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
Delicious77 [7]
3 years ago
8

Use the Fundamental Theorem of Calculus to find the area of the region between the graph of the function x5 + 8x4 + 2x2 + 5x + 1

5 and the x-axis on the interval [–6, 6]. Round off your answer to the nearest integer.
Question 15 options:

25,351 units2

149,473 units2

3,758 units2

2,362 units2
Mathematics
1 answer:
belka [17]3 years ago
6 0

Answer:

The area of the region between the graph of the given function and the x-axis = 25,351 units²

Step-by-step explanation:

Given  x⁵ + 8 x⁴ + 2 x² + 5 x + 15

If 'f' is a continuous on [a ,b] then the function

            F(x) = \int\limits^a_b {f(x)} \, dx

By using integration formula

\int{x^n} \, dx = \frac{x^{n+1} }{n+1} +c

Given  x⁵ + 8 x⁴ + 2 x² + 5 x + 15 in the interval [-6,6]

 \int\limits^6_^-6} (x^{5}  + 8 x^{4}  + 2 x^{2}  + 5 x + 15) )dx

<em>On integration , we get</em>

=   (\frac{x^{6} }{6} + \frac{8 x^{5} }{5} + 2 \frac{x^{3} }{3} +\frac{5 x^{2} }{2} + 15 x)^{6} _{-6}

F(x) = \int\limits^a_b {f(x)} \, dx = F(b) -F(a)

= (\frac{6^{6} }{6} + \frac{8 6^{5} }{5} + 2 \frac{6^{3} }{3} +\frac{5 6^{2} }{2} + 15X 6) - ((\frac{(-6)^{6} }{6} + \frac{8 (-6)^{5} }{5} + 2 \frac{(-6)^{3} }{3} +\frac{5 (-6)^{2} }{2} + 15 (-6))

After simplification and cancellation we get

 =  \frac{2 X 8 X (6)^{5} }{5} + \frac{2 X 2 X (6)^3}{3} + 2 X 15 X 6

on calculation , we get

= \frac{124,416}{5} + \frac{864}{3} + 180

On L.C.M  15

= \frac{124,416 X 3 + 864 X 5 + 180 X 15}{15}

= 25 351.2 units²

<u><em>Conclusion</em></u>:-

<em>The area of the region between the graph of the given function and the x-axis = 25,351 units²</em>

You might be interested in
Evaluate -3.25 - 2.75z for z = -4.
Usimov [2.4K]
The answer is $7.75 -3.25-2.75x-4=$7.75
4 0
3 years ago
Dan earns $42.75 per month working as a tutor. He worked as a tutor for 4 months. He also earned $52.45 walking dogs in the neig
hjlf

Answer:

step 3

Step-by-step explanation:

in step 3, he wrote that the sum of 171+ 52.45-18.50 was 241.95 when it is 199.95

3 0
2 years ago
Help please it’s a quiz
Sidana [21]
......................
3 0
3 years ago
The acceleration, in meters per second per second, of a race car is modeled by A(t)=t^3−15/2t^2+12t+10, where t is measured in s
oksian1 [2.3K]

Answer:

The maximum acceleration over that interval is A(6) = 28.

Step-by-step explanation:

The acceleration of this car is modelled as a function of the variable t.

Notice that the interval of interest 0 \le t \le 6 is closed on both ends. In other words, this interval includes both endpoints: t = 0 and t= 6. Over this interval, the value of A(t) might be maximized when t is at the following:

  • One of the two endpoints of this interval, where t = 0 or t = 6.
  • A local maximum of A(t), where A^\prime(t) = 0 (first derivative of A(t)\! is zero) and A^{\prime\prime}(t) (second derivative of \! A(t) is smaller than zero.)

Start by calculating the value of A(t) at the two endpoints:

  • A(0) = 10.
  • A(6) = 28.

Apply the power rule to find the first and second derivatives of A(t):

\begin{aligned} A^{\prime}(t) &= 3\, t^{2} - 15\, t + 12 \\ &= 3\, (t - 1) \, (t + 4)\end{aligned}.

\displaystyle A^{\prime\prime}(t) = 6\, t - 15.

Notice that both t = 1 and t = 4 are first derivatives of A^{\prime}(t) over the interval 0 \le t \le 6.

However, among these two zeros, only t = 1\! ensures that the second derivative A^{\prime\prime}(t) is smaller than zero (that is: A^{\prime\prime}(1) < 0.) If the second derivative A^{\prime\prime}(t)\! is non-negative, that zero of A^{\prime}(t) would either be an inflection point (ifA^{\prime\prime}(t) = 0) or a local minimum (if A^{\prime\prime}(t) > 0.)

Therefore \! t = 1 would be the only local maximum over the interval 0 \le t \le 6\!.

Calculate the value of A(t) at this local maximum:

  • A(1) = 15.5.

Compare these three possible maximum values of A(t) over the interval 0 \le t \le 6. Apparently, t = 6 would maximize the value of A(t)\!. That is: A(6) = 28 gives the maximum value of \! A(t) over the interval 0 \le t \le 6\!.

However, note that the maximum over this interval exists because t = 6\! is indeed part of the 0 \le t \le 6 interval. For example, the same A(t) would have no maximum over the interval 0 \le t < 6 (which does not include t = 6.)

4 0
3 years ago
Help asap please!!!!
Alexxx [7]

Answer:

(3, 1)

Step-by-step explanation:

Let the coordinates of B be (x, y)

Since, M is the midpoint of AB.

Therefore, by mid point formula:

1 =  \frac{ - 1 + x}{2},  \:  \:  \: 3 =  \frac{5 + y}{2}  \\  \\ 2 =  - 1 + x, \:  \:  \: 6 = 5 + y \\  \\ 2 + 1 = x, \:  \:  \: 6 - 5 = y \\  \\ 3 = x, \:  \:  \: 1 = y \\  \\ x = 3, \:  \:  \: y = 1 \\  \\ B = (3, \:  \: 1)

3 0
3 years ago
Other questions:
  • A mother is 1 and 3/4 times as tall as her son. Her son is 3 feet tall. How tall is the mother?
    13·2 answers
  • Write a real-world problem that is solved by an equation and then solve (show work).
    5·1 answer
  • All of the following are ways to write 25 percent of N EXCEPT 0.25N 25N/100 1/4N 25N
    12·2 answers
  • Please help ASAP. Please show ur work
    13·1 answer
  • Can 10/5 be a whole number ?
    6·2 answers
  • I need help with this please
    15·1 answer
  • Which of the following best describes the expression 6(y + 3)?<br><br><br> Please !
    14·1 answer
  • What is the missing reason in the proof shown ?
    13·1 answer
  • 11 is 0.6 percent of what number
    14·1 answer
  • The radioisotope Mo-91 has a t1/2 of 15.5min. A sample decays at the rate of 954counts/min (954cpm). After how many minutes will
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!