# The population of a bacterial colony

1. The population of a bacterial colony after t hours is given by
n(t) = 48t −t
3 +100.
(a) (3 pts) Determine the growth rate as a function of time.
(b) (3 pts) Find the growth rate after 2 hours.
(c) (3 pts) Find the time t at which the population starts diminishing.
pts: /9
2. Compute the following limits. Each limit is worth 5 points.
(a) lim
x→π/6
3 sin(−x)

(b) lim
x→0
cos2
(3x)−1
x

(c) lim
x→2
sin(x−2)
x

# 2 −x−2

pts: /15

1. Compute the derivatives of the following functions. Each derivative is worth 5 points.
(a) If y = π
2 +x
2
sin(8x) then y
0 =
(b) If y = cos√
x then y
0 =
(c) If y = tan2
x−tan(x
2
) then y
0 =
(d) If y =
cos x
x−1
then y
0 =
pts: /20
2. The volume of a ball is increasing at a rate of 10 cm3/min.
How fast is the surface area increasing when the radius is 30 cm?
pts: /10
3. Each problem is worth 4 points
(a) Find the second derivative of f(x) = √
1−x.
(b) If g is a twice differentiable function, find the second derivative of f(x) = g(x
2+1) in terms
of g,g
0
,g
00
.
pts: /8
4. Calculate the derivatives of the following functions. Each derivative is worth 5 points.
(a) If F(x) = (x
3 −5)
3
then F
0
(x) =
(b) If F(x) = √
x−4x
5
then F
0
(x) =
(c) If F(x) = sin(cos(sinx)) then F
0
(x) =
(d) If F(x) = sin
1−x
1+x

then F
0
(x) =
pts: /20
5. Each problem is worth 5 points.
(a) Find the equation of the tangent line to the curve y
3 −2xy+x
3 = 0 at the point P(1,1).
(b) Express the derivative of y with respect to x in terms of x and y if y
2 =
x−1
y−1
.
pts: /10
6. Each part is worth 4 points.
(a) Find the linearization L(x) of f(x) = √3 x at a = 27.
(b) Estimate the value of √3
28.
Note: A calculator solution is not an acceptable answer.
pts: /8 