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.
   Note: Remember to simplify your answers!
   (a) lim
   x→π/6
   3 sin(−x)

cos2(2x)

(b) lim
x→0
cos2
(3x)−1
x

2

(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.
   Do not simplify your answers.
   (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.
   Do not simplify your answers.
   (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