The document contains an individual assignment with 14 math problems. The assignment includes problems on calculus topics such as derivatives, limits, implicit differentiation, and optimization. The solutions show the steps and work to arrive at the answers for each problem.

1. Name: xxxxxxxxxxxxxxx
ID: xxxxxxxxxxx
Class: xxxx
INDIVIDUAL ASSIGNMENT
1. The balance of a bank account after 3 years with respect to the interest rate r% is given by
A= 1000(1+r/1000)36 Find the rate of change of the balance with respect to r when r = 6
SOLUTION
A=1000(1+6/1000)
=1000(1+0.006)
36
36
= 1000(1.24)
=1240.3
36= 3 YEARS
12= 1 YEAR
A=1000(1+6/1000)
12
=1074.42
AVERAGE RATE OF CHANGE IS GIVEN BY: Y3-Y1/X3-X1
1240.3-1074.42/36-12
ARC = 6.912
2- Find f (a) for given value of a. f(x) = 5x2- 4x + 7; and a = 4:
SOLUTION
2
F(x) =5x -4x+7
F’(x) =10x-4

2. F’ (4) =10(4)-4
F’ (a) =36
3- Find f (g(x)) and g (f(x)) for f(x) = 4x + 3; and g(x) = -2x + 1:
SOLUTION
F (G(X)) = f (-2x+1)
= 4(-2x+1) +3
= -8x+7
G (f(x)) = g (4x+3)
= -2(4x+3) +1
=-8x-5
4- Let s be a distance given by
S (t) = 3t3 + 4t2 + 8t:
Find the acceleration.
SOLUTION
3
2
S (t) = 3t +4t +8t
To find the acceleration we derive the function twice
2
S’ (t) = 9t +8t
S’’ (t) = 18t+8
5- Find f0(x) for f(x) = 10/x
Solution
6
F (x) = 10/x =10x
-6
6

3. F’ (x) = -60x
-7
6- For the function f(x) = 5x2 + 4x. Find f(x + h) - f(x)/h
For:
a. x = 2; h = 0:1.
b. x = 1; h = 0:5.
c. x = 3; h = 1.
Solution
2
a- f (2) =5(2) +4(2) =28
2
F (2+0.1) = 5(2+0.1) +4(2+0.1) =30.09
30.09-28/0.1=20.9
b- F (1) =5(1) +4(1) =9
2
F (1+0.5) =f (1.5) +4(1.5) =17.25
17.25-9/0.5=16.5
2
c- F (3) =5(3) +4(3) =57
2
F (3+1) =5(3+1) +4(3+1) =96
96-57/1=39
7- Find the limit if it exists lim 10x+3
X
3
Lim 10x+3 = lim 10x+3 = 33
+
x 3
x
3
3
8- Differentiate the function y = 5x (2x - 7x)

4. SOLUTION).
USING CHAIN RULE: F(X).G(X) = F’(X).G(X) + G’(X).F(X)
F(X) =5X
F’(X) =5
3
2
G’(X) = 6X -7
G(X) =2X -7X
3
2
Y’= 5. (2x -7x)+ (6x -7)5x
3
Y’= 4x -7x
2
9- Differentiate the function y= 2x/x +4
F(x) =2x
f’(x) =2
2
G(x) =x +4
g’(x) = 2x
Y’ = g(x).f’(x)-f(x).g’(x)/g(x)
2
2
2
Y’ = (x +4).2 – 2x (2x) / (x +4)
2
2
Y’ = -2(x +8) / (x +4)
2
2
10- Find an expression for dy/dx: y = 3u2 and u = 3x + 1:
Dy/du= 6u,
and
du/dx= 3
Dy/dx= dy/du.U + du/dx.y
2
= 6u. (3x + 1) + 3. (3u )
= 18ux + 6u + 9u
2
2
= 3u + 6ux + 2u
3
2
12- Find the absolute maximum and minimum of the function f(x) = 2x - 3x + 4 on the interval [-1; 2].
SOLUTION

5. 2
F’(x) = 6x -6x
F’(x) =0
2
6x -6x =0
6x(x-1) =0
6x=0
or x-1=0
X=0, X=1
{-1, 0, 1, 2}
F (-1) =-1
F (0) =4
F (1) =3
F (2) =8
The absolute maximum is 8= f (2)
The absolute minimum is -1= f (-1)
3
13- Find dy for given values of x and dx: y = x ;
Dy/dx=3x
2
x = 4 and dx = 0:1
2
dy= 3x .dx
Dy= 3(4). (0.1)
Dy=4.8
3
14- Determine where the function is concave up and down: f(x) = x - 3x + 2:
Concavity can be found using the second derivative.
2
F’ (x) = 3x -3
F’’(x) = 6x
-to find where the function is concave up, we find where the second derivative is positive
6x>0
x>0: the function is concave up for all x- values > 0.
Similarly, the function is concave down for all x-values < 0.

6. 3
2
16- The cost of producing x items is given by: C(x) = x -34x + 5600x: Find the production level that
minimizes the average cost.
3
C(x)/x
2
2
x -34x +5600x/x
= x -34x+5600
3
2
3
17- Find dy=dx by implicit differentiation, given that x + xy + y = 3:
3
2
3
D/dx(x +xy +y ) =d/dx (3)
3
2
3
D/dx(x ) +d/dx (xy ) +d/dx (y ) =0
2
2
2
3x +x.2y.dy/dx+y +3y d/dx=0
2
2
2
2
2
3y d/dx+2yxdy/dx=-3x -y
2
(3y +2yx) dy/dx=-3x -y
2
2
2
Dy/dx= -3x -y /3y +2yx
Find the slope of the tangent line at point (1; 1).
y - y0 = m(x - x0)
18- The cost and the revenue with respect to the number of items is given by C(x) = 200x + 2000 and
2
R(x) = 5x + 100x + 80:
Find the marginal profit
Solution
P(X) = R(x) – C(x)
2
P(x) = (5x +100x+80) – (200x +200)
2
P(x) = 5x – 100x -120
P’(x) = 10x-100