
8
Domain
Range
Domain
Range
5
4
2
→8
7
2
Answers
Answer:
Top left
Step-by-step explanation:
If an x value (domain) leads to more than one y-value (range), it is not a function. In other words, if each input value leads to only one output value, it is a function. It's still a function if the y-value leads to multiple x-values.
Find the perimeter of the figure if the following
is true:
a = x2
b = 4x + 8
c = 2x²
d = x
Answers
Which sequence can be defined by the recursive formula f (1) = 4, f (n + 1) = f (n) – 1.25 for n ≥ 1?
1, –0.25, –1.5, –2.75, –4, . . .
1, 2.25, 3.5, 4.75, 6, . . .
4, 2.75, 1.5, 0.25, –1, . . .
4, 5.25, 6.5, 7.75, 8, . . .
Answers
Answer:
4, 2.75, 1.5, 0.25, –1, . . .
Step-by-step explanation:
edge2020
James has $36.42 if he can only spend 1/6 of his money, how much money can he spend? 6 1
Answers
Answer:6.07
Step-by-step explanation:
Answer:
$6.07
Step-by-step explanation:
36 / 6 = 6
.42 / 6 = 7
6.07
bc we skip two decimal spaces to put it before the 0, there for the answer is $6.07!
Find the slope of the line
graphed below.
Answers
Answer:
[tex]\frac{3}{5}[/tex] or 0.6
Step-by-step explanation:
This problem requires the slop formula which is [tex]\frac{y2-y1}{x2-x1}[/tex]
You start with the first point which is (-1,1). This will be x1 and y1.
The next point is (4,4). This will be x2 and y2.
You plug these values into your equation which gives you [tex]\frac{4-1}{4-(-1)}[/tex]
To solve, you evaluate, [tex]\frac{4-1}{4-(-1)}[/tex] = [tex]\frac{3}{5}[/tex] or 0.6
What is the slope and y-intercept of the linear equation y = 5x − 4?
m =
b =
Answers
y = -4
the formula is y=mx+b so just look at what numbers are placed where and you should be able to understand next time :)
Answer:
I got it right on edge 2020/2021
Step-by-step explanation:
m=5
b=-4
How do I solve for X?
Answers
Answer
50
Step-by-step explanation:
60 is supplementary to 120 so 60+70=130
and a triangle adds up to 180 so 180-130 would be 50
LUCILLE'S PENCIL POUCH WOULD HOLD 3/8
OF THE 48 PENCILS THAT SHE PURCHASED AT
THE BEGINNING OF THE NEW SCHOOL YEAR. How many pencils will fit in her pouch
HOW MANY PENCILS WILL FIT IN HER POUCH?
Answers
Answer:
The pouch would be able to hold 12.8 pencils :)
Step-by-step explanation:
because 3/8 is 37.5 and if you do that divided by 48 and add a decimal infront of the eight you will get your answer
In 2001, a school population was 1696. By 2007 the population had grown to 2614.
1) How much did the population grow between the year 2001 and 2007?
students
2) How long did it take the population to grow from 1696 students to 2614 students?
years
3) What is the average population growth per year?
students/year
4) What was the population in the year 2000?
students
5) Find an equation for the population,
P
, of the school
t
years after 2000.
P
=
6) Using your equation, predict the population of the school in 2011.
students
Answers
Answer:
grew by 6
Step-by-step explanation:
In the figure, <6 and <2 are
Answers
Answer:
d
Step-by-step explanation:
consecutive angles aren't a thing and the others there would have to be an angle on the other side.
6 ≤ -3x + 12
can somone solve this for me *correctly* ill give u brainlest only if ur right
Answers
Answer:
x ≤ 2
Step-by-step explanation:
-3x+12≥6
-3x≥6
3x≤6
x ≤ 2
Answer:
x ≤ 2 is the correct answer!
Step-by-step explanation:
Hope this helps!
Jayden's family watched 8 hours of TV last week. This week, they watchec
25% more than that. How many hours of TV did they watch this week? *
Answers
Answer:
I THINK that it is 10 hours
Step-by-step explanation:
(Sorry if I am wrong)
25% = 1/4 SO 8 divided by 4 = 2
Since they watched 25% MORE you add 2 to 8 and get 10
Evaluate 12x−3y when x=−14 and y=3.
Answers
Answer:
-177
Step-by-step explanation:
12(-14) -3(3)
-168-9
-177
Answer:
-177
Step-by-step explanation:
Plug in -14 for x and 3 for y:
12(-14) - 3(3)
Note that when you multiply a positive and a negative number, your answer will be negative.
Multiply:
12 * -14 = -168
-3 * 3 = -9
Combine the terms:
-168 + (-9) = -168 - 9 = -177
-177 is your answer.
~
If a fuel efficient car can go 210 miles
on 4 gallons of fuel, how far can it go
on 12 gallons?
Answers
Answer:
The distance covered by the car in 12 gallons of fuel is 630 miles.
Step-by-step explanation:
Distance covered by car in 4 gallons of fuel = 210 miles.
Let us use unitary method to find distance covered by car in 1 gallon of fuel.
Distance covered by car in 1 gallon of fuel = 210/4 miles.
Let us use unitary method to find distance covered by car in 12 gallon of fuel.
Distance covered by car in 12 gallons of fuel = (210/4)*12 = 630 miles.
The function h(x)=1/x^2+1 is the result of the composition f(g(x)). If g(x) = x^2+1,what is f(x)? A f(x)=1/square root x B f(x)=1/x C f(x)= 1/x+1 D f(x)=1/x^2+1
Answers
Answer:
Option B is correct
Step-by-step explanation:
Given: [tex]h(x)=\frac{1}{x^2+1}[/tex] is the result of the composition [tex]f(g(x))[/tex].
Also, [tex]g(x)=x^2+1[/tex]
To find: [tex]f(x)[/tex]
Solution:
Take [tex]f(x)=\frac{1}{x}[/tex]
Now check whether [tex]h(x)[/tex] is equal to [tex]f(g(x))[/tex] or not.
First find [tex]f(g(x))[/tex]
[tex]f(g(x))=f(x^2+1)=\frac{1}{x^2+1}[/tex]
Also, [tex]h(x)=\frac{1}{x^2+1}[/tex]
Therefore,
[tex]h(x)=f(g(x))[/tex]
So,
Option B is correct
Answer: B. f(x)=1/x
Step-by-step explanation:
Edge
How to solve this without calculator?
sin-1(2/3)
Answers
Answer:-0.560
Step-by-step explanation:
I am even. The sum of my digits is sixteen. Who am I?
Answers
Answer:
2, 4, 6, 8, 10, 12, 14
Step-by-step explanation:
8+8=16
6+10=16
4+12=16
2+14=16