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
Solnce55 [7]
3 years ago
12

If f(x)=2x+sinx and the function g is the inverse of f then g'(2)=

Mathematics
1 answer:
Alexxx [7]3 years ago
5 0
\bf f(x)=y=2x+sin(x)
\\\\\\
inverse\implies x=2y+sin(y)\leftarrow f^{-1}(x)\leftarrow g(x)
\\\\\\
\textit{now, the "y" in the inverse, is really just g(x)}
\\\\\\
\textit{so, we can write it as }x=2g(x)+sin[g(x)]\\\\
-----------------------------\\\\

\bf \textit{let's use implicit differentiation}\\\\
1=2\cfrac{dg(x)}{dx}+cos[g(x)]\cdot \cfrac{dg(x)}{dx}\impliedby \textit{common factor}
\\\\\\
1=\cfrac{dg(x)}{dx}[2+cos[g(x)]]\implies \cfrac{1}{[2+cos[g(x)]]}=\cfrac{dg(x)}{dx}=g'(x)\\\\
-----------------------------\\\\
g'(2)=\cfrac{1}{2+cos[g(2)]}

now, if we just knew what g(2)  is, we'd be golden, however, we dunno

BUT, recall, g(x) is the inverse of f(x), meaning, all domain for f(x) is really the range of g(x) and, the range for f(x), is the domain for g(x)

for inverse expressions, the domain and range is the same as the original, just switched over

so, g(2) = some range value
that  means if we use that value in f(x),   f( some range value) = 2

so... in short, instead of getting the range from g(2), let's get the domain of f(x) IF the range is 2

thus    2 = 2x+sin(x)

\bf 2=2x+sin(x)\implies 0=2x+sin(x)-2
\\\\\\
-----------------------------\\\\
g'(2)=\cfrac{1}{2+cos[g(2)]}\implies g'(2)=\cfrac{1}{2+cos[2x+sin(x)-2]}

hmmm I was looking for some constant value... but hmm, not sure there is one, so I think that'd be it
You might be interested in
What can you conclude about the signs of two integers whos product i a positive and negative
KengaRu [80]

If two integers product is:

  1. <u>Positive:</u> then the integers were both negative or both positive (same sign)
  2. <u>Negative:</u> then the integers consisted of one negative and one positive (opposite signs)
7 0
3 years ago
Ft?
VARVARA [1.3K]

Answer:

15 5/8 ft^3

Step-by-step explanation:

Volume of a cube = l x b x h

Given the Length of sides is

2 1/2

Change 2 1/2 to an improper fraction.

It becomes 5/2

Volume = 5/2 x 5/2 x 5/2

Multiply through

= 125/8

= 15 5/8 ft^3

5 0
3 years ago
An HMO charges $35 per doctor visit (d) and $15 per prescription (p). Which expression shows the total cost for prescriptions an
Alecsey [184]

Answer:

90% of people marry there 7th grade love. since u have read this, u will be told good news tonight. if u don't pass this on nine comments your worst week starts now this isn't fake. apparently if u copy and paste this on ten comments in the next ten minutes you will have the best day of your life tomorrow. you will either get kissed or asked out in the next 53 minutes someone will say i love u

Step-by-step explanation:

6 0
2 years ago
The 17th term of an Arithmetic sequence is 51. If the commons difference is 7, what is the first term ?
vovikov84 [41]

Answer:

The first term is:

  • a_1=-61

Step-by-step explanation:

The arithmetic sequence is defined by

a_n=a_1+\left(n-1\right)d

where

a_1 is the first term

d is the common difference

as

the 17th term of an arithmetic sequence is 51.

i.e. a_{17}=51

so

a_n=a_1+\left(n-1\right)d

a_{17}=a_1+\left(17-1\right)d        

51=a_1+\left(17-1\right)7         ∵ n = 17, a_{17}=51 , d=7

a_1+112=51                   ∵ \left(17-1\right)\cdot \:\:7=112

a_1=-61

Therefore, the first term is:

  • a_1=-61
8 0
3 years ago
A recent research study randomly divided participants into groups who were given different levels of Vitamin E to take daily. On
horrorfan [7]

Answer: D. To produce treatment groups with similar characteristics

Step-by-step explanation:

By using randomization in sampling, the Sample would be more representative of the Population it is based off of because different demographic characteristics may be picked.

This leads to a situation where the groups have similar characteristics between themselves thereby making it easier for comparison. For example, Group 1 would have certain types of people that will be represented in Group 2 and Group 3 as well. That way the effects of the drug can be properly studied as it affects different people. For instance, say there are 4 obese people in a sample of 10, instead of group one having all obese people, randomization may be able to give group one, 2 obese people and 2 obese people to group 2 as well. That way when comparing, the effects of the drug on the two groups is easier to be compared because the two groups have similar people.

5 0
3 years ago
Other questions:
  • A graph titled Monthly Sales and Advertising Costs has Advertising Costs (1,000 dollars) on the x-axis and sales (1,000 dollars)
    15·2 answers
  • Sal is making bracelets for a fundraiser.he uses 6 inches of yarn per bracelets how many bracelets can sal make with 5 feet of y
    15·2 answers
  • Someone please help me with this question? Thanks if you do!
    14·1 answer
  • Rename the number 805 tens?
    9·2 answers
  • Adult tickets to a play cost $22.00 Tickets for children cost $15.00. Tickets for a group of 11 people cost a total of $228.00.
    8·1 answer
  • A scientist has two kinds of solutions. The first solution is Solution A, which is an $80\%$ acid solution. The second solution
    15·1 answer
  • Marcus borrows $3,000 from his local credit union, to be repaid with 3.5% interest over 4 years. What is the simple interest?
    9·1 answer
  • 7x-49=y<br><br> Solve for x?
    13·1 answer
  • latrell wants to keep his test average in his class at 85 or higher. so far, he has earned score of 85, 76, 94 and 81 on his tes
    6·1 answer
  • Below is a right cone with a radius of 3 cm and a slant height of 7.6 cm. Find both the volume and surface area of the cone. Rou
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!