Friday, 26 February
Math 105, section 207
Quiz 7
Name:
Student number:
Time: 25 minutes
Rxp
1. Find the derivative of the function F (x) = 1 4 + 2t + sin(t) dt at the point
x = 1:
p
p
a) 0
b) 6 + sin(1) − 2
c) 6 + sin(1)
d) √2+cos(1)
6+sin(1)
R ln x p
2. (Midterm 2, 2015) Find the derivative of the function G(x) = 1
4 + 2t + sin(t) dt
at the point x = 1:
√
(1)
a) 2(2 + cos(1))
b) 2+cos
c)
2
d)
6
2
p
R 2x
3. What is the derivative of the function H(x) = x
4 + 2t + sin(t) dt?
p
p
a) 2p 4 + 4x + sin(2x) −p 4 + 2x + sin(x)
b) 4 + 4x + sin(2x)
p
p − 4 + 2x + sin(x)
c) 2(2 + 2cos(2x))p 4 + 4x + sin(2x) − (2 + cos x)p 4 + 2x + sin(x)
d) (2 + 2cos(2x)) 4 + 4x + sin(2x) − (2 + cos x) 4 + 2x + sin(x)
R1
4. −1 (x + 1)(x2 + 2x)6 dx?
1
1
1
27
a)
b) (27 − 1)
c) (37 + 1)
d)
7
14
14
7
5.
R x2 + x
dx?
x3 − x
1
a)−
+C
x−1
b) ln
|x + 1|
+C
|x − 1|
c) ln|x − 1| + C
6. (Midterm 2, 2014, with a small change)
tution
x = 2sinθ)√
√
3
a) 4
b)1 − 43
R2
√
c)1 −
3
2
1
x2
√
d) ln(x3 − x) + C
1
dx? (Hint: use the substi4 − x2
√
d)
2
2
7. Use the technique of integration by parts to compute
a)0
b) π2
c)π
d) π 2
Rπ
0
x sin x dx?
Friday, 26 February
Math 105, section 207
Quiz 7
Some formulae:
1 + cos2x
1 − cos2x
, sin2 x =
,
2
2
1
1
secx =
, (tanx)0 = 1 + tan2 x =
= sec2 x,
cosx
cos2 x
Z
Z
tanx dx = −ln|cos x| + C = ln|sec x| + C,
cotx dx = ln|sinx| + C,
cos2 x =
Z
Z
secx dx = ln|secx + tanx| + C,
cscx dx = −ln|cscx + cotx| + C,
1 − sin2 x = cos2 x,
1 + tan2 x = sec2 x,
sec2 x − 1 = tan2 x,
sin 2x = 2sinx cosx,
cos2x = cos2 x − sin2 x = 2cos2 x − 1 = 1 − 2sin2 x