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.
    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