Remedial plan
Grade/Cluster
12 ES
Subject
Section
Date
Question 1:
Find the derivative of each of the following functions:


 2 3
1)
f ( x)  ( x  1)
2)
f ( x)  4 x 2  3x  10
3)
2
f ( x)  x 2 ( x 2  1)( x 5 )
4)
f ( x)  ( x 3  x  1) 5
5)
f ( x)  x 3
4x  1
8x
f
(
x
)

6)
( x 3  1) 2
7)
f ( x)  ( x  4  3 x )
8)
f ( x)  x 3 e 2 x
9)
f ( x)  5 ln x
2
10) f ( x)  e 2 x (ln xex  1)
2
3
2
Mathematics
11) f ( x)  log 6 (1  x  x 2 )
12) f ( x )  cos(3 x  5)
13) f ( x)  sin(ln 3x 4 )
14) f ( x)  esin 2 x
15) f ( x)  sin(ln(cos x 3 ))
16) f ( x )  ln(sec x  tan x )
17) f ( x)  xsin x
18) f ( x)  e 2 x cos 4 x
19) f ( x) 
ln x 2
x
20) f ( x)  4 tan x
21) f ( x)  ln( x  5 x )
4
22) f ( x)  3 10 x
2
3
2
2
23) f ( x)  3(sin 2 x  cos2 x)10
24) f ( x)  cos 4 x  3 sin

12
Question 2:
The distance of a moving particle from the origin at time t
S (t )  sin t  2 cos t
Find the velocity at time t 

4
Question 3:
Given that
h( x )   f ( x ) 
3
and
f (3)  1 and f (3)  2
Find h (3)
Question 4:
Let
f ( x)  x 2 and g ( x)  3x  6
Find
 f ( g( x))
Question 5:
Find the equation of tangent line to the graph of equation
f ( x)  x 4  5 x 3  2
At x = 2
Question 6:
Find all points on the graph of f ( x)  x 3  5x 2  2 x  1 where the tangent line to
the graph is horizontal
Question 7:
If the cost function in dollars for q units is
C (q )  100q  100
Demand function for q units
p  D( q )  100 
50
ln q
a) Find the marginal revenue
b) Find the marginal cost
c) Find the marginal profit from one more unit when 8 units are sold
Question 8:
Assume that the total revenue received from the sale of x items is given by
R( x)  30 ln( 2 x  1)
While the total cost to produce x items is
C ( x) 
x
2
Find the number of items that should be manufactures so that profit is
maximum
Question 9:
The sales of a new personal computer (in thousands) is given by
S (t )  100  90e 0.3t
Where t represent time in years
Find the rate of change of sales at each time
a) After 1 year
b ) after 5 years
c) What is happening to the rate of change of sales as time?
Goes on?
d) Does the rate of change of sales ever equal zero
Question 10:
Find the equation of the tangent line to the curve
xe x
At x = 1
Question 11:
Find the rate of change of the calcium level in the blood stream is given by:
1
1
C (t )  (2t  1) 2 with respect to time after 4 days
2
Question 12:
The figure is a graph of
function f and a tangent
line to the graph at
x=2. Find
f (2) 
Question 13:
Find a relative maximum about the graph
Question 14:
Find the horizontal asymptote of the given function
2 x 2  5x  1
f ( x) 
x2  2
Question 15:
Let
f ( x)  5 x  9
Compute
and
x2
g ( x) 
1
5
( f g ( x))
Question 16:
A piece of carton 12 cm by 10 cm is to be used to make an open box for
doughnuts. Squares of equal sides x are cut out of each corner then the
sides are folded to make the box
1. Prove that the volume of the open box is V(x) = 4x3– 44x2+ 120x.
2. When does the volume reach its maximum?
What is the maximum volume?
3. If the volume of a doughnut is 5cm3, how many doughnuts will
you be able to fit in the box?
4. What is the rate of change of the volume at the maximum?
5. Check the concavity of the volume function?
Question 17:
f ( x)  ln x
Compute
f ( x), f ( x), f ( x), f ( 4) ( x), f (5) ( x)
f ( n ) ( x)
n is a positive integer
Question 18:
f ( x)  e x
Compute
f ( x), f ( x), f ( x), f ( 4) ( x), f (5) ( x)
f ( n ) ( x)
n is a positive integer
Question 19:
f ( x)  sin x
Compute
f ( x), f ( x), f ( x), f ( 4) ( x), f (5) ( x)
f ( n ) ( x)
n is a positive integer
Question 20:
The percent of concentration of a certain drug in the bloodstream x hours after
the drug is administrated is given by:
K ( x) 
3x
x2  4
1. Find the time at which concentration is maximum
2. Find the maximum concentration
Question 21:
When an object is dropped straight down, the distance (in feet) that travels in t
seconds is given by
s(t )  16t 2
1. Find the velocity after 3 seconds , 5 seconds , 8 seconds
2. Find the acceleration
Question 22:
A ball is fired straight up; its position equation is given by:
s(t )  16t 2  140t  37
1. Find the maximum height of the ball
2. Find the time and the velocity when the ball hits the ground
Question 23:
A car rolls down a hill; its distance from its starting point is given by
s(t )  1.5t 2  4t
1. How far will the car move in 10 seconds
2. What is the velocity at 5 seconds , at 10 seconds
3. How can you tell that the car will not stop
4. What is the acceleration at 5 seconds , at 10 seconds
5. What is happening o the velocity and acceleration as t increases
Question 24:
the total profit (in thousands of dollars) from the sale of x units of a certain
prescription drug is given by
P( x)  ln(  x 3  3x 2  72 x  1)
0  x  10
1. Find the number of units that should be sold in order to maximize the total
profit
2. What is the maximum profit
Question 25:
In the summer the activity level of a certain type of lizard varies according
to the time of day
A biologist has determined that the activity level is given by the function
a(t )  0.08t 3  0.288t 2  2.340t  7
Where t is the number of hours after 12 noon
When is the activity level highest, when it is lowest?
Question 26:
The percent of concentration of a certain drug in the bloodstream x hours after
the drug is administrated is given by:
K ( x) 
4x
3x 2  27
1. On what time intervals is the concentration of the drug increasing
2. On what intervals it is decreasing
Question 27:
Find
1.
2.
3.
4.
5.
The critical points
The intervals where the function is increasing or decreasing
The values of the relative extrema
The intervals where the function is concave up or down
The inflection points
Question 28:
Find all the relative extrema as well as where the function is increasing or
decreasing, critical points
2 x2
f ( x)  xe
Question 29:
A small company manufactures and sells bicycles
The production manager has determined that the cost and demand
functions for q ( q  0 ) bicycles per week are
1 3
C (q)  10  5q  q
60
p  D(q)  90  q
1. Find the maximum weekly revenue
2. Find the maximum weekly profit
3. Find the price the company should charge to realize maximum profit
Question 30: