Answers
Answer:
Part A)
[tex]f(1)=2, \; f^{-1}(1)=0, \; f^\prime(1)=1.4, \; (f^{-1})^\prime(1)=\frac{10}{7}[/tex]
Part B)
[tex]y=\frac{5}{14}x+\frac{4}{7}[/tex]
Step-by-step explanation:
Please refer to the table of values.
Part A)
A. 1)
We want to find f(1).
According to the table, when x=1, f(x)=2.
Hence, f(1)=2.
A. 2)
We want to find f⁻¹(1).
Notice that when x=0, f(x)=1.
So, f(0)=1.
Then by definition of inverses, f⁻¹(1)=0.
A. 3)
We want to find f’(1).
According to the table, when x=1, f’(x)=1.4.
Hence, f’(1)=1.4.
A. 4)
We will need to do some calculus.
Let g(x) equal to f⁻¹(x). Then by the definition of inverses:
[tex]f(g(x))=x[/tex]
Take the derivative of both sides with respect to x. On the left, this will require the chain rule. Therefore:
[tex]f^\prime(g(x))\cdot g^\prime(x)=1[/tex]
Solve for g’(x):
[tex]g^\prime(x)=\frac{1}{f^\prime(g(x))}[/tex]
Substituting back f⁻¹(x) for g(x) yields:
[tex](f^{-1})^\prime(x)=\frac{1}{f^\prime({f^{-1}(x)})}[/tex]
Therefore:
[tex](f^{-1})^\prime(1)=\frac{1}{f^\prime({f^{-1}(1)})}[/tex]
We already determined previously that f⁻¹(1) is 0. Therefore:
[tex](f^{-1})^\prime(1)=\frac{1}{f^\prime(0)}[/tex]
According to the table, f’(0) is 0.7. So:
[tex](f^{-1})^\prime(1)=\frac{1}{0.7}=\frac{10}{7}[/tex]
Hence, (f⁻¹)’(1)=10/7.
Part B)
We want to find the equation of the tangent line of y=f⁻¹(x) at x=4.
First, let’s determine the points. Since f(2)=4, this means that f⁻¹(4)=2.
Hence, our point is (4, 2).
We will now need to find our slope. This will be the derivative at x=4. Therefore:
[tex](f^{-1})^\prime(4)=\frac{1}{f^\prime({f^{-1}(4)})}[/tex]
We know that f⁻¹(4)=2. So:
[tex](f^{-1})^\prime(4)=\frac{1}{f^\prime(2)}[/tex]
Evaluate:
[tex](f^{-1})^\prime(4)=\frac{1}{f^\prime(2)}=\frac{1}{2.8}=\frac{10}{28}=\frac{5}{14}[/tex]
Now, we can use the point slope form. Our point is (4, 2) and our slope at that point is 5/14.
So:
[tex]y-2=\frac{5}{14}(x-4)[/tex]
Solve for y:
[tex]y-2=\frac{5}{14}x-\frac{20}{14}[/tex]
Adding 2 to both sides yields:
[tex]y=\frac{5}{14}x-\frac{20}{14}+\frac{28}{14}[/tex]
Hence, our equation is:
[tex]y=\frac{5}{14}x+\frac{4}{7}[/tex]
Related Questions
Which statement is true about the expression?
√18 + √2
A. It is rational and equal to 4 times the square root of 2.
B. It is irrational and equal to 4 times the square root of 2.
C. It is rational and equal to 3 times the square root of 2.
D. It is irrational and equal to 3 times the square root of 2.
Answers
Answer:
Your number is (3 sqrt(2)) / sqrt(2) = 3, and is a rational number indeed. I don't know exactly how to interpret the rest of the question. If r is a positive rational number and p is some positive real number, then sqrt(r^2 p) / sqrt(p) is always rational, being equal r. Possibly your question refers to situtions in which sqrt(c) is not uniquely determined, as for c negative real number or complex non-real number. In those situations a discussion is necessary. Also, in general expressions the discussion is necesary, because the denominator must be different from 0, and so on.
Step-by-step explanation:
Answer:
it is rational and equal to 4/2
i took the test. I hope i recieve brainliest. glad i could help you today
Six times a number plus twice a second number is four. Twice the first number plus four times the second number is 18. Find the numbers.
Answers
Answer:
Step-by-step explanation:
equation 1. 6a + 2b = 14 ( times this equation by -2)
equation 2. 2a + 4b = 18
equation 1 becomes -12a - 4b = -28
equation 2. remains. 2a + 4b = 18
Cancel out the b’s
-10 a = -10 (Divide both sides by -10)
a = 1
substitute 1 for a to find b
i think your numbers in the problem you gave is not accurate though
please check it again
b = 4
8/15 divided by ( -0.35) hurry pls it’s for a test
Answers
Answer:
-1.52380952381
Step-by-step explanation:
Answer:
-1.5238...
Step-by-step explanation:
I just did it on my calculator. Hope it helps!
Nicole is trying to make extra money on her summer vacation from school. She works two different jobs, earning $19 per hour lifeguarding and $8 per hour landscaping, How much money would Nicole earn in a week in which she worked 2 hours lifeguarding and 4 hours landscaping? How much money would Nicole earn in a week in which she worked 2 hours lifeguarding and y hours landscaping?
Answers
Answer:
This would mean Makayla worked a total of 15 hours and made $264. Which meets her requirements.
Step-by-step explanation:
You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N):
λ(n) = 0 , λ(x + 1) = λ(x) , λ(n+1/2)=1
Find two functions p, q : R → R with q(x)6=0 for all x such that λ(x) = q(x)(p(x) + 1).
Answers
Answer:
One possible combination:
[tex]p(x) = \sin(2\, \pi \, x) - 1[/tex].
[tex]\displaystyle q(x) = \frac{1}{2}[/tex].
[tex]\displaystyle \lambda(x) = \frac{1}{2}\, \sin(2\, \pi\, x) - \frac{1}{2} + 1 = \frac{1}{2}\, \sin(2\, \pi\, x) + \frac{1}{2}[/tex].
Step-by-step explanation:
The requirement that [tex]\lambda(x + 1) = \lambda(x)[/tex] for all [tex]x \in \mathbb{R}[/tex] suggests that [tex]\lambda[/tex] should be a cyclic function with a period of [tex]1[/tex].
One such example is the sine function. Let [tex]\omega[/tex] denote a non-zero real number. Consider the expression [tex]\sin(\omega\, x)[/tex].
[tex]\begin{aligned}\sin(\omega\, x) &= \sin(\omega\, x + 2\, \pi) \\ &= \sin\left(\omega\, \left(x + \frac{2\, \pi}{\omega}\right)\right)\end{aligned}[/tex].
Note how replacing [tex]x[/tex] with [tex]\displaystyle \left(x + \frac{2\, \pi}{\omega}\right)[/tex] does not change the value of [tex]\sin(\omega\, x)[/tex]. Therefore, the period of [tex]\sin(\omega\, x)\![/tex] would be [tex]\displaystyle \frac{2\, \pi}{\omega}[/tex].
The question requires that replacing the [tex]x[/tex] with [tex](x + 1)[/tex] should not change the value of [tex]\lambda(x)[/tex]. In other words, the period of [tex]\lambda[/tex] should be [tex]1[/tex]. If [tex]\lambda(x)\![/tex] is indeed in the form [tex]\sin(\omega\, x)[/tex], the value of [tex]\omega[/tex] should ensure that [tex]\displaystyle \frac{2\, \pi}{\omega} = 1[/tex]. Therefore, [tex]\omega = 2\,\pi[/tex].
It would take a few more transformations to ensure that for all integers [tex]n[/tex],[tex]\displaystyle \lambda(n) = 0[/tex] and [tex]\displaystyle \lambda\left(n + \frac{1}{2}\right) = 1[/tex].
One possible approach is to make each [tex]n[/tex] a trough of this function, and each [tex]\displaystyle\left(n + \frac{1}{2}\right)[/tex] a crest of this function. The corresponding sine wave would have a range of only [tex]1[/tex]. However, without any transformation, [tex]y = \sin(2\, \pi\, x)[/tex] would have a range of two.
Therefore, it would be necessary to compress [tex]y = \sin(2\, \pi\, x)[/tex] vertically by a factor of [tex]2[/tex], so that its range meets the needs.
[tex]y = \displaystyle \frac{1}{2}\, \sin(2\, \pi\, x)[/tex] is the output of one such compression and has the correct period and range. However, its troughs would be at [tex]y = -\displaystyle \frac{1}{2}[/tex] (rather than [tex]y = 0[/tex]) whereas its crests are at [tex]\displaystyle y = \frac{1}{2}[/tex] (rather than [tex]y = 1[/tex].) It would be necessary to shift this function upwards by [tex]\displaystyle \frac{1}{2}[/tex] unit for the troughs and crests to meet match the ones in the requirements.
[tex]\begin{aligned} y &= \displaystyle \frac{1}{2}\, \sin(2\, \pi\, x) + \frac{1}{2} \\ &= \frac{1}{2}\, \sin(2\, \pi\, x) - \frac{1}{2} + 1 = \frac{1}{2}\left(\sin(2\, \pi\, x) - 1\right) + 1\end{aligned}[/tex].
Convert 7 pounds 3 ounces to kilograms.
Conversion ratios:
1 lb = 16 oz
1 kg = 2.2 lb
HINT: Covert 3oz to lbs, then add that number to the 7lb. Then convert to kg.
Round your final answer to the nearest tenth.
Answers
Answer:
3.3
Step-by-step explanation:
7lb 3oz
3oz/16oz=.1875lb
7lb+.1875lb=7.1875lb
7.1875lb/2.2kg=3.27kg
The final answer is 3.3 kg (round to the nearest tenth).
What is ratio?Ratio basically compares quantities, that means it shows the value of one quantity with respect to the other quantity.
If a and b are two values, their ratio will be a:b,
Given that,
1 lb = 16 ounces
1 kg = 2.2 lb
To solve this type of questions, use ratio property,
Since,
16 ounces = 1 lb
⇒1 ounce = 1/16 lb
⇒3 ounces = 3/16 lb
Add 7 lb to 3/16,
= 3/16+7 = 7.1875 lb
Since,
2.2 lb = 1 kg
⇒1 lb = 1/2.2 kg
⇒7.1875 lb = 7.1875/2.2 kg = 3.267 kg
In round of the answer is 3.3 kg
To learn more about Ratio on :
https://brainly.com/question/13419413
#SPJ2
PLEASE HELP ME WITH THIS QUESTION ASAP
Answers
Answer: $801
Step-by-step explanation:
to find the slope, you can divide 445÷5=89. (now we know that the rate is $89/1 month)
to find the cost of 9 months, multiply 89•9=801
so since it's asking the cost after 9 months, i think the answer should be $801.
Mr. Scott want to tip his taxicab driver 18%. If his commute costs $25, what is the tip?
Answers
Answer:
4.5$
Step-by-step explanation:
25 times .18 = 4.5$
144:128 convert into simplest form
Answers
simplify each od the following ratios a) 15/24 :9/7
b) 2/3 : 3/2 : 5/8
Answers
Answer:
Fourth Option
Step-by-step explanation:
In order to find the answer to this question you must find the common number and then divide the numerator and the denominator by that number.
[tex]\frac{15}{24}[/tex]
[tex]CN=3[/tex]
[tex]15\div3=5[/tex]
[tex]24\div3=8[/tex]
[tex]=\frac{5}{8}[/tex]
Hope this helps.
The yard care staff can mow 45 lawns in a 10-hour work day. Each of the 9 workers can mow the same number of lawns per hour. How many lawn can one worker mow per hour
Answers
If it takes 10 hours for 9 workers to mow 45 lawns, dividing the 45 lawns by 9 we see that in 10 hours one worker can mow 5 lawns. This means that in one hour, a single worker can mow 5/10=1/2 of a lawn
The expression -7y is a _____. pls answer quickly i'm giving 40 for an answer.
variable
term
constant
Answers
Answer: Term
In math, a "term" is another word for "monomial".
If you add a bunch of monomials (or terms) together, you get a polynomial.
It takes Louis 1 2/3 hours to write 5/6 of a short story. How long will it take Louis to complete the story?
Answers
Answer:
2 hours
Step-by-step explanation:
Answer:
do it in thirds so 10/6 and 5/6 now simplify = 2/6= 1/6 of the story so full story would be 12/6=6/3=2 hours
Step-by-step explanation:
The mean number of cars owned is 2 and the standard deviation is 0.80. Fiona owns 3 cars. What is her z score?a. 1.50b. 2.00c. 1.25d. 2.25
Answers
Answer:
Her Z score is 1.25
Step-by-step explanation:
You have26 library books. Then you take out 15 more books. Now you have 27 books. How many books did you return.
Answers
What decimal equivalent to 5/6
Answers
Answer:
0.83333333
Step-by-step explanation:
The 3 is repeating
Answer:
0.83333333...
Step-by-step explanation:
Which angle pairs are congruent angle pairs? Select all that apply.
alternate exterior
same-side (consecutive exterior)
same-side (consecutive interior)
alternate interior
corresponding
linear pair
vertical
Answers
All are angle pair relationship, except consecutive exterior.
i need help my people
Answers
Answer:
no
Step-by-step explanation:
If all of their coresponding angles will have the same measure (AAA Th. of symilarity) then thhey will be symilar
84.6+30.4+66=181 not 180
(i'll be giving brainliest on this one!! :) )
Solve seven times four and five eighths.
A) twenty eight and three eighths
B) twenty eight and five eighths
C) thirty two and three eighths
D) thirty two and five eighths
(Here is an image if needed :D) :
Answers
Answer:
Third option
Step-by-step explanation:
In order to find the answer to this question you must change the mixed number into a improper fraction and then multiply the numerators to the denominators.
[tex]7\times4\frac{5}{8}[/tex]
Remember that all numbers have a invisible one as the denominator:
[tex]7=\frac{7}{1}[/tex]
[tex]\frac{7}{1} \times4\frac{5}{8}[/tex]
Turn the mixed number into a improper fraction:
[tex]8\times4=32+5=37[/tex]
(Remember to keep the denominator the same)
[tex]=\frac{37}{8}[/tex]
Now multiply:
[tex]7\times37=259[/tex]
[tex]1\times8=8[/tex]
[tex]=\frac{259}{8}[/tex]
Turn the improper fraction into a mixed number:
[tex]\frac{259}{8} =32\frac{3}{8}[/tex]
[tex]=32\frac{3}{8}[/tex]
Hope this helps.
What is the completely factored form of 5k^2 - 46k + 77?
Answers
Answer:
The completely factored form of 5k^2 - 46k + 77 is (−7) (5−11)
Step-by-step explanation:
May I have brainliest please? :)
Answer:
(k-7)(5k-11)
Step-by-step explanation:
А
Learning Task 4 : Do the following in your notebook.
A. Find the equation of the line given the graph.
Answers
Answer:
Question 3: y =x+2
Step-by-step explanation:
to find the equation use y=mx+c where m is the gradient/slope and c is the y intercept. In example 3, the y intercept is 2 so thats why the equation ends in +2, and the slope which is calculated by rise over run, is 2 over 2, or 1 so x doesnt need a coefficient.
the descriptive method:
{4, 8, 12, 16, 20)
Answers
Answer:
The set of the first five positive numbers multiples of 4.
Step-by-step explanation:
Find the area of a circle with diameter 14 cm use 3.14
Answers
Answer:
153.94
Step-by-step explanation:
What percent of 41.7 is 39?
Answers
if you can answer this question right ill mark you brainiest thanks
Answers
In the year 2005, a person bought a new car for $27500. For each consecutive year after that, the value of the car depreciated by 12%. How much would the car be worth in the year 2008, to the nearest hundred dollars?
Answers
Answer:
Getting my point back lol
Step-by-step explanation:
The art teacher bought 6 boxes of colored pencils. Each box contains 12 colored pencils. She wants to give each of her 10 students as many colored pencils as possible and keep the rest. Based on this information, which statement is true
Answers
Answer:
60
Step-by-step explanation:
(6x12)-(6x2)
72-12
60
Answer:
I need the statement before I answer
Step-by-step explanation:
Find an equation parallel to x = 0 and passing through (5. - 1).
Answers
Answer:
x=5
Both are vertical lines and parallel to each other.
Yvonne wants to rent a kayak for 7 hours. How much would this cost at each company? Which one should she choose?
Answers
Answer: raging river kayaks -63
paddlers-47
Step-by-step explanation: Hope its right
Answer: is, cost for, raging river kayaks -63 & paddlers-47.
What is multiplication?In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.
here, we have,
from the given chart we get,
the equation for raging river kayaks is, x= 9y
the equation for paddlers is, x= n+1
i.e. for 7 hrs. cost is, 63 & 47 respectively.
Hence, Answer: is, cost for, raging river kayaks -63 & paddlers-47.
To learn more on multiplication click:
brainly.com/question/5992872
#SPJ2
Kiley had a piece of bamboo skewer that measured 14Three-fifths inches long. She wanted to cut it into toothpicks that were each 3One-fifth inches long. How many toothpicks can she make?
Answers
Answer:
69
Step-by-step explanation: