Question

Please help! Will FAN AND MEDAL!

F of x equals the integral from 1 to x of the natural logarithm of t squared. Use your calculator to find F ′(3).

a. 2.197
b. 2.079
c. 1.099
d. 0.693

Answer 1

Can you please help? @agent0smith @jim_thompson5910 @mathmale @Directrix @Kainui @Photon336

Answer 2

@jim_thompson5910 is your go to guy

Answer 3

Use the fundamental theorem of calculus (FTC)

If
\[\Large f(x) = \int_{a}^{x}g(t)dt\]
then
\[\Large f \ '(x) = g(x)\]

Answer 4

So this means
\[\Large f(x) = \int_{1}^{x}\ln(t^2)dt\]
\[\Large f \ '(x) = \ln(x^2)\]

Answer 5

afterwards, just plug in x = 3

Answer 6

I got 2.197? right?

Answer 7

correct

Answer 8

Can you help me with two more?

Answer 9

sure

Answer 10

Thank you.

The graph of y = f ′(x) is shown below. List the intervals where the graph of f is concave up.

I don't understand what concave up really means?

Answer 11

on a concave up region, the tangent slopes are below the curve
on a concave down region, the tangent slopes are above the curve

http://mathsfirst.massey.ac.nz/Calculus/Sign2ndDer/images/ConcaveUpDown.png

Answer 12

one way to think of it is like this

Feeling down means you have a frowny face
so a concave down region looks like a sad frowny face



while feeling up and positive means you have a happy face
concave up region has a happy smiley face look

Answer 13

This is the graph and these are the answers.

a. (-4,-2)
b. (-2, 2)
c. (-4, 0)
d. (0, 4)

Answer 14

So a concave up would be c. (-4, 0)? right?

Answer 15

Rule:

if `f ' (x)` is increasing on some interval, then `f '' (x) > 0` making `f(x) concave up` on this same interval

if `f ' (x)` is decreasing on some interval, then `f '' (x) < 0` making `f(x) concave down` on this same interval

Answer 16

look at the graph of `f ' (x)`

on what interval(s) is the function increasing?

Answer 17

(-2,2)

Answer 18

yep `-2 < x < 2`

Answer 19

so f(x) is concave up on the interval -2 < x < 2

Answer 20

so that would be where it concaves up? So the correct answer is b. (-2,2)

Answer 21

yes `-2 < x < 2` in interval notation would be `(-2,2)`

Answer 22

One more question, if you don't mind?

Answer 23

go ahead

Answer 24

A particle moves on the x-axis so that its position is continuous on the interval [3, 13] with some of its values for its velocity v(t) given in the table below. Use a trapezoidal sum with 4 intervals to approximate the total distance the particle travelled in the time interval [3, 13].



a. 277.50
b. 138.75
c. 115.50
d. 87.50

Answer 25

Step 1) Plot the points on the same coordinate grid. See attached.

Answer 26

Okay. I understand that.

Answer 27

Step 2) Form trapezoids between the points like you see in the attached image.

Answer 28

Step 3) Find the area of each trapezoid. Use this formula

\[\Large A = \frac{h*(b_1+b_2)}{2}\]
A = area
h = height
b1 = base1
b2 = base2
b1 and b2 are parallel to each other