1 a. Find lim limx0. Defend your answer.

b. Find lim sin x infinity. Defend your answer.

2. Find the derivative.

a. y =sin x^5

b. y =sin^5 x

c. y = cos ( 4 x + 9 )

d. y = cos^2 ( x ^2 + 3 )

e. y = x ^2 sin ( x ^2 + 5 )

f. y =5 e^x

g. y = e ^5^ x

h. y = xe ^3 ^x^ +^ 1

i. y = ln ( x + 3 )

j. y = 5 ln ( x ^2 + 4 )

k. y =5 ^x

l. y = ( x ^2 + 2 )^2( x^2+ 2 x + 1 )^3

m. y = ln ( sin x )

n. y = ( x + ln x )^5

3. Use logarithmic differentiation to find the derivative of y=(x+3)^2/(x+2)^5(x+4)^7

4. Find the equation of the tangent line to the function y = ln ( x ^2 - 2 ) when x =3

5. Find the equation of the tangent to the curve y=cos2x when x=3.14 / 3.Your solution should use radian measure.

6. Find the equation of the tangent line to the curve y = e^x when it is parallel to 2x-y= 7

7. Find the intervals of increasing/decreasing for the function f ( x ) = x^3e^x.

8. Find the minimum value(s) of the function f ( x ) = e ^x -2 x.

9. Determine the maximum/minimum point(s) for y = sin 2 x - cos x - 4 x on the interval 0 < x < 2 p . The value(s) should be in expressed in radians.

10. A ladder 15 m long leans against the wall. The ladder slides down the wall at a rate of 1.5 m/s. What is the rate of change of the angle at the ground when the ladder is 6 m from the wall?

11. The position of a particle as it moves horizontally is described by the given equation s=sin t cos t 0 t 2 . If s is the displacement in metres and t is the time in seconds, find the maximum and minimum displacements.

12.Show that a rectangle with given perimeter has maximum area when it is a square.

13.Solve the problem by following the procedure for solving an optimization problem. Show all your steps.

a.The perimeter of an isosceles triangle is 38 m. Find the length of the sides of the

triangle of maximum area.

b.An apple orchard now has 80 trees per hectare and the average yield is 400 apples

per tree. For each additional tree planted per hectare the average yield per tree is reduced by approximately four apples. How many trees per hectare will give the largest crop of apples?

c. A rectangular piece of paper with perimeter 100 cm is to be rolled to form a

cylindrical tube. Find the dimensions of the paper that will produce a tube with maximum volume.

d. A farmer has 1800 m of fencing to make two equal rectangular pens that share

a common side. Determine the dimensions of each pen so that the area is a maximum.

e. New Horizons Travel advertises a package plan for a Florida vacation. The fare

for the flight is $400/person plus $8/person for each unsold seat on the plane. The plane holds 120 passengers and the flight will be cancelled if there are fewer than 50 passengers. What number of passengers will maximize revenue?