 Sample Third Exam: Calculus 1 (MAC 2311)

```Sample Third Exam: Calculus 1 (MAC 2311)
Be sure to show all work and justify your answers; otherwise you
will lose points. NO CALCULATORS OR NOTES. WRITE YOUR
NAME ON THIS TEST!
1.
Evaluate the following integrals (5 points each):
(a)
R
x2 + 2x + 1 dx =
(c)
R
√1
x
(e)
R
2
(1−2x)3
(g)
R
2x cos( x7 ) dx =
+
1
x
+
1
x2
R
(b) cos2 x sin x dx =
dx =
dx =
(f)
2
ex
12x2
R
1
1+36x2
R
1
sin t+1 dt
(h)
1
1
R
(d)
dx =
dx
=
1.(continued) Evaluate the following integrals ( 5 points each):
R
(i) cos3 y dy =
(j)
R
tan x dx =
√
2. Approximate the value of 3 by applying Newton’s method to the function
f (x) = x2 − 3. Start with x0 = 1 and calculate x2 . (15 points)
2
3. (i) Approximate the area under the curve y = x3 between the lines x = 0
and x = 2 by using the sum of the areas under 3 rectangles of equal width and
height equal to x3 where x is the position of the right side of each rectangle. (5
points)
(ii) (b) Use the formula
n
X
k=1
k3 =
(n)(n+1) 2
to compute an expression giving
2
the value for the total area under the sum of n rectangles of width n2 and height
equal to x3 where x is the position of the right side of each rectangle between
x = 0 and x = 2 and show what the limit as n → ∞ is. (10 points)
3
4. What is the average value of the function f (x) = sin x between 0 and
(10 points)
4
π
2?
5.
(a.)
Evaluate the following integrals (5 points each):
R3
2
4 − x2 dx =
R3
(b.)
|4 − x2 | dx =
1
(notice the lower limit is different in (b.) from that in (a.))
5
``` 