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
miskamm [114]
3 years ago
15

Calculus 2

Mathematics
1 answer:
FinnZ [79.3K]3 years ago
6 0

Answer:

See Below.

Step-by-step explanation:

We want to estimate the definite integral:

\displaystyle \int_1^47\sqrt{\ln(x)}\, dx

Using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with six equal subdivisions.

1)

The trapezoidal rule is given by:

\displaystyle \int_{a}^bf(x)\, dx\approx\frac{\Delta x}{2}\Big(f(x_0)+2f(x_1)+...+2f(x_{n-1})+f(x_n)\Big)

Our limits of integration are from x = 1 to x = 4. With six equal subdivisions, each subdivision will measure:

\displaystyle \Delta x=\frac{4-1}{6}=\frac{1}{2}

Therefore, the trapezoidal approximation is:

\displaystyle =\frac{1/2}{2}\Big(f(1)+2f(1.5)+2f(2)+2f(2.5)+2f(3)+2f(3.5)+2f(4)\Big)

Evaluate:

\displaystyle =\frac{1}{4}(7)(\sqrt{\ln(1)}+2\sqrt{\ln(1.5)}+...+2\sqrt{\ln(3.5)}+\sqrt{\ln(4)})\\\\\approx18.139337

2)

The midpoint rule is given by:

\displaystyle \int_a^bf(x)\, dx\approx\sum_{i=1}^nf\Big(\frac{x_{i-1}+x_i}{2}\Big)\Delta x

Thus:

\displaystyle =\frac{1}{2}\Big(f\Big(\frac{1+1.5}{2}\Big)+f\Big(\frac{1.5+2}{2}\Big)+...+f\Big(\frac{3+3.5}{2}\Big)+f\Big(\frac{3.5+4}{2}\Big)\Big)

Simplify:

\displaystyle =\frac{1}{2}(7)\Big(f(1.25)+f(1.75)+...+f(3.25)+f(3.75)\Big)\\\\ =\frac{1}{2}(7) (\sqrt{\ln(1.25)}+\sqrt{\ln(1.75)}+...+\sqrt{\ln(3.25)}+\sqrt{\ln(3.75)})\\\\\approx 18.767319

3)

Simpson's Rule is given by:

\displaystyle \int_a^b f(x)\, dx\approx\frac{\Delta x}{3}\Big(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+4f(x_{n-1})+f(x_n)\Big)

So:

\displaystyle =\frac{1/2}{3}\Big((f(1)+4f(1.5)+2f(2)+4f(2.5)+...+4f(3.5)+f(4)\Big)

Simplify:

\displaystyle =\frac{1}{6}(7)(\sqrt{\ln(1)}+4\sqrt{\ln(1.5)}+2\sqrt{\ln(2)}+4\sqrt{\ln(2.5)}+...+4\sqrt{\ln(3.5)}+\sqrt{\ln(4)})\\\\\approx 18.423834

You might be interested in
Please Help! Which of the following properties was used in the expression shown here?
Elena L [17]

Answer:

Distributive Property

Step-by-step explanation:

Multiply 5 with each number in the parentheses

Then add the variables after

3 0
3 years ago
Find a new function g(x) transformed from f(x)=-x-2 such that g(x) is perpendicular to f(x) .
sineoko [7]

The new function is g(x) =-1(1/-x) +2.

The transformation of function f(x) to function g(x) is given by g(x)=A f(Bx+ H)+K. where the constant A scales the function vertically. (A negative A reflects the function about the x-axis B scales the function horizontally. (A negative B reflects the function on the y-axis.) H shifts the function horizontally, and K shifts the function vertically.

Convert f(x) to g(x)

where the conversion is g(x)=f(x)+2.

A perpendicular line has negative reciprocal slopes, so for g(x) to be perpendicular to f(x), we need to get the negative reciprocal of -x, which is -1(1/-x)

So, f(x)=-x-2  ; m = -1

From the above information, we get g(x) as follows.

g(x) =-1(1/-x) +2

More problems related to finding new functions for a given function are below.

brainly.com/question/13825802?referrer=searchResults

#SPJ4

6 0
1 year ago
Find the equation of the graphed line.
Alex_Xolod [135]

example: the line is y = ax + b

The line go through 2 points ( -4 ; 0 ) and ( 0 ; -2 )

So:

  • -2 = 0a + b => b = -2
  • 0 = -4a + (-2) => a = 2/-4 = -1/2

The line is y = -1/2x - 2 ( i'm not sure)

Answer: a

Ok done. Thank to me :>

5 0
3 years ago
$13,000 for 8 years at<br> 3.6% annual<br> simple<br> interest
Vadim26 [7]
$16,744.00

Link: http://www.moneychimp.com/features/simple_interest_calculator.htm
3 0
3 years ago
What is equivalent decimal for 0.85
Alex_Xolod [135]

Answer:

17/20

Step-by-step explanation:

0.85 = 85/100 = 17/20

The number 0.85 can be written using the fraction 85/100 which is equal to 17/20 when reduced to lowest terms.

5 0
3 years ago
Other questions:
  • A line passes through (2, −1) and (4, 5).<br><br> Which answer is the equation of the line?
    5·1 answer
  • Find the distance between A (0,1) and B (-4,6) to the nearest tenth
    14·2 answers
  • Please Help If you Can​
    15·1 answer
  • What is the area of a rectangle with length 12 meters and width 6 meters? 0.5 m² 2 m² 36 m² 72 m²
    5·1 answer
  • Can somebody help me find the value of x
    14·1 answer
  • In ΔKLM, the measure of ∠M=90°, LK = 97, ML = 65, and KM = 72. What is the value of the sine of ∠L to the nearest hundredth?
    8·1 answer
  • Can side lengths 6 , 8 and 10 forms a right triangle .
    7·1 answer
  • How is an equation different from an inequality
    10·2 answers
  • Only answer if yk how to do this
    12·2 answers
  • Plis help me here :(
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!