- 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 - 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

- 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 - 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 - 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 - 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 - 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 - 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