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
Kay [80]
3 years ago
14

Consider functions of the form f(x)=a^x for various values of a. In particular, choose a sequence of values of a that converges

to e. (like a=1, a=1.5, a=2, a=2.5, a=2.6, a=2.7...a=3). Grpah f(x), f'(x), and the line tangent to f(x) for x=2 for each of your values of a. Describe the relationship between f and its derivatives as the values of the parameter a pass through e.

Mathematics
1 answer:
sleet_krkn [62]3 years ago
3 0

Answer:

A. As "a"⇒e, the function f(x)=aˣ tends to be its derivative.

Step-by-step explanation:

A. To show the stretched relation between the fact that "a"⇒e and the derivatives of the function, let´s differentiate f(x) without a value for "a" (leaving it as a constant):

f(x)=a^{x}\\ f'(x)=a^xln(a)

The process will help us to understand what is happening, at first we rewrite the function:

f(x)=a^x\\ f(x)=e^{ln(a^x)}\\ f(x)=e^{xln(a)}\\

And then, we use the chain rule to differentiate:

f'(x)=e^{xln(a)}ln(a)\\ f'(x)=a^xln(a)

Notice the only difference between f(x) and its derivative is the new factor ln(a). But we know  that ln(e)=1, this tell us that as "a"⇒e, ln(a)⇒1 (because ln(x) is a continuous function in (0,∞) ) and as a consequence f'(x)⇒f(x).

In the graph that is attached it´s shown that the functions follows this inequality (the segmented lines are the derivatives):

if a<e<b, then aˣln(a) < aˣ < eˣ < bˣ < bˣln(b)  (and below we explain why this happen)

Considering that ln(a) is a growing function and ln(e)=1, we have:

if a<e<b, then ln(a)< 1 <ln(b)

if a<e, then aˣln(a)<aˣ

if e<b, then bˣ<bˣln(b)

And because eˣ is defined to be the same as its derivative, the cases above results in the following

if a<e<b, then aˣ < eˣ < bˣ (because this function is also a growing function as "a" and "b" gets closer to e)

if a<e, then aˣln(a)<aˣ<eˣ ( f'(x)<f(x) )

if e<b, then eˣ<bˣ<bˣln(b) ( f(x)<f'(x) )

but as "a"⇒e, the difference between f(x) and f'(x) begin to decrease until it gets zero (when a=e)

You might be interested in
Help me with this! i’ll give brainliest
kondor19780726 [428]

Answer:

first box:6

second box: 1

Step-by-step explanation:

divide bothe 18b and 3y by 3 and you get those values.

3 0
3 years ago
Read 2 more answers
Write the point-slope form of the equation of the line through the given point woth the given slope
Andrej [43]

Answer:

3.) \displaystyle y - 4 = 4[x - 1]

2.) \displaystyle y + 5 = 2[x + 2]

1.) \displaystyle y + 3 = 1\frac{1}{5}[x + 5]

Step-by-step explanation:

In the Point-Slope Formula, \displaystyle y - y_1 = m[x - x_1],all the negative symbols give the OPPOSITE terms of what they really are, so be EXTREMELY CAREFUL inserting the coordinates into the formula with their CORRECT SIGNS.

* \displaystyle 1\frac{1}{5} = \frac{6}{5}

I am joyous to assist you anytime.

3 0
2 years ago
Write 3/100 as a decimal​
Vlada [557]

Answer:

0.03

Step-by-step explanation:

There are two 0s so it is 0.03

7 0
3 years ago
It is your first semester at college and you take out a $5000 loan and in order to pay off that loan by the end of the semester
kakasveta [241]

Answer:

5000-1250x

Step-by-step explanation:

All you need to do is find the amount of money you get paid a month and use variable x to represent the months and then subtract that from the money you need to pay off.

8 0
3 years ago
Figure this out please Asap.
weeeeeb [17]

If the polygons are similar then the sides are in proportion.

\dfrac{32}{24}=\dfrac{32:8}{24:8}=\dfrac{4}{3}

Therefore we have the equations:

\dfrac{x-1}{6}=\dfrac{4}{3}\qquad\text{cross multiply}\\\\3(x-1)=(4)(6)\\\\3x-3=24\qquad\text{add 3 to both sides}\\\\3x=27\qquad\text{divide both sides by 3}\\\\\boxed{x=9}


\dfrac{y+1}{21}=\dfrac{4}{3}\qquad\text{cross multiply}\\\\3(y+1)=(21)(4)\\\\3y+3=84\qquad\text{subtract 3 from both sides}\\\\3y=81\qquad\text{divide both sides by 3}\\\\\boxed{y=27}

6 0
3 years ago
Other questions:
  • How can I solve this two step equation: 2(n+5)=-2
    12·2 answers
  • Z is centroid of triangle RST. What is RW, if RV=4x+3, WS=5x-1, and VT=2x+9?
    6·1 answer
  • What is the answer to this problem because I need help with the problem
    5·1 answer
  • Elena walked 20 minutes more than Lin. Jada walked twice as long as Elena. Jada walked for 90 minutes. The equation 2(x +20) = 9
    8·2 answers
  • Find the area of the surface cut from the bottom of the paraboloid by the plane
    10·1 answer
  • Swifty Corporation started the year with total assets of $300000 and total liabilities of $240000. During the year the business
    11·1 answer
  • If Jonny had 3 apples and he gave 1 away how much does he have​
    15·1 answer
  • Solve the system: 4x - y = 5 x + 2y = 8
    5·2 answers
  • Square root of 24 using perfect squares
    9·1 answer
  • Thaddeus shakes his head as he pays $12.37 to the clerk for lunch. Complaining to a friend, Thaddeus says, “Do you realize that
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!