Stuck on some calculus problems

[CalcStudent003]

1. If f(3) = 7 and f'(x) = (sin(1/x^2))/(x^3-2x) then f(5) =

(a) -16.006
(b) -9.006
(c) -0.008
(d) 6.992
(e) 7.008

2. Suppose that f(x), f'(x), and f''(x) are continuous for all real numbers x, and the f has the following properties.

I. f is negative on (-inf, 6) and positive on (6,inf).
II. f is increasing on (-inf, 8) and positive on 8, inf).
III. f is concave down on (-inf, 10) and concave up on (10, inf).

Of the following, which has the smallest numerical value?

(a) f'(0)
(b) f'(6)
(c) f''(4)
(d) f''(10)
(e) f''(12)

3. If 0 <= k <= pi/2 and the area of the region in the first quadrant under the graph of y = 2x-sinx from 0 to k is 0.1, then k =

(a) 0.444
(b) 0.623
(c) 0.883
(d) 1.062
(e) 1.571

4. Region R is bounded by the functions f(x) = 2(x-4) + pi, g(x) = cos^-1(x/2 - 3), and the x axis.

a. What is the area of the region R?

b. Find the volume of the solid generated when region R is rotated about the x axis.

c. Find all values c for f(x) and g(x) in the closed interval p <= c <= q for which each function equals the average value in the indicated interval.

If some could help with these questions, I'd really appreciate it. I've tried some things but I can't seem to get anywhere. Thanks!

 

[Deleted member 4993]

1. If f(3) = 7 and f'(x) = (sin(1/x^2))/(x^3-2x) then f(5) =

(a) -16.006
(b) -9.006
(c) -0.008
(d) 6.992
(e) 7.008

2. Suppose that f(x), f'(x), and f''(x) are continuous for all real numbers x, and the f has the following properties.

I. f is negative on (-inf, 6) and positive on (6,inf).
II. f is increasing on (-inf, 8) and positive on 8, inf).
III. f is concave down on (-inf, 10) and concave up on (10, inf).

Of the following, which has the smallest numerical value?

(a) f'(0)
(b) f'(6)
(c) f''(4)
(d) f''(10)
(e) f''(12)

3. If 0 <= k <= pi/2 and the area of the region in the first quadrant under the graph of y = 2x-sinx from 0 to k is 0.1, then k =

(a) 0.444
(b) 0.623
(c) 0.883
(d) 1.062
(e) 1.571

4. Region R is bounded by the functions f(x) = 2(x-4) + pi, g(x) = cos^-1(x/2 - 3), and the x axis.

a. What is the area of the region R?

b. Find the volume of the solid generated when region R is rotated about the x axis.

c. Find all values c for f(x) and g(x) in the closed interval p <= c <= q for which each function equals the average value in the indicated interval.

If some could help with these questions, I'd really appreciate it. I've tried some things but I can't seem to get anywhere. Thanks!

I've tried some things but .... can show us some of those things that you tried?

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...217#post322217

We can help - we only help after you have shown your work - or ask a specific question (not a statement like "Don't know any of these")

Please share your work with us indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[CalcStudent003]

Oops... I'm sorry. I didn't mean to violate any rules.

1. For any x > 3, f'(x) is just about zero, so I concluded that f(x) has to remain somewhat constant around 7 if x > 3, but how would I determine if it's a little higher or lower than 7 at 6.992 or 7.008?

2. & 3. Nevermind about these, I figured them out.

4. My main problems with this question are parts b and c. Would you use the disk or washer method for part b? And what would be the interval I used for the definite integral? For c... I don't even know where to begin.

 

[Deleted member 4993]

Oops... I'm sorry. I didn't mean to violate any rules.

1. For any x > 3, f'(x) is just about zero, so I concluded that f(x) has to remain somewhat constant around 7 if x > 3, but how would I determine if it's a little higher or lower than 7 at 6.992 or 7.008?

Have you been taught Taylor's series? I would use that for this problem

2. & 3. Nevermind about these, I figured them out.

4. My main problems with this question are parts b and c. Would you use the disk or washer method for part b? And what would be the interval I used for the definite integral? For c... I don't even know where to begin.

I would first obtain a graph of the region - using a graphing calculator.

I would determine the limits of integration - by looking at the graph and finding the point of intersection of the given functions.

To find the area (and the volume) - I would use differential areas parallel to y-axis (similar to disk method).

.