Download QUIZ 2

Prof. A. Suciu
MTH 1108
Fundamentals of Integral Calculus
Spring 1999
QUIZ 2
1. A colony of giant bacteria is cultured for a period of time, but eventually dies out, for lack
of sufficient nutrients. Its population at time t in hours is given by P (t) = 10et − tet , as
long as P (t) ≥ 0.
(8)
(a) What is the initial population?
(b) What is the rate of growth of the bacteria population at time t = 2 hours?
(c) When is the rate of growth zero?
(d) What is the maximum size of the bacteria population?
(e) When does the population die out?
MTH 1108
Quiz 2
2. Compute:
Z
(a)
x3 (x4 + 7)5 dx =
Z
(b)
4x
dx =
(1 − 5x2 )3
Z √
(c)
3 ln x
dx =
x
Spring 1999
(4)
(4)
(4)
Z
(d)
esin(2x) cos(2x) dx =
(4)
MTH 1108
Quiz 2
3.
Spring 1999
Z
(a) Sketch the area represented by the integral
1
(b) Use a left-approximating sum
this integral.
Pn−1
i=0
(6)
3
1
dx.
x
f (xi )∆x with n = 4 rectangles to approximate
MTH 1108
Quiz 2
Spring 1999
Table of Derivatives
(f g)0 = f 0 g + f g 0
(product rule)
µ ¶0
f
f 0g − f g0
=
g
g2
(quotient rule)
f (g(x))0 = f 0 (g(x)) · g 0 (x)
(chain rule)
(xn )0 = nxn−1
(ex )0 = ex
1
x
0
(sin x) = cos x
(ln x)0 =
(cos x)0 = − sin x
(arctan x)0 =
x2
1
+1
Table of Antiderivatives
Z
a dx = ax + C
Z
xn dx =
Z
xn+1
+C
n+1
(n 6= −1)
1
dx = ln |x| + C
x
Z
eax dx =
1 ax
e +C
a
Z
sin(ax) dx = −
Z
(a 6= 0)
1
cos(ax) + C
a
(a 6= 0)
1
sin(ax) + C
(a 6= 0)
a
Z
Z
Z
(af (x) + bg(x)) dx = a f (x) dx + b g(x) dx
Z
where F 0 = f
f (g(x))g 0 (x) dx = F (g(x)) + C,
cos(ax) dx =